Some basic topics related to partial differential equations

Stephen Semmes Rice University

Preface

Some aspects of history related to calculus and the theory behind it may be found in [25, 49, 50, 51, 52, 93, 94, 108, 128, 129, 136, 137, 138, 139, 140, 143, 158, 198, 305]. See also [44, 126, 227], in connection with Fourier series in particular. Some songs related to calculus may be found in [231, 232, 233], and some remarks concerning the clarity of explanations in mathematics may be found in [142]. The reader may be interested in [42] as well.

Some aspects of history related to multivariable calculus and differential equations may be found in [14, 15, 60, 61, 62, 82, 118, 127, 134, 141, 143, 144, 158, 196, 219, 220, 221, 227, 229, 240, 241]. A number of other texts in the bibliography include some discussion of history too.

Some additional perspectives concerning partial differential equations may be found in [3, 45, 46, 79, 150, 154, 181, 187, 188, 192, 200, 264, 269, 272, 281, 302, 328, 345, 351].

The study of differential equations is closely related to that of mathematical analysis, as indicated in particular in some of the references concerning history mentioned earlier. Some familiarity with analysis would be helpful here, although it is not required. Of course, there are many textbooks in analysis, some of which are mentioned in the bibliography. My colleague Frank Jones’ book [186] includes some of the theory related to multivariable calculus, which may be helpful. The reader may also be interested in the brief introductions in [145, 341], as well as Gouvêa’s recent text [135] on infinite series. Some aspects of analysis that are related to the topics discussed here may be found in the appendices. Some details are included for the sake of completeness, although some results are merely stated, or are briefly explained somewhat informally.

Some basic familiarity with ordinary differential equations and linear algebra would be helpful here too, but is not required. Some relevant notions may be reviewed as needed, and a number of related references may be found in the bibliography as well. Of course, many treatments of ordinary differential equations include some linear algebra, as does Frank Jones’ book [186].

Contents

1 Some basic notions 1

1.1 What is a differential equation? . . . . . . . . . . . . . . . . . . . 1 1.1.1 First-order ordinary differential equations . . . . . . . . . 2 1.1.2 Some questions about F (x, p, q) . . . . . . . . . . . . . . . 2 1.1.3 A simpler type of ordinary differential equation . . . . . . 3

1.2 Some other types of first-order equations . . . . . . . . . . . . . . 4 1.2.1 Linear first-order equations . . . . . . . . . . . . . . . . . 4 1.2.2 Some first-order differential operators . . . . . . . . . . . 5

1.3 Invariance under translations . . . . . . . . . . . . . . . . . . . . 6 1.3.1 Some equations of this type . . . . . . . . . . . . . . . . . 7 1.3.2 Constant coefficients . . . . . . . . . . . . . . . . . . . . . 8

1.4 First-order partial differential equations . . . . . . . . . . . . . . 8 1.4.1 Equations in two variables . . . . . . . . . . . . . . . . . . 8 1.4.2 Linear first-order equations in two variables . . . . . . . . 9

1.5 More on invariance under translations . . . . . . . . . . . . . . . 10

1.5.1 More on constant coefficients . . . . . . . . . . . . . . . . 11 1.6 Second-order partial differential equations . . . . . . . . . . . . . 11

1.6.1 Equations in two variables again . . . . . . . . . . . . . . 12

Appendices 12

A A bit of analysis 13

A.1 The real line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

A.1.1 Intervals in R . . . . . . . . . . . . . . . . . . . . . . . . . 13 A.2 Euclidean spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

A.2.1 The standard Euclidean norm and metric on R^{n} . . . . . 15 A.2.2 Open and closed balls . . . . . . . . . . . . . . . . . . . . 15

A.3 The dot product on R^{n} . . . . . . . . . . . . . . . . . . . . . . . 16 A.3.1 The Cauchy–Schwarz inequality . . . . . . . . . . . . . . . 17 A.3.2 Using the Cauchy–Schwarz inequality . . . . . . . . . . . 17

A.4 Proving the Cauchy–Schwarz inequality . . . . . . . . . . . . . . 17

A.4.1 Another approach to the Cauchy–Schwarz inequality … 18 A.5 Some other norms on R^{n} . . . . . . . . . . . . . . . . . . . . . . 19

A.5.1 Comparing these three norms . . . . . . . . . . . . . . . . 19 A.6 Metrics associated to norms . . . . . . . . . . . . . . . . . . . . . 20

A.6.1 The metrics associated to ∥ · ∥_{1}, ∥ · ∥_{2}, and ∥ · ∥_{∞} . . . . . 21 A.6.2 Open and closed balls with respect to a norm . . . . . . . 21 A.6.3 Convex subsets of R^{n} . . . . . . . . . . . . . . . . . . . . 22

A.7 Complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 22 A.7.1 Complex conjugates . . . . . . . . . . . . . . . . . . . . . 22 A.7.2 Absolute values . . . . . . . . . . . . . . . . . . . . . . . . 23 A.7.3 Some inequalities . . . . . . . . . . . . . . . . . . . . . . . 23 A.7.4 Comparison with R^{2} . . . . . . . . . . . . . . . . . . . . . 24

A.8 Convergent sequences . . . . . . . . . . . . . . . . . . . . . . . . 24 A.8.1 Convergent sequences in R^{n} . . . . . . . . . . . . . . . . . 24 A.8.2 Bounded sequences in R^{n} . . . . . . . . . . . . . . . . . . 25 A.8.3 Monotone sequences of real numbers . . . . . . . . . . . . 25 A.8.4 Convergent sequences of complex numbers . . . . . . . . . 26

A.9 More on convergent sequences . . . . . . . . . . . . . . . . . . . . 26 A.9.1 A helpful lemma . . . . . . . . . . . . . . . . . . . . . . . 27 A.9.2 Sequences of reciprocals . . . . . . . . . . . . . . . . . . . 27 A.9.3 More on sequences in R^{n} . . . . . . . . . . . . . . . . . . 27

A.10 Some particular sequences . . . . . . . . . . . . . . . . . . . . . . 28 A.10.1 {1/j^{p}}_{j=1}^{∞} . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 A.10.2 {j^{α} a^{j}}_{j=1}^{∞} . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 A.10.3 {p^{1/j}}_{j=1}^{∞} . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 A.10.4 {j^{1/j}}_{j=1}^{∞} . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

A.11 Infinite series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 A.11.1 Some properties of convergent series . . . . . . . . . . . . 30 A.11.2 Geometric series . . . . . . . . . . . . . . . . . . . . . . . 31 A.11.3 Infinite series with nonnegative terms . . . . . . . . . . . 31 A.11.4 Absolute convergence . . . . . . . . . . . . . . . . . . . . 32

A.12 More on infinite series . . . . . . . . . . . . . . . . . . . . . . . . 32 A.12.1 Power series . . . . . . . . . . . . . . . . . . . . . . . . . . 33 A.12.2 The radius of convergence . . . . . . . . . . . . . . . . . . 34

A.13 Rearrangements of infinite series . . . . . . . . . . . . . . . . . . 34 A.14 Cauchy products of infinite series . . . . . . . . . . . . . . . . . . 35

A.14.1 Cauchy products and power series . . . . . . . . . . . . . 36 A.14.2 Convergence of Cauchy products . . . . . . . . . . . . . . 36

A.15 The binomial theorem . . . . . . . . . . . . . . . . . . . . . . . . 37

B A bit of analysis, 2 38

B.1 The complex exponential function . . . . . . . . . . . . . . . . . 38 B.1.1 Exponentials of sums . . . . . . . . . . . . . . . . . . . . . 38 B.1.2 Exponentials and complex conjugates . . . . . . . . . . . 39

B.2 Continuous complex-valued functions . . . . . . . . . . . . . . . . 39 B.2.1 More on continuous functions . . . . . . . . . . . . . . . . 40 B.2.2 Continuity of compositions . . . . . . . . . . . . . . . . . 40 B.2.3 Real and imaginary parts . . . . . . . . . . . . . . . . . . 41 B.3 Pointwise and uniform convergence . . . . . . . . . . . . . . . . . 42

B.3.1 An example with E = [0, 1] . . . . . . . . . . . . . . . . . 42 B.3.2 Uniform convergence and continuity . . . . . . . . . . . . 42

B.4 Weierstrass’ criterion for uniform convergence . . . . . . . . . . . 43

B.4.1 Continuity of the sum . . . . . . . . . . . . . . . . . . . . 43 B.5 Continuity of functions defined by power series . . . . . . . . . . 44

B.5.1 Continuity on open disks . . . . . . . . . . . . . . . . . . 44 B.6 Bounded functions . . . . . . . . . . . . . . . . . . . . . . . . . . 45

B.6.1 A version of the extreme value theorem . . . . . . . . . . 45 B.7 Uniformly bounded sequences of functions . . . . . . . . . . . . . 46

B.7.1 More on uniform convergence . . . . . . . . . . . . . . . . 46 B.7.2 Boundedness and uniform convergence . . . . . . . . . . . 47

B.8 Uniform convergence and integration . . . . . . . . . . . . . . . . 47

B.8.1 The bounded convergence theorem . . . . . . . . . . . . . 48 B.9 Uniform convergence and differentiation . . . . . . . . . . . . . . 48

B.9.1 Differentiating power series . . . . . . . . . . . . . . . . . 49 B.9.2 Differentiating exp(a t) . . . . . . . . . . . . . . . . . . . . 49 B.9.3 More on differentiating power series . . . . . . . . . . . . 50

B.10 Open and closed sets . . . . . . . . . . . . . . . . . . . . . . . . . 50 B.10.1 Limit points and closed sets . . . . . . . . . . . . . . . . . 51 B.10.2 Complements of subsets of R^{n} . . . . . . . . . . . . . . . 51 B.10.3 Bounded sets and the extreme value theorem . . . . . . . 52 B.10.4 Closures and boundaries . . . . . . . . . . . . . . . . . . . 52

B.11 Partial derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . 53 B.11.1 Continuous differentiability . . . . . . . . . . . . . . . . . 54 B.11.2 Directional derivatives . . . . . . . . . . . . . . . . . . . . 54 B.11.3 Twice continuous differentiability . . . . . . . . . . . . . . 54

B.12 Connected sets in R^{n} . . . . . . . . . . . . . . . . . . . . . . . . 55 B.12.1 Connectedness in the real line . . . . . . . . . . . . . . . . 56 B.12.2 Locally constant functions on subsets of R^{n} . . . . . . . . 56 B.12.3 Locally constant functions on open sets . . . . . . . . . . 57

B.13 Vector-valued functions . . . . . . . . . . . . . . . . . . . . . . . 57 B.13.1 More on continuity of compositions . . . . . . . . . . . . . 58 B.13.2 More on partial derivatives . . . . . . . . . . . . . . . . . 58 B.13.3 Compositions and continuous differentiability . . . . . . . 59 B.13.4 More on twice continuous differentiability . . . . . . . . . 59

B.14 More on connectedness . . . . . . . . . . . . . . . . . . . . . . . . 60 B.14.1 Path connectedness . . . . . . . . . . . . . . . . . . . . . . 61 B.14.2 Nonemptiness of the boundary of an open set . . . . . . . 61

C A bit of analysis, 3 62

Bibliography 63

Index 87 vi CONTENTS

Chapter 1

Some basic notions

1.1 What is a differential equation?

The title of Section 1.1 of [335] is “What is a partial differential equation?” It seems helpful to begin by asking what an ordinary differential equation is. Part of the answer is that ordinary differential equations deal with functions of one independent variable, while partial differential equations are concerned with functions of more than one independent variable.

A function of a single real variable might frequently be taken to be defined on something like an interval in the real line R, particularly when considering solutions of an ordinary differential equation. Various types of intervals are mentioned in Subsection A.1.1. Sometimes one might consider functions defined on other subsets of the real line, such as the function 1/x, defined for real numbers x ≠ 0.

Let u be a real-valued function defined on an interval I in R, and suppose that u is differentiable at every point in I. If

u′(x) = 0 (1.1.1)

for every x ∈ I, then it is well known that

(1.1.2) u is constant on I,

by the mean value theorem.

If x is an endpoint of I that is also an element of I, then the derivative u′(x) of u at x should be considered as a one-sided deriviative, defined using a one-sided limit, when it exists. Sometimes we may be concerned with functions that are differentiable at points in the interior of an interval or half-line, and continuous at an endpoint. This is sufficient for the result mentioned in the preceding paragraph.

1.1.1 First-order ordinary differential equations

A first-order ordinary differential equation may be expressed as

F (x, u(x), u′(x)) = 0. (1.1.3)

Here F is a function of three variables, and u is a real-valued function of one variable.

These remarks are intended to be a bit informal, and in practice one should probably be a bit more precise about where these functions are defined. A basic possibility is that F (x, p, q) (1.1.4)

should be defined for x in an interval I in the real line, and for all real numbers p and q. In this case, the differential equation (1.1.3) would make sense for functions u defined on a subinterval of I.

Of course, one would also like u to be differentiable, so that u′(x) is defined. Sometimes one may be interested in functions as in (1.1.4) that are defined on other sets of (x, p, q), and this may lead to additional restrictions on u.

If x_{0} is an element of I and p_{0} is a real number, then the corresponding initial value problem for (1.1.3) concerns the existence and uniqueness of a solution u defined on a subinterval of I that contains x_{0} such that

u(x_{0}) = p_{0}. (1.1.5)

Of course, this means in particular that

F (x_{0}, u(x_{0}), u′(x_{0})) = 0. (1.1.6)

Let us express this as F (x_{0}, p_{0}, q_{0}) = 0, (1.1.7)

where q_{0} = u′(x_{0}).

1.1.2 Some questions about F (x, p, q)

In order to have a solution of (1.1.4) that satisfies (1.1.5), there has to be a real number q_{0} that satisfies (1.1.7).

Without additional hypotheses about F, it could be that (1.1.4) does not depend on q. This means that

F (x, p, q) = F̃(x, p) (1.1.8)

for some function F̃(x, p) of two variables. In this case, the differential equation (1.1.3) reduces to F̃(x, u(x)) = 0, (1.1.9)

and does not involve the derivative of u.

Similarly, (1.1.4) might not depend on p or q, so that

F (x, p, q) = F̂(x) (1.1.10)

for some function F̂(x) of one variable. This means that the differential equation (1.1.3) reduces to F̂(x) = 0, (1.1.11)

and does not involve u or its derivative.

If (1.1.8) holds, then (1.1.7) reduces to

F̃(x_{0}, p_{0}) = 0. (1.1.12)

One can still look for solutions of (1.1.9) that satisfy (1.1.5). This is related to the implicit function theorem, at least for x near x_{0}. Part of the hypothesis of the implicit function theorem is that the derivative of F̃(x, p) in p at (x_{0}, p_{0}) not be equal to 0.

1.1.3 A simpler type of ordinary differential equation

Often one considers first-order ordinary differential equations of the form

u′(x) = f(x, u(x)), (1.1.13)

where f is a function of two variables. We may take

f(x, p) (1.1.14)

to be defined for x in an interval in the real line again, and for all real numbers p, although sometimes one may want to consider other restrictions on (x, p), as before.

Observe that (1.1.13) is the same as (1.1.3), with

F (x, p, q) = q − f(x, p). (1.1.15)

In this case, (1.1.7) is the same as saying that

q_{0} = f(x_{0}, p_{0}). (1.1.16)

Suppose that we have a differential equation of the form (1.1.3) again, with the initial condition (1.1.5), and a real number q_{0} that satisfies (1.1.7). Under some conditions, we may be able to transform (1.1.3) into a differential equation of the form (1.1.13), at least for x close to x_{0}, using the implicit function theorem. Part of the hypothesis of the implicit function theorem is that

\frac{∂F}{∂q} (x_{0}, p_{0}, q_{0}) ≠ 0, (1.1.17)

as before.

4 CHAPTER 1. SOME BASIC NOTIONS

1.2 Some other types of first-order equations

Consider a first-order ordinary differential equation of the form a(x, u(x)) u′(x) = b(x, u(x)), (1.2.1) where a and b are functions of two variables. We may take a(x, p) and b(x, p) to be defined for x in an interval in the real line, and all real numbers p, although one may consider other restrictions on (x, p), as usual.

This equation is the same as (1.1.3), with F (x, p, q) = a(x, p) q − b(x, p). (1.2.2) Similarly, (1.1.7) is the same as saying that a(x_{0}, p_{0}) q_{0} = b(x_{0}, p_{0}). (1.2.3)

Of course, (1.2.1) corresponds to (1.1.13) with f(x, p) = b(x, p)/a(x, p), (1.2.4) as long as the denominator is not zero. In particular, if a(x_{0}, p_{0}) ≠ 0, (1.2.5) then (1.2.3) is the same as saying that q_{0} = b(x_{0}, p_{0})/a(x_{0}, p_{0}), (1.2.6) as in (1.1.16). Note that (1.2.5) is the same as (1.1.17) when F is as in (1.2.2).

Otherwise, if a(a_{0}, p_{0}) = 0, (1.2.7) then (1.2.3) reduces to b(x_{0}, p_{0}) = 0. (1.2.8) If a ≡ 0, (1.2.9) then (1.2.1) is the same as (1.1.9), with F̃(x, p) = b(x, p). (1.2.10)

1.2.1 Linear first-order equations

A linear first-order ordinary differential equation may be expressed as α(x) u′(x) + β(x) u(x) = γ(x), (1.2.11) where α(x), β(x), and γ(x) are functions of one variable. We may take these functions to be defined on an interval in R, as before.

1.2. SOME OTHER TYPES OF FIRST-ORDER EQUATIONS 5

This equation is the same as in (1.2.1), with a(x, p) = α(x) and b(x, p) = −β(x) p + γ(x). (1.2.12) Thus (1.2.3) is the same as saying that α(x_{0}) q_{0} = −β(x_{0}) p_{0} + γ(x_{0}) (1.2.13) in this case.

Suppose that γ ≡ 0, so that (1.2.11) becomes α(x) u′(x) + β(x) u(x) = 0. (1.2.14) This type of equation is said to be homogeneous. If u satisfies this equation and c is a real number, then c u satisfies the analogous equation α(x) (c u)′(x) + β(x) (c u)(x) = 0. (1.2.15) Similarly, if v is another real-valued function defined and differentiable on the same interval as u that satisfies the analogous equation α(x) v′(x) + β(x) v(x) = 0, (1.2.16) then u + v satisfies the analogous equation α(x) (u + v)′(x) + β(x) (u + v)(x) = 0. (1.2.17)

1.2.2 Some first-order differential operators

Put L(u) = α u′ + β u. (1.2.18) More precisely, if u is a differentiable real-valued function defined on an interval on which α and β are defined, then this defines a real-valued function on the same interval. We may also use the notation

L = α \frac{d}{dx} + β, (1.2.19)

and refer to this as a first-order differential operator,

It is easy to see that this defines a linear mapping from the space of differentiable real-valued functions on an interval on which α and β are defined into the space of all real-valued functions on that interval. This means that if u is such a function and c is a real number, then L(c u) = c L(u). (1.2.20) This also means that if v is another such function on the same interval, then L(u + v) = L(u) + L(v). (1.2.21)

If one is familiar with the notion of a vector space over the real numbers, then one may know that the space of all real-valued functions on an interval is a vector spaces, with respect to the usual definitions of addition and scalar multiplication of such functions, using pointwise addition and scalar multiplication of functions. The space of differentiable real-valued functions on the same interval is a linear subspace of the space of all real-valued functions on that interval, and is another vector space over the real numbers in particular. The differential operator L determines a linear mapping from this subspace into the space of all real-valued functions on the interval, in the sense of linear algebra, as in the previous paragraph.

Using L, the homogeneous equation (1.2.14) may be expressed as

L(u) = 0, (1.2.22)

and the inhomogeneous equation (1.2.11) may be expressed as

L(u) = γ. (1.2.23)

The space of solutions to the homogeneous equation on a suitable interval is a linear subspace of the space of all differentiable real-valued functions on that interval, as in the previous subsection. This also follows from the linearity of the corresponding mapping between the appropriate vector spaces, by a standard argument in linear algebra.

This is related to some remarks beginning on p2 of [335]. If u satisfies the inhomogeneous equation (1.2.23), and v is a differentiable real-valued functio on the same interval that satisfies the homogeneous equation

L(v) = 0, (1.2.24)

then L(u + v) = L(u) + L(v) = γ, (1.2.25)

so that u + v satisfies the analogous inhomogeneous equation, as mentioned on p3 of [335]. We shall consider analogous notions for functions of more variables, as in [335].

1.3 Invariance under translations

Let us say that a first-order ordinary differential equation as in (1.1.3) is invariant under translations if the corresponding function (1.1.4) does not depend on x. This means that (1.1.4) may be expressed as

F (x, p, q) = \overline{F} (p, q) (1.3.1)

for some function \overline{F} (p, q) of two variables. In this case, the corresponding ordinary differential equation (1.1.3) reduces to

\overline{F} (u(x), u′(x)) = 0. (1.3.2)

1.3. INVARIANCE UNDER TRANSLATIONS 7

Let u be a real-valued function on an interval I in the real line that is differentiable at every point in I. Also let ξ be a real number, and put I + ξ = {t + ξ : t ∈ I}. (1.3.3) This is another interval in R, and u(x − ξ) (1.3.4) defines a real-valued of x on (1.3.3). Note that (1.3.4) is differentiable at every point in (1.3.3), with

\frac{d}{dx}(u(x − ξ)) = u′(x − ξ). (1.3.5)

Using this, it is easy to see that u satisfies (1.3.2) on I if and only if (1.3.4) satisfies the same equation on (1.3.3), or equivalently \overline{F} (u(x − ξ), u′(x − ξ)) = 0 (1.3.6) on (1.3.3).

If c is any real number, then the first-order ordinary differential equation F (x, u(x), u′(x)) = c (1.3.7) corresponds to the function F (x, p, q) − c (1.3.8) as in Subsection 1.1.1. Invariance under translations in the sense considered here means that this differential equation is invariant under translations as well.

1.3.1 Some equations of this type

Let ϕ be a real-valued function on the real line, and consider the differential equation ϕ(u(x)) u′(x) = 1. (1.3.9) This is the same as (1.3.2), with \overline{F} (p, q) = ϕ(p) q − 1. (1.3.10)

Suppose that Φ is a differentiable real-valued function on the real line such that Φ′ = ϕ (1.3.11) on R. Thus (1.3.9) is the same as saying that

\frac{d}{dx}(Φ(u(x))) = 1, (1.3.12)

by the chain rule. A differentiable real-valued function on an interval in R satisfies this differential equation exactly when (1.3.13) Φ(u(x)) = x + a constant on that interval.

1.3.2 Constant coefficients

A differential operator L as in (1.2.19) is said to have constant coefficients if the coefficients α and β are constants. If γ is also a constant, then the corresponding inhomogeneous differential equation (1.2.23) is invariant under translations.

Even if γ is not a constant, the inhomogeneous differential equation (1.2.23), or equivalently (1.2.11), may be described as having constant coefficients when α and β are constants. One may also be concerned with linear ordinary differential equations of higher order with constant coefficients, as well as linear partial differential equations with constant coefficients. Many of the equations that we shall consider here are of this type.

1.4 First-order partial differential equations

Let u(x, y) be a real-valued function of two variables x, y. More precisely, u should be a real-valued function defined on a subset of the plane R^{2}. This is the set of ordered pairs of real numbers, as in Section A.2, with n = 2.

In fact, we might typically ask that u be defined on an open set in R^{2}. We shall not get into the definition of an open set here, although it is mentioned in Section B.10. The basic idea should be clear in many examples, such as an open disk, an open rectangle, or something like that.

Suppose that the partial derivatives ∂u/∂x and ∂u/∂y exist at every point in the domain of u. It is often convenient to use subscripts to denote these partial derivatives, so that

u_{x} = \frac{∂u}{∂x} and u_{y} = \frac{∂u}{∂y}, (1.4.1)

as on p1 of [335], and as in Section B.11. Let us also ask that the partial derivatives of u be continuous on the domain of u, so that u is continuously differentiable, as in Subsection B.11.1. This implies in particular that u is continuous on its domain, as mentioned in Subsection B.11.1. Note that derivatives under consideration in [335] are normally asked to be continuous, as mentioned in item 2 on p4 of [335].

1.4.1 Equations in two variables

A first-order partial differential equation for functions of two variables may be expressed as F (x, y, u(x, y), u_{x}(x, y), u_{y}(x, y)) = 0, (1.4.2)

as in (1) on p1 of [335]. Here

F (x, y, p, q, r) (1.4.3)

is a function of the five real variables x, y, p, q, and r. As in Subsection 1.1.1, a basic possibility is that (1.4.3) is defined for all (x, y) in a subset of R^{2} that contains the domain of u, and all real numbers p, q, r. As before, one may be interested in functions as in (1.4.3) that are defined on other sets of (x, y, p, q, r), and this may lead to additional restrictions on u.

It is sometimes convenient to express (1.4.2) more succinctly as

F (x, y, u, u_{x}, u_{y}) = 0, (1.4.4)

as in (1) on p1 of [335] again. It is implicit here that u and its derivatives are supposed to be evaluated at (x, y), as in (1.4.2). In particular, partial differential equations in more variables, or involving more derivatives of u, may be expressed in this way.

1.4.2 Linear first-order equations in two variables

Let a(x, y), b(x, y), c(x, y), and d(x, y) be real-valued functions defined on a subset of R^{2}. Using these functions, we get the linear first-order partial differential equation

a(x, y) u_{x}(x, y) + b(x, y) u_{y}(x, y) + c(x, y) u(x, y) = d(x, y). (1.4.5)

This is the same as (1.4.2), with

F (x, y, p, q, r) = a(x, y) q + b(x, y) r + c(x, y) p − d(x, y). (1.4.6)

Consider the corresponding first-order differential operator

L = a \frac{∂}{∂x} + b \frac{∂}{∂y} + c, (1.4.7)

as in Subsection 1.2.2. If the domain of u is contained in a subset of R^{2} on which a, b, and c are defined, then

L(u) = a u_{x} + b u_{y} + c u (1.4.8)

is defined as a real-valued function on the domain of u. This defines a linear mapping between suitable vector spaces of functions, as before.

Using L, (1.4.5) may be expressed as

L(u) = d. (1.4.9)

If d ≡ 0, then we get the corresponding homogeneous equation

a(x, y) u_{x}(x, y) + b(x, y) u_{y}(x, y) + c(x, y) u(x, y) = 0. (1.4.10)

This may be expressed as L(u) = 0, (1.4.11)

as before. This determines a linear subspace of the space of continuouslydifferentiable real-valued functions on a fixed open set in R^{2}, as long as a, b, and c are defined on that open set, as before.

1.5 More on invariance under translations

Let us say that a first-order partial differential equation as in (1.4.2) is invariant under translations if the corresponding function (1.4.3) does not depend on x or y, so that it may be expressed as

F (x, y, p, q, r) = \overline{F} (p, q, r) (1.5.1)

for some function \overline{F} (p, q, r) of three variables. This means that the corresponding partial differential equation (1.4.2) reduces to

\overline{F} (u(x, y), u_{x}(x, y), u_{y}(x, y)) = 0. (1.5.2)

Let u be a continuously-differentiable real-valued function on an open subset U of R^{2}, and let ξ, η be real numbers. Put

U + (ξ, η) = {(x, y) + (ξ, η) = (x + ξ, y + η) : (x, y) ∈ U}, (1.5.3)

which is another open set in R^{2}. If U is an open disk in R^{2}, for instance, then this is an open disk of the same radius, with the center of the initial disk translated by (ξ, η). Similarly, if U is an open rectangle in R^{2}, then this is an open rectangle with the same sidelengths, and the center of the initial rectangle translated by (ξ, η).

Observe that u(x − ξ, y − η) (1.5.4) defines a real-valued function of (x, y) on (1.5.3). The partial derivatives of this function are equal to the corresponding translates of the partial derivatives of u, so that

\frac{∂}{∂x}(u(x − ξ, y − η)) = u_{x}(x − ξ, y − η) (1.5.5)

and

\frac{∂}{∂y} (u(x − ξ, y − η)) = u_{y}(x − ξ, y − η) (1.5.6)

for all (x, y) in (1.5.3), as in the one-variable case.

In particular, (1.5.4) is continuously differentiable on (1.5.3), because u is continuously differentiable on U, by hypothesis. Using (1.5.5) and (1.5.6), we also have that u satisfies (1.5.2) on U if and only if (1.5.4) satisfies the same equation on (1.5.3), so that

\overline{F} (u(x − ξ, y − η), u_{x}(x − ξ, y − η), u_{y}(x − ξ, y − η)) = 0 (1.5.7)

on (1.5.3).

If c is any real number, then the first-order partial differential equation

F (x, y, u(x, y), u_{x}(x, y), u_{y}(x, y)) = c (1.5.8)

corresponds to the function

F (x, y, p, q, r) − c (1.5.9)

in the same way as before. Invariance under translations in the sense considered here means that this partial differential equation is invariant under translations too.

1.5.1 More on constant coefficients

A differential operator L as in (1.4.7) is said to have constant coefficients if the coefficients a, b, and c are constants. If d is a constant as well, then the corresponding inhomogeneous partial differential equation (1.4.9) is invariant under translations.

The inhomogeneous equation (1.4.9), or equivalently (1.4.5), may be described as having constant coefficients when a, b and c are constants, even if d is not a constant.

1.6 Second-order partial differential equations

Let u(x, y) be a real-valued function of two variables x and y defined on an open subset U of R^{2} again. Suppose that the first partial derivatives ∂u/∂x and ∂u/∂y exist at every point in U, as well as the second derivatives

\frac{∂^{2}u}{∂x^{2}}, \frac{∂}{∂x}(\frac{∂u}{∂y}), \frac{∂}{∂y} (\frac{∂u}{∂x}), and \frac{∂^{2}u}{∂y^{2}}. (1.6.1)

More precisely, we ask that all of these second derivatives be continuous on U, so that u is twice continuously differentiable on U, as in Subsection B.11.3.

This is the same as saying that the partial derivatives ∂u/∂x and ∂u/∂y are continuously differentiable on U, as in Subsection B.11.3. This implies that ∂u/∂x and ∂u/∂y are continuous on U, as mentioned in Subsection B.11.1, and near the beginning of the previous section. This means that u is continuously differentiable on U, so that u is continuous on U as well, as before.

It is well known that

\frac{∂}{∂x}(\frac{∂u}{∂y}) = \frac{∂}{∂y} (\frac{∂u}{∂x}) (1.6.2)

on U, because u is twice continuously differentiable on U, as mentioned in Subsection B.11.3. These second derivatives may be denoted

\frac{∂^{2}u}{∂x ∂y} = \frac{∂^{2}u}{∂y ∂x} (1.6.3)

or

u_{xy} = u_{yx}. (1.6.4)

Similarly, we put

u_{xx} = \frac{∂^{2}u}{∂x^{2}} and u_{yy} = \frac{∂^{2}u}{∂y^{2}}, (1.6.5)

as in Subsection B.11.3.

1.6.1 Equations in two variables again

A second-order partial differential equation for functions of two variables may be expressed as F (x, y, u, u_{x}, u_{y}, u_{xx}, u_{xy}, u_{yy}) = 0, (1.6.6)

as in (2) on p1 of [335]. Here

F (x, y, p, q, r, t, v, w) (1.6.7)

is a function of the eight real variables x, y, p, q, r, t, v, and w. As before, a basic possibility is that (1.6.7) is defined for all (x, y) in a subset of R^{2} that contains U, and for all real values of the other variables. Sometimes one may be interested in functions as in (1.6.7) that are defined on other sets of (x, y, p, q, r, t, v, w), and this may lead to additional restrictions on u.

As in Subsection 1.4.1, it is implicit here that u and its derivatives are supposed to be evaluated at (x, y) in (1.6.7). Partial differential equations in more variables, or involving more derivatives of u, may be expressed similarly, as before. Of course, for equations involving more derivatives of u, one would normally ask that U satisfy appropriate differentiability properties on U.

Appendix A

A bit of analysis

A.1 The real line

Let R be the real line, which is to say the set of real numbers. If x ∈ R, then the absolute value is defined by

(A.1.1)  |x| = x when x ≥ 0
= −x when x ≤ 0.

It is easy to see that |x y| = |x| |y| (A.1.2)

for every x, y ∈ R. One can also check that

|x + y| ≤ |x| + |y| (A.1.3)

for every x, y ∈ R. This is a version of the triangle inequality.

If x, y ∈ R, then the distance from x to y with respect to the standard Euclidean metric is defined by

d(x, y) = |x − y|. (A.1.4)

If z is another real number, then

(A.1.5)  d(x, z) = |x − z| = |(x − y) + (y − z)|
≤ |x − y| + |y − z| = d(x, y) + d(y, z).

This is another version of the triangle inequality.

A.1.1 Intervals in R

Let a, b be real numbers with a < b. The open interval in R from a to b is defined by (a, b) = {x ∈ R : a < x < b}. (A.1.6)

14 APPENDIX A. A BIT OF ANALYSIS

Similarly, the closed interval from a to b is defined by [a, b] = {x ∈ R : a ≤ x ≤ b}, (A.1.7) which may also be used when a = b. Sometimes one may wish to consider the corresponding half-open, half-closed intervals [a, b) = {x ∈ R : a ≤ x < b} (A.1.8) and (a, b] = {x ∈ R : a < x ≤ b} (A.1.9) as well.

It is sometimes convenient to use the notation (a, +∞) = {x ∈ R : a < x} (A.1.10) and (−∞, b) = {x ∈ R : x < b}. (A.1.11) These may be described as open half-lines in R. We may also put (−∞, +∞) = R. (A.1.12) These may be considered as unbounded open intervals in R.

Similarly, we may use the notation [a, +∞) = {x ∈ R : a ≤ x} (A.1.13) and (−∞, b] = {x ∈ R : x ≤ b}. (A.1.14) These may be described as closed half-lines in R. Sometimes +∞, −∞ are considered as extended real numbers, with −∞ < x < +∞ (A.1.15) for every x ∈ R.

A.2 Euclidean spaces

Let n be a positive integer, and let R^{n} be the usual space of (ordered) n-tuples of real numbers. This means that each x ∈ R^{n} may be expressed as x = (x_{1}, …, x_{n}), (A.2.1) with x_{j} ∈ R for j = 1, …, n. This is the same as the real line when n = 1.

If x, y ∈ R^{n}, then x + y is defined as an element of R^{n} by x + y = (x_{1} + y_{1}, …, x_{n} + y_{n}). (A.2.2)

If t ∈ R, then t x is defined as an element of R^{n} by

t x = (t x_{1}, …, t x_{n}). (A.2.3)

It is well known that R^{n} is a vector space over the real numbers, with addition and sclalar multiplication defined in this way. This basically means that addition and scalar multiplication on R^{n} satisfy some standard properties, such as commutativity and associaitvity of addition and distributivity of scalar multiplication, that follow easily from analogous properties of addition and multiplication of real numbers.

A.2.1 The standard Euclidean norm and metric on R^{n}

If x ∈ R^{n}, then the standard Euclidean norm is the nonnegative real number defined by

|x| = ( ∑_{j=1}^{n} x_{j}^{2})^{1/2}. (A.2.4)

The is the same as the absolute value of a real number when n = 1. Note that |x| = 0 if and only if x = 0, which means that x_{j} = 0 for j = 1, …, n.

It is easy to see that |t x| = |t| |x| (A.2.5)

for every t ∈ R, where |t| is the absolute value of t, as in the previous section. If y is another element of R^{n}, then it is well known that

|x + y| ≤ |x| + |y|. (A.2.6)

This is another version of the triangle inequality, which is more complicated when n ≥ 2. A proof of this will be discussed in the next section.

If x, y ∈ R^{n}, then the distance from x to y with respect to the standard Euclidean metric is defined by

d(x, y) = |x − y|, (A.2.7)

which is the same as before when n = 1. If z ∈ R^{n} too, then

d(x, z) ≤ d(x, y) + d(y, z), (A.2.8)

because of the triangle inequality for the standard Euclidean norm on R^{n}, as before. This is the triangle inequality for the standard Euclidean metric on R^{n}.

A.2.2 Open and closed balls

Let x ∈ R^{n} and a positive real number r be given. The open ball in R^{n} centered at x with radius r with respect to the standard Euclidean metric is defined by

B(x, r) = {y ∈ R^{n} : |x − y| < r}. (A.2.9)

Similarly, the closed ball in R^{n} centered at x with radius r with respect to the standard Euclidean metric is defined by

\overline{B}(x, r) = \{y ∈ R^{n} : |x − y| ≤ r\}, (A.2.10)

which may also be used when r = 0.

If n = 1, then (A.2.9) is the same as the open interval (x − r, x + r), and (A.2.10) is the same as the closed interval [x − r, x + r].

Observe that the standard Euclidean metric on R^{n} is invariant under translations, in the sense that

d(x + a, y + a) = |(x + a) − (y + a)| = |x − y| = d(x, y) (A.2.11)

for all a, x, y ∈ R^{n}. This implies that translates of open balls in R^{n} are open balls of the same radius. Similarly, translates of closed balls in R^{n} are closed balls of the same radius.

A.3 The dot product on R^{n}

Let n be a positive integer, and let x, y ∈ R^{n} be given. The dot product of x and y is the real number defined by

x · y = ∑_{j=1}^{n} x_{j} y_{j}. (A.3.1)

This is also known as the standard inner product on R^{n}. Note that the dot product is symmetric in x and y, which is to say that

x · y = y · x. (A.3.2)

We also have that

x · x = ∑_{j=1}^{n} x_{j}^{2} = |x|^{2}, (A.3.3)

where |x| is the standard Euclidean norm of x, as in (A.2.4).

If t is a real number, then it is easy to see that

(t x) · y = x · (t y) = t (x · y). (A.3.4)

If w is another element of R^{n}, then

(x + w) · y = x · y + w · y (A.3.5)

and x · (y + w) = x · y + x · w. (A.3.6)

Of course, x · y = 0 (A.3.7)

when x = 0 or y = 0.

A.3.1 The Cauchy–Schwarz inequality

The Cauchy–Schwarz inequality states that

|x · y| ≤ |x| |y|, (A.3.8)

using the absolute value of x · y as a real numbers on the left side, and the standard Euclidean norms on x and y on the right side. Note that (A.3.8) holds trivially when x = 0 or y = 0. Two proofs of the Cauchy–Schwarz inequality when x, y ≠ 0 will be given in the next section.

It is easy to see that equality holds in (A.3.8) when x = t y for some t ∈ R. The proof will show that this is the only way to get equality in (A.3.8) when y ≠ 0.

A.3.2 Using the Cauchy–Schwarz inequality

Let us see how to use the Cauchy–Schwarz inequality to get the triangle inequality (A.2.6) for the standard Euclidean norm on R^{n}.

Observe that

(A.3.9)  |x + y|^{2} = (x + y) · (x + y) = x · x + 2 x · y + y · y
= |x|^{2} + 2 x · y + |y|^{2}.

Using the Cauchy–Schwarz inequality, we get that

|x + y|^{2} ≤ |x|^{2} + 2 |x| |y| + |y|^{2} = (|x| + |y|)^{2}. (A.3.10)

Of course, this implies (A.2.6).

More precisely, equality holds in (A.3.10) if and only if

x · y = |x| |y|. (A.3.11)

This means that equality holds in (A.2.6) if and only if (A.3.11) holds. The proof of the Cauchy–Schwarz inequality will show that (A.3.11) holds only when y = 0 or x = t y for some nonnegative real number t.

A.4 Proving the Cauchy–Schwarz inequality

Let n be a positive integer again, and let x, y ∈ R^{n} be given. One way to prove the Cauchy–Schwarz inequality is to use the fact that

(x − t y) · (x − t y) = |x − t y|^{2} ≥ 0 (A.4.1)

for every t ∈ R. This implies that

x · x − 2 t (x · y) + t^{2} (y · y) ≥ 0, (A.4.2)

which is the same as saying that

|x|^{2} − 2 t (x · y) + t^{2} |y|^{2} ≥ 0. (A.4.3)

18 APPENDIX A. A BIT OF ANALYSIS Equivalently, this means that 2 t (x · y) ≤ |x|^{2} + t^{2} |y|^{2}. (A.4.4) Observe that these inequalities are strict when x ≠ t y.

We may as well suppose that x, y ≠ 0, as before. If we choose t so that |t| = |x|/|y|, (A.4.5) then we get that 2 (t/|t|) (x · y) ≤ |x|^{2}/|t| + |t| |y|^{2} = 2 |x| |y|. (A.4.6) We may also choose t so that (t/|t|) (x · y) = |x · y|. (A.4.7) Using this, (A.3.8) follows from (A.4.7).

More precisely, this argument shows that if x, y ≠ 0 and equality holds in (A.3.8), then x = t y (A.4.8) for some t ∈ R.

A.4.1 Another approach to the Cauchy–Schwarz inequal-

ity

If a and b are real numbers, then a^{2} − 2 a b + b^{2} = (a − b)^{2} ≥ 0. (A.4.9) This implies the well-known inequality a b ≤ (1/2) (a^{2} + b^{2}). (A.4.10) These inequalities are strict when a ≠ b.

It follows that

x · y ≤ ∑_{j=1}^{n}(1/2) (x_{j}^{2} + y_{j}^{2}) = (1/2) (|x|^{2} + |y|^{2}). (A.4.11)

This inequlity is strict when x_{j} ≠ y_{j} for some j, so that x ≠ y.

In particular, x · y ≤ 1 (A.4.12) when |x|, |y| ≤ 1. (A.4.13) The inequality in (A.4.12) is strict when x ≠ y, and when |x| < 1 or |y| < 1.

A.5. SOME OTHER NORMS ON R^{N} 19

This shows that x · y ≤ |x| |y| (A.4.14)

when |x| = |y| = 1. If x, y ≠ 0, then one can reduce to this case, using x/|x| and y/|y|. We also get that the inequality is strict when x/|x| ≠ y/|x|.

Observe that

−x · y = (−x) · y ≤ | − x| |y| = |x| |y|. (A.4.15)

The Cauchy–Schwarz inequality is the same as the combination of this and the previous inequality.

A.5 Some other norms on R^{n}

Let n be a positive integer, and let N be a nonnegative real-valued function on R^{n}. We say that N is a norm on R^{n} if it satisfies the following three conditions. First, for each x ∈ R^{n}, we have that

(A.5.1) N(x) = 0 if and only if x = 0.

Second, N(t x) = |t| N(x) (A.5.2)

for every t ∈ R and x ∈ R^{n}. Third,

N(x + y) ≤ N(x) + N(y) (A.5.3)

for every x, y ∈ R^{n}, which is the triangle inequality for a norm.

The standard Euclidean norm on R^{n} is a norm on this sense. One can check that

∥x∥_{1} = ∑_{j=1}^{n} |x_{j}| (A.5.4)

also defines a norm on R^{n}. One can verify that

∥x∥_{∞} = max_{1≤j≤n} |x_{j}| (A.5.5)

defines a norm on R^{n} as well. The standard Euclidean norm on R^{n} may be denoted ∥x∥_{2}, to be consistent with this notation. Note that these three norms are the same as the absolute value of a real number when n = 1.

A.5.1 Comparing these three norms

If x ∈ R^{n}, then it is easy to see that

∥x∥_{∞} ≤ ∥x∥_{1} (A.5.6)

and ∥x∥_{∞} ≤ ∥x∥_{2}. (A.5.7)

20 APPENDIX A. A BIT OF ANALYSIS One can also check that ∥x∥_{1} ≤ n ∥x∥_{∞} (A.5.8) and ∥x∥_{2} ≤ n^{1/2} ∥x∥_{∞}. (A.5.9)

Observe that

∥x∥_{2}^{2} = ∑_{j=1}^{n} x_{j}^{2} ≤ ∥x∥_{1} ∥x∥_{∞} ≤ ∥x∥_{1}^{2}, (A.5.10)

using (A.5.6) in the third step. This implies that ∥x∥_{2} ≤ ∥x∥_{1}. (A.5.11) One can use the Cauchy–Schwarz inequality to get that

∥x∥_{1} ≤ ∥x∥_{2} ( ∑_{j=1}^{n} 1)^{1/2} = n^{1/2} ∥x∥_{2}. (A.5.12)

A.6 Metrics associated to norms

Let n be a positive integer, and let N be a norm on R^{n}. If x, y ∈ R^{n}, then the distance from x to y with respect to N may be defined by d_{N}(x, y) = N(x − y). (A.6.1) Observe that (A.6.2) d_{N}(x, y) = 0 if and only if x = y, because of (A.5.1). Similarly, d_{N}(x, y) = N(x − y) = N(y − x) = d_{N}(y, x), (A.6.3) using (A.5.2) in the second step, with t = −1. If z ∈ R^{n} too, then d_{N}(x, z) = N(x − z) ≤ N(x − y) + N(y − z) = d_{N}(x, y) + d_{N}(y, z), (A.6.4) using (A.5.3) in the second step.

In fact, one can define the notion of a metric on any set. The remarks in the preceding paragraph imply that d_{N} defines a metric on R^{n}.

Note that d_{N} is invariant under translations on R^{n}, in the sense that d_{N}(x + a, y + a) = N((x + a) − (y + a)) = N(x − y) = d_{N}(x, y) (A.6.5) for all a, x, y ∈ R^{n}.

A.6. METRICS ASSOCIATED TO NORMS 21

A.6.1 The metrics associated to ∥ · ∥_{1}, ∥ · ∥_{2}, and ∥ · ∥_{∞}

In particular, d_{1}(x, y) = ∥x − y∥_{1} (A.6.6) and d_{∞}(x, y) = ∥x − y∥_{∞} (A.6.7) are the metrics on R^{n} associated to the norms ∥ · ∥_{1} and ∥ · ∥_{∞} in (A.5.4) and (A.5.5), respectively. The metric d_{2}(x, y) = ∥x − y∥_{2} (A.6.8) on R^{n} associated to the standard Euclidean norm ∥ · ∥_{2} is the same as the standard Euclidean metric.

Observe that d_{∞}(x, y) ≤ d_{2}(x, y) ≤ d_{1}(x, y) (A.6.9) for all x, y ∈ R^{n}, because of (A.5.7) and (A.5.11). We also have that d_{2}(x, y) ≤ n^{1/2} d_{∞}(x, y) (A.6.10) and d_{1}(x, y) ≤ n^{1/2} d_{2}(x, y) (A.6.11) for every x, y ∈ R^{n}, by (A.5.9) and (A.5.12).

A.6.2 Open and closed balls with respect to a norm

Let N be a nrom on R^{n} again, and let x ∈ R^{n} and a positive real number r be given. The open ball in R^{n} centered at x with radius r with respect to N, or equivalently with respect to the metric d_{N} associated to N, is defined by B_{N}(x, r) = B_{d_{N}} (x, r) = {y ∈ R^{n} : N(x − y) < r}. (A.6.12) Similarly, the closed ball in R^{n} centered at x with radius r with respect to N, or equivalently with respect to d_{N}, is defined by \overline{B}_{N}(x, r) = \overline{B}_{d_{N}} (x, r) = \{y ∈ R^{n} : N(x − y) ≤ r\}. (A.6.13)

Let us use B_{d_{1}}(x, r), B_{d_{2}}(x, r), and B_{d_{∞}}(x, r) for the open balls corresponding to the metrics d_{1}, d_{2}, and d_{∞} associated to the norms ∥·∥_{1}, ∥·∥_{2}, and ∥·∥_{∞}, respectively, and similarly for closed balls. It is easy to see that B_{d_{1}}(x, r) ⊆ B_{d_{2}}(x, r) ⊆ B_{d_{∞}}(x, r) (A.6.14) and \overline{B}_{d_{1}}(x, r) ⊆ \overline{B}_{d_{2}}(x, r) ⊆ \overline{B}_{d_{∞}}(x, r), (A.6.15) using (A.6.9).

We also have that B_{d_{∞}}(x, r) ⊆ B_{d_{2}}(x, n^{1/2} r) (A.6.16)

and \overline{B}_{d_{∞}}(x, r) ⊆ \overline{B}_{d_{2}}(x, n^{1/2} r), (A.6.17)

because of (A.6.10). Similarly,

B_{d_{2}}(x, r) ⊆ B_{d_{1}}(x, n^{1/2} r) (A.6.18)

and \overline{B}_{d_{2}}(x, r) ⊆ \overline{B}_{d_{1}}(x, n^{1/2} r), (A.6.19)

because of (A.6.11).

A.6.3 Convex subsets of R^{n}

A subset E of R^{n} is said to be convex if for any pair of elements w, z of E and real number t with 0 < t < 1, we have that

(1 − t) w + t z ∈ E. (A.6.20)

If N is a norm on R^{n}, then one can check that the open and closed balls in R^{n} with respect to N are convex.

A.7 Complex numbers

A complex number z can be expressed in a unique way as z = x + y i, where x, y ∈ R and i^{2} = −1. In this case, x and y are known as the real and imaginary parts of z, and may be denoted Re z and Im z, respectively.

Addition and multiplication of real numbers can be extended to the set C of complex numbers in a standard way.

A.7.1 Complex conjugates

The complex conjugate of z is the complex number defined by

\overline{z} = x − y i. (A.7.1)

If w is another complex number, then one can check that

\overline{z + w} = \overline{z} + \overline{w} (A.7.2)

and \overline{(z w)} = \overline{z} \overline{w}. (A.7.3)

Note that the complex conjugate of \overline{z} is z.

It is easy to see that Re z = (1/2) (z + \overline{z}) (A.7.4)

and Im z = (1/(2 i)) (z − \overline{z}) (A.7.5)

A.7. COMPLEX NUMBERS 23

A.7.2 Absolute values

The absolute value or modulus is the nonnegative real number defined by |z| = (x^{2} + y^{2})^{1/2}. (A.7.6) Clearly |\overline{z}| = |z|. (A.7.7) It is easy to see that z \overline{z} = |z|^{2}. (A.7.8)

It follows that |z w|^{2} = z w \overline{(z w)} = z w \overline{z} \overline{w} = z \overline{z} w \overline{w} = |z|^{2} |w|^{2}. (A.7.9) This means that |z w| = |z| |w|. (A.7.10)

If z ≠ 0, then |z| > 0, and z (\overline{z}/|z|^{2}) = (z \overline{z})/|z|^{2} = 1. (A.7.11) This implies that z has a multiplicative inverse in C, with 1/z = \overline{z}/|z|^{2}. (A.7.12)

A.7.3 Some inequalities

Observe that | Re z|, | Im z| ≤ |z|. (A.7.13) More precisely, |z| is equal to | Re z| or | Im z| exactly when z is real or purely imaginary, as appropriate.

It follows that | Re(z \overline{w})| ≤ |z \overline{w}| = |z| |\overline{w}| = |z| |w|. (A.7.14) Equality holds if and only if z \overline{w} is a real number, as in the preceding paragraph.

We also have that

|z + w|^{2} = (z + w) \overline{(z + w)} = (z + w) (\overline{z} + \overline{w})
(A.7.15)  = z \overline{z} + z \overline{w} + \overline{z} w + w \overline{w} = |z|^{2} + 2 Re(z\overline{w}) + |w|^{2}.

Combining this with (A.7.14), we get that |z + w|^{2} ≤ |z|^{2} + 2 |z| |w| + |w|^{2} = (|z| + |w|)2. (A.7.16) Of course, this means that |z + w| ≤ |z| + |w|. (A.7.17) One can verify that equality holds if and only if z \overline{w} is a nonnegative real number.

A.7.4 Comparison with R^{2}

If z = x + y i with x, y ∈ R, as before, then z corresponds in a simple way to (x, y) as an element of R^{2}. This correspondence is compatible with addition on C and R^{2}, as well as multiplication by a real number. Similarly, the absolute value of a complex number corresponds exactly to the standard Euclidean norm on R^{2} in this way.

Suppose that w = u + v i with u, v ∈ R, so that w corresponds to (u, v) as an element of R^{2} in the same way. Observe that

Re(z\overline{w}) = x u + y v = (x, y) · (u, v), (A.7.18)

where the right side uses the dot product on R^{2}. Thus (A.7.14) corresponds to the Cauchy–Schwarz inequality on R^{2}.

Note that d(z, w) = |z − w| (A.7.19) corresponds to the standard Euclidean metric on R^{2} in this way.

A.8 Convergent sequences

Let n be a positive integer, and let x(1), x(2), x(3), … be an infinite sequence of elements of R^{n}. Thus, for each positive integer l, x(l) may be expressed as

x(l) = (x_{1}(l), x_{2}(l), …, x_{n}(l)), (A.8.1)

where x_{j}(l) is a real number for each j = 1, 2, …, n.

A.8.1 Convergent sequences in R^{n}

We say that {x(l)}_{l=1}^{∞} converges to an element x of R^{n} if for every positive real number ϵ there is a positive integer L such that

|x(l) − x| < ϵ (A.8.2)

for every l ≥ L.

In this case, x is said to be the limit of {x(l)}_{l=1}^{∞}, which may be expressed by

lim_{l→∞} x(l) = x (A.8.3)

or x(l) → ∞ as l → ∞. (A.8.4) More precisely, it is well known and not too difficult to show that the limit of a convergent sequence is unique.

In particular, this reduces to the usual definition of a convergent sequence of real numbers when n = 1. It is also well known and not too difficult to show that (A.8.3) holds if and only if

lim_{l→∞} x_{j}(l) = x_{j} (A.8.5)

for each j = 1, 2, …, n, as a sequence of real numbers.

A.8.2 Bounded sequences in R^{n}

A sequence {x(l)}_{l=1}^{∞} of elements of R^{n} is said to be bounded if there is a nonnegative real number C such that

|x(l)| ≤ C (A.8.6)

for every l ≥ 1. It is well known and not difficult to check that convergent sequences in R^{n} are bounded.

If {x(l)}_{l=1}^{∞} converges to x ∈ R^{n}, then it is well known and not too difficult to verify that

lim_{l→∞} |x(l)| = |x|, (A.8.7)

as a sequence of real numbers.

A.8.3 Monotone sequences of real numbers

Let {x_{j}}_{j=1}^{∞} be a sequence of real numbers. A real number a is said to be a lower bound for {x_{j}}_{j=1}^{∞} if a ≤ x_{j} (A.8.8)

for each j ≥ 1. Similarly, a real number b is said to be an upper bound for {x_{j}}_{j=1}^{∞} if x_{j} ≤ b (A.8.9)

for each j ≥ 1. Note that {x_{j}}_{j=1}^{∞} is bounded in R, as in the previous subsection, if and only if {x_{j}}_{j=1}^{∞} has both an upper and a lower bound in R.

Suppose for the moment that {x_{j}}_{j=1}^{∞} is monotonically increasing, so that

x_{j} ≤ x_{j+1} (A.8.10)

for each j ≥ 1. In particular, this implies that x_{1} is a lower bound for {x_{j}}_{j=1}^{∞}. If {x_{j}}_{j=1}^{∞} has an upper bound in R, then it is well known that {x_{j}}_{j=1}^{∞} converges in R. In this case, the limit of {x_{j}}_{j=1}^{∞} is an upper bound for {x_{j}}_{j=1}^{∞}, and in fact

(A.8.11)  lim_{j→∞} x_{j} is the smallest upper bound for {x_{j}}_{j=1}^{∞}.

This is known as the least upper bound or supremum of the set of x_{j}’s, j ≥ 1, which may be expressed as

sup_{j≥1} x_{j}. (A.8.12)

Suppose now that {x_{j}}_{j=1}^{∞} is monotonically decreasing, so that

x_{j} ≥ x_{j+1} (A.8.13)

for each j ≥ 1. This implies that x_{1} is an upper bound for {x_{j}}_{j=1}^{∞}. If {x_{j}}_{j=1}^{∞} has a lower bound in R, then {x_{j}}_{j=1}^{∞} converges in R, and

(A.8.14)  lim_{j→∞} x_{j} is the largest lower bound for {x_{j}}_{j=1}^{∞}.

This is known as the greatest lower bound or of the set of x_{j}’s, j ≥ 1, which may be expressed as

inf_{j≥1} x_{j}. (A.8.15)

These statements can be obtained from those in the preceding paragraph, because {−x_{j}}_{j=1}^{∞} is monotonically increasing.

A.8.4 Convergent sequences of complex numbers

An infinite sequence {z_{j}}_{j=1}^{∞} of complex numbers is said to converge to a complex number z if for every ϵ > 0 there is a positive integer L such that

|z_{j} − z| < ϵ (A.8.16)

for each j ≥ L. This is equivalent to the convergence of the corresponding sequence in R^{2}, as in Subsection A.7.4. In this case, we have that

lim_{j→∞} |z_{j}| = |z|, (A.8.17)

as before.

Note that

lim_{j→∞} z_{j} = z (A.8.18)

if and only if

(A.8.19)  lim_{j→∞} Re z_{j} = Re z

and

(A.8.20)  lim_{j→∞} Im z_{j} = Im z,

as sequences of real numbers, as before. This is also equivalent to

lim_{j→∞} \overline{z_{j}} = \overline{z}. (A.8.21)

A.9 More on convergent sequences

Let {z_{j}}_{j=1}^{∞} and {w_{j}}_{j=1}^{∞} be sequences of complex numbers that converge to z, w ∈ C, respectively. It is well known and not difficult to show that the corresponding sequence {z_{j} + w_{j}}_{j=1}^{∞} of sums converges to z + w. This means that

(A.9.1)  lim_{j→∞}(z_{j} + w_{j}) = lim_{j→∞} z_{j} + lim_{j→∞} w_{j},

where more precisely the limits on the right exist by hypothesis, and the existence of the limit on the left is part of the conclusion.

It is also well known that the sequence {z_{j} w_{j}}_{j=1}^{∞} of products converges to z w, although this is a bit more complicated. This means that

(A.9.2)  lim_{j→∞}(z_{j} w_{j}) = ( lim_{j→∞} z_{j}) ( lim_{j→∞} w_{j}),

where the limits on the right exist by hypothesis, and the existence of the limit on th left is part of the conclusion.

A.9.1 A helpful lemma

Let {a_{j}}_{j=1}^{∞} be a sequence of complex numbers that converges to 0, and let {b_{j}}_{j=1}^{∞} be a bounded sequence of complex numbers. Under these conditions, one can check that

lim_{j→∞}(a_{j} b_{j}) = 0. (A.9.3)

In particular, one can use this to get that

lim_{j→∞}((z_{j} − z) w_{j}) = 0. (A.9.4)

One can also verify that

lim_{j→∞}(z w_{j}) = z w, (A.9.5)

directly from the definitions. It is easy to obtain (A.9.2) from (A.9.4) and (A.9.5).

A.9.2 Sequences of reciprocals

If z_{j} ≠ 0 for each j, and z ≠ 0, then it is well known that {1/z_{j}}_{j=1}^{∞} converges to 0. This means that

lim_{j→∞}(1/z_{j}) = 1/( lim_{j→∞} z_{j}), (A.9.6)

whwre the limit on the right exists by hypohtesis, and the existence of the limit on the left is part of the conclusion, as usual.

In order to show this, it is helpful to first use the convergence of {z_{j}}_{j=1}^{∞} to z to get that there is a positive integer L_{1} such that

|z_{j} − z| < |z|/2 (A.9.7)

when j ≥ L_{1}. One can check that this implies that

|z_{j}| > |z|/2 (A.9.8)

when j ≥ L_{1}.

A.9.3 More on sequences in R^{n}

Let n be a positive integer, and let {x(l)}_{l=1}^{∞}, {y(l)}_{l=1}^{∞} be sequences of elements of R^{n} that converge to x, y ∈ R^{n}, respectively. Under these conditions, one can check that {x(l) + y(l)}_{l=1}^{∞} converges to x + y. This means that

(A.9.9)  lim_{l→∞}(x(l) + y(l)) = lim_{l→∞} x(l) + lim_{l→∞} y(l),

where the limits on the right exist by hypothesis, and the existence of the limit on the left is part of the conclusion, as before. This can be shown using an argument like that for sequences of real or complex numbers, or by reducing to the analogous statement for sequences of real numbers.

One can also verify that the sequence {x(l) · y(l)}_{l=1}^{∞} converges to the dot product x · y of the limits. This means that

(A.9.10)  lim_{l→∞} x(l) · y(l) = ( lim_{l→∞} x(l)) · ( lim_{l→∞} y(l)),

where the limits on the right existe by hypothesis, and the existence of the limit on the left is part of the conclusion. This follows from the earlier statements about sums and products of convergent sequences of real numbers.

Let {t_{l}}_{l=1}^{∞} be a sequence of real numbers that converges to t ∈ R. In this case, we have that {t_{l} x(l)}_{l=1}^{∞} converges to t x in R^{n}. This means that

(A.9.11)  lim_{l→∞}(t_{l} x(l)) = ( lim_{l→∞} t_{l}) ( lim_{l→∞} x(l)),

where the limits on the right exist by hypothesis, and the existence of the limit on the left is part of the conclusion. This can be obtained from the analogous statement for products of convergent sequences of real numbers, or shown in a similar way.

A.10 Some particular sequences

We consider the convergence of some particular sequences of real and complex numbers in this section. Some of the arguments for obtaining convergence are at least sketched, although other approaches are sometimes used as well.

A.10.1 {1/j^{p}}_{j=1}^{∞}

If p is a positive real number, then it is well known and not difficult to show that

lim_{j→∞} 1/j^{p} = 0. (A.10.1)

One might as well take p = 1/k for some positive integer k here, which is a bit simpler.

A.10.2 {j^{α} a^{j}}_{j=1}^{∞}

If a is a complex number with |a| < 1 (A.10.2)

and α is any real number, then it is well known that

lim_{j→∞} j^{α} a^{j} = 0. (A.10.3)

Of course, this is trivial when a = 0, and one can reduce to the case where a is a positive real number. One can also reduce to the case where α < 1, using a positive integer k such that α/k < 1. (A.10.4)

A.10. SOME PARTICULAR SEQUENCES 29 More precisely, if one can show that

lim_{j→∞} j^{α/k} a^{j/k} = 0, (A.10.5)

then it follows that j^{α} a^{j} = (j^{α/k} a^{j/k})^{k} → 0 (A.10.6) as j → ∞, as before. Note that a^{1/k} < 1 when 0 < a < 1.

A.10.3 {p^{1/j}}_{j=1}^{∞}

If p is a positive real number, then it is well known that

lim_{j→∞} p^{1/j} = 1. (A.10.7)

One can reduce to the case where p ≥ 1, using the remarks in Subsection A.9.2. If ϵ is a positive real number, then p^{1/j} < 1 + ϵ (A.10.8) if and only if p < (1 + ϵ)^{j}. (A.10.9) It is not too difficult to show that (A.10.9) holds when j is sufficiently large. This is basically the same as (A.10.3), with α = 0, and a = 1/(1 + ϵ).

A.10.4 {j^{1/j}}_{j=1}^{∞}

It is also well known that

lim_{j→∞} j^{1/j} = 1. (A.10.10)

If ϵ is a positive real number again, then j^{1/j} < 1 + ϵ (A.10.11) if and only if j < (1 + ϵ)^{j}. (A.10.12) One can check that this holds when j is sufficiently large. Note that (A.10.12) is the same as saying that j (1 + ϵ)^{−j} < 1. (A.10.13) That this holds when j is sufficiently large corresponds to (A.10.3) with α = 1 and a = 1/(1 + ϵ).

A.11 Infinite series

Let ∑_{j=1}^{∞} a_{j} be an infinite series of complex numbers. If n is a positive integer, then

s_{n} = ∑_{j=1}^{n} a_{j} (A.11.1)

is called the nth partial sum of ∑_{j=1}^{∞} a_{j}. If the sequence {s_{n}}_{n=1}^{∞} of partial sums converges, then we say that ∑_{j=1}^{∞} a_{j} converges as an infinite series, and that the value of the sum is

∑_{j=1}^{∞} a_{j} = lim_{n→∞} s_{n}. (A.11.2)

If ∑_{j=1}^{∞} a_{j} converges, then it is well known that

lim_{j→∞} a_{j} = 0. (A.11.3)

One way to see this is to observe that

a_{n} = s_{n} − s_{n−1} (A.11.4)

when n ≥ 2, and that the right side tends to 0 as n → ∞, because s_{n} and s_{n−1} converge to the same limit.

A.11.1 Some properties of convergent series

If ∑_{j=1}^{∞} a_{j} converges and ∑_{j=1}^{∞} b_{j} is another convergent series of complex numbers, then ∑_{j=1}^{∞}(a_{j} + b_{j}) converges too, with

∑_{j=1}^{∞}(a_{j} + b_{j}) = ∑_{j=1}^{∞} a_{j} + ∑_{j=1}^{∞} b_{j}. (A.11.5)

Indeed,

∑_{j=1}^{n}(a_{j} + b_{j}) = ∑_{j=1}^{n} a_{j} + ∑_{j=1}^{n} b_{j} (A.11.6)

for each n, so that

(A.11.7)  lim_{n→∞} ∑_{j=1}^{n}(a_{j} + b_{j}) = lim_{n→∞} ∑_{j=1}^{n} a_{j} + lim_{n→∞} ∑_{j=1}^{n} b_{j}.

Similarly, if ∑_{j=1}^{∞} a_{j} converges and c is a complex number, then ∑_{j=1}^{∞} c a_{j} converges, with

∑_{j=1}^{∞} c a_{j} = c ∑_{j=1}^{∞} a_{j}. (A.11.8)

This is because

∑_{j=1}^{n} c a_{j} = c ∑_{j=1}^{n} a_{j} (A.11.9)

for each n, so that

lim_{n→∞} ∑_{j=1}^{n} c a_{j} = c ( lim_{n→∞} ∑_{j=1}^{n} a_{j}). (A.11.10)

A.11.2 Geometric series

If z is a complex number, then ∑_{j=0}^{∞} z^{j} is called a geometric series. Here z^{j} is interpreted as being equal to 1 when j = 0, even when z = 0. Of course, infinite series beginning with j = 0, or any other integer, can be handled in the same way as before.

Let n be a nonnegative integr, and observe that

(1 − z) ∑_{j=0}^{n} z^{j} = ∑_{j=0}^{n} z^{j} − ∑_{j=0}^{n} z^{j+1} = ∑_{j=0}^{n} z^{j} − ∑_{j=1}^{n+1} z^{j} = 1 − z^{n+1}. (A.11.11)

If z ≠ 1, then it follows that

∑_{j=0}^{n} z^{j} = (1 − z^{n+1})/(1 − z). (A.11.12)

If |z| < 1, then

lim_{n→∞} z^{n} = 0, (A.11.13)

as in Subsection A.10.2, with α = 0. This implies that ∑_{j=0}^{∞} z^{j} converges, with

∑_{j=0}^{∞} z^{j} = 1/(1 − z), (A.11.14)

because z^{n} → 0 as n → ∞,

If |z| ≥ 1, then |z^{j}| = |z|^{j} ≥ 1 (A.11.15)

for each j ≥ 0. This implies that {z^{j}}_{j=0}^{∞} does not converge to 0, so that ∑_{j=0}^{∞} z^{j} does not converge.

A.11.3 Infinite series with nonnegative terms

If ∑_{j=1}^{∞} a_{j} is a convergent series of complex numbers, then the corresponding sequence of partial sums is bounded, because it converges.

Suppose now that ∑_{j=1}^{∞} a_{j} is an infinite series whose terms are nonnegative real numbers. This implies that

s_{n} ≤ s_{n} + a_{n+1} = s_{n+1} (A.11.16) for each n, so that the corresponding sequence of partial sums is monotonically increasing. If {s_{n}}_{n=1}^{∞} has an upper bound in R, then it follows that ∑_{j=1}^{∞} a_{j} converges, as in Subsection A.8.3.

If p is a positive real number, then it is well known that

∑_{j=1}^{∞} 1/j^{p} (A.11.17)

converges if and only if p > 1.

A.11.4 Absolute convergence

An infinite series ∑_{j=1}^{∞} a_{j} of complex numbers is said to converge absolutely if

∑_{j=1}^{∞} |a_{j}| (A.11.18)

converges. It is well known that this implies that ∑_{j=1}^{∞} a_{j} converges. In this case, it is not too difficult to show that

|∑_{j=1}^{∞} a_{j}| ≤ ∑_{j=1}^{∞} |a_{j}|. (A.11.19)

If ∑_{j=1}^{∞} a_{j} is an absolutely convergent series of real numbers, then one can show that the corresponding sums of the positive and negative parts of the a_{j}’s converge, using the remarks in the previous subsection. One can use this to get that ∑_{j=1}^{∞} a_{j} converges.

If ∑_{j=1}^{∞} a_{j} is an absolutely convergent series of complex numbers, then it is easy to see that the corresponding series of real and imaginary parts of the a_{j}’s are absolutely convergent. One can use this and the remarks in the preceding paragraph to get that ∑_{j=1}^{∞} a_{j} converges.

Let ∑_{j=1}^{∞} a_{j} be an infinite series of complex numbers, and let ∑_{j=1}^{∞} b_{j} be an infinite series of nonnegative real numbers. If

|a_{j}| ≤ b_{j} (A.11.20)

for each j, and ∑_{j=1}^{∞} b_{j} converges, then the comparison test says that ∑_{j=1}^{∞} a_{j} converges absolutely. This can be obtained from the remarks in the previous subsection.

A.12 More on infinite series

Let {a_{j}}_{j=0}^{∞} be a sequence of complex numbers such that the sequence of sums

A_{n} = ∑_{j=0}^{n} a_{j} (A.12.1)

is bounded. Also let {b_{j}}_{j=0}^{∞} be a monotonically decreasing sequence of nonnegative real numbers that converges to 0. Under these conditions, it is well known that

∑_{j=0}^{∞} a_{j} b_{j} (A.12.2)

converges. This is Theorem 3.42 on p70 of [309].

If a_{j} = (−1)^{j} for each j, then it is easy to see that A_{n} = 1 when n is even, and A_{n} = 0 when n is odd. In this case, this criterion for convergence corresponds to Leibniz’ alternating series test. This corresponds to Theorem 3.3 A on p72 of [125], and to Theorem 3.43 on p71 of [309].

Suppose that z is a complex number with |z| = 1 and z ≠ 1. If

a_{j} = z^{j} (A.12.3)

for each j, then one can check that {A_{n}}_{n=0}^{∞} is a bounded sequence in C, using some of the remarks in Section A.11.2. This corresponds to Theorem 3.44 on p71 of [309].

A.12.1 Power series

Let a_{0}, a_{1}, a_{2}, a_{3}, … be an infinite sequence of complex numbers, and consider the corresponding power series

∑_{j=0}^{∞} a_{j} z^{j}. (A.12.4)

Suppose for the moment that (A.12.4) converges absolutely for some z ∈ C, and that w ∈ C satisfies |w| ≤ |z|. (A.12.5)

One can check that

∑_{j=0}^{∞} a_{j} w^{j} (A.12.6)

converges absolutely, using the comparison test.

Suppose for the moment again that (A.12.4) converges for some z ∈ C with z ≠ 0. If w ∈ C satisifes |w| < |z|, (A.12.7)

then it is not too difficult to show that (A.12.6) converges absolutely. More precisely, the convergence of (A.12.4) implies that

lim_{j→∞} a_{j} z^{j} = 0, (A.12.8)

and in particular that {a_{j} z^{j}}_{j=0}^{∞} is a bounded sequence. This means that there is a nonnegative real number C such that

|a_{j} z^{j}| ≤ C (A.12.9) for each j. Using this, it is easy to see that

|a_{j} w^{j}| ≤ C (|w|/|z|)^{j} (A.12.10)

for each j. If (A.12.7) holds, then ∑_{j=0}^{∞}(|w|/|z|)^{j} is a convergent geometric series. In this case, one can use the comparison test to get that (A.12.6) converges asbolutely.

A.12.2 The radius of convergence

Suppose for the moment that there is a z ∈ C such that (A.12.4) does not converge. In this case, it is well known that there is a nonnegative real number ρ with the following two properties. First, if w ∈ C and

|w| > ρ, (A.12.11)

then (A.12.6) does not converge. Second, if w ∈ C and

|w| < ρ, (A.12.12)

then (A.12.6) converges absolutely.

If (A.12.4) converges for every z ∈ C, then (A.12.6) converges absolutely for every w ∈ C, as in the previous subsection. In this case, we can take ρ = +∞.

One can check that ρ is uniquely determined by these properties. This is called the radius of convergence of the power series (A.12.4).

A.13 Rearrangements of infinite series

Let π be a one-to-one mapping from the set Z_{+} of positive integers onto itself. This means that π(j) ∈ Z_{+} for every positive inetger j, and that every positive integer k can be expressed as π(j) for exactly on j ∈ Z_{+}, which may be expressed as π^{−1}(k).

If ∑_{j=1}^{∞} a_{j} is an infinite series of complex numbers, then the infinite series

∑_{j=1}^{∞} a_{π(j)} (A.13.1)

is called a rearrangement of ∑_{j=1}^{∞} a_{j}. If

(A.13.2) a_{j} = 0 for all but finitely many j,

then it is easy to see that

(A.13.3) a_{π(j)} = 0 for all but finitely many j

as well, with

∑_{j=1}^{∞} a_{π(j)} = ∑_{j=1}^{∞} a_{j}. (A.13.4)

Suppose for the moment that a_{j} is a nonnegative real number for each j. Under these conditions, it is well known and not too difficult to show that ∑_{j=1}^{∞} a_{j} converges if and only if (A.13.1) converges, in which case (A.13.4) holds. This is because

(A.13.5) every partial sum of ∑_{j=1}^{∞} a_{j} is less than or equal to

some partial sum of (A.13.1),

and vice-versa. This corresponds to Lemma 3.5 E on p77 of [125].

If ∑_{j=1}^{∞} a_{j} is an absolutely convergent series of complex numbers, then it follows that (A.13.1) is absolutely convergent too. More precisely, the convergence of ∑_{j=1}^{∞} |a_{j}| implies the convergence of

∑_{j=1}^{∞} |a_{π(j)}|, (A.13.6)

as in the preceding paragraph.

If a_{j} is a real number for each j, then one can get that (A.13.4) holds, by considering the positive and negative parts of a_{j} separately. This corresponds to Theorem 3.5 F on p77 of [125]. Otherwise, one can reduce to this case, by considering the real and imaginary parts of a_{j} for each j.

Another argument is used in the proof of Theorem 3.55 on p78 of [309]. Basically, one can show more directly that

lim_{n→∞} ( ∑_{j=1}^{n} a_{j} − ∑_{j=1}^{n} a_{π(j)}) = 0 (A.13.7)

under these conditions.

Suppose now that ∑_{j=1}^{∞} a_{j} is an infinite series of real numbers that converges, and does not converge absolutely. In this case, it is well known that there are rearrangements of the series that do not converge, as in Exercises 4, 5 on p80 of [125]. There are also rearrangements that converge, with the sum equal to any real number, as in Theorem 3.5 D on p77 of [125]. See Theorem 3.54 on p76 of [309] as well.

A.14 Cauchy products of infinite series

If ∑_{j=0}^{∞} a_{j} and ∑_{l=0}^{∞} b_{l} are infinite series of complex numbers, then one might like to multiply these two series, and arrange the terms into a single series. A nice way to do this is to take

c_{n} = ∑_{j=0}^{n} a_{j} b_{n−j} (A.14.1)

for each nonnegative integer n. The corresponding infinite series ∑_{n=0}^{∞} c_{n} is called the Cauchy product of ∑_{j=0}^{∞} a_{j} and ∑_{l=0}^{∞} b_{l}.

It is not difficult to see that

∑_{n=0}^{∞} c_{n} = ( ∑_{j=0}^{∞} a_{j}) ( ∑_{l=0}^{∞} b_{l}) (A.14.2)

formally. In fact, both sides of the equation correspond to

∑_{j,l≥0} a_{j} b_{l} (A.14.3)

formally, where the sum is taken over all pairs of nonnegative integers j, l.

Suppose for the moment that a_{j} = 0 for all but finitely many j, and that b_{l} = 0 for all but finitely many l. This implies that a_{j} b_{l} = 0 for all but finitely many pairs of nonnegative integers j, l. One can also check that c_{n} = 0 for all but finitely many n. In this case, one can verify that (A.14.2) holds. One can interpret (A.14.3) as reducing to a finite sum, that is equal to both sides of (A.14.2).

A.14.1 Cauchy products and power series

The definition of the Cauchy product works well with power series. Indeed, the Cauchy product of the power series ∑_{j=0}^{∞} a_{j} z^{j} and ∑_{l=0}^{∞} b_{l} z^{l}, is the power series

∑_{n=0}^{∞} c_{n} z^{n}, (A.14.4)

where c_{n} is as in (A.14.1) for each n.

A.14.2 Convergence of Cauchy products

Suppose for the moment that a_{j} is a nonnegative real number for each j, and that b_{l} is a nonnegative real number for each l. This implies that c_{n} is a nonnegative real number for each n. If N is a nonnegative integer, then one can check that

∑_{n=0}^{N} c_{n} ≤ ( ∑_{j=0}^{N} a_{j}) ( ∑_{l=0}^{N} b_{l}). (A.14.5)

Similarly, if N_{1} and N_{2} are nonnegative integers, then one can verify that

( ∑_{j=0}^{N_{1}} a_{j}) ( ∑_{l=0}^{N_{2}} b_{l}) ≤ ∑_{n=0}^{N_{1}+N_{2}} c_{n}. (A.14.6)

If ∑_{j=0}^{∞} a_{j} and ∑_{l=0}^{∞} b_{l} converge, then one can use these inequalities to get that ∑_{n=0}^{∞} c_{n} converges, with sum as in (A.14.2).

Observe that

|c_{n}| ≤ ∑_{j=0}^{n} |a_{j}| |b_{n−j}| (A.14.7)

for each n when the a_{j}’s and b_{l}’s are any complex numbers. The right side of this inequality is the same as the nth term of the Cauchy product of the series ∑_{j=0}^{∞} |a_{j}| and ∑_{l=0}^{∞} |b_{l}|. If these series converge, then their Cauchy product converges, as in the previous paragraph. This implies that ∑_{n=0}^{∞} |c_{n}| converges, by the comparison test.

It is not difficult to show that (A.14.2) holds under these conditions, by reducing to convergent series of nonnegative real numbers. This corresponds to the theorem on p78 of [125], which uses another argument.

If one of ∑_{j=0}^{∞} a_{j} and ∑_{l=0}^{∞} b_{l} converges absolutely, and the other series converges, then it is well known that ∑_{n=0}^{∞} c_{n} converges, with sum as in (A.14.2). This is Theorem 3.50 on p74 of [309]. However, the Cauchy product of convergent series may not converge, as in Example 3.49 on p73 of [309].

A.15 The binomial theorem

If j is a positive integer, then j! is j factorial, the product of the positive integers from 1 to j, as usual. This is interpreted as being equal to 1 when j = 0.

Let n be a nonnegative integer, and let z, w be complex numbers. The binomial theorem states that

(z + w)^{n} = ∑_{j=0}^{n} \binom{n}{j} z^{j} w^{n−j}, (A.15.1)

where

\binom{n}{j} = \frac{n!}{j! (n − j)!} (A.15.2)

is the usual binomial coefficient.

It is easy to see that (z + w)^{n} (A.15.3)

can be expanded into a sum of terms of the form

z^{j} w^{n−j}, (A.15.4)

0 ≤ j ≤ n, where each of these terms occurs a positive number of times. One can show that the number of times that these terms occur is equal to (A.15.2), using induction.

These coefficients can also be obtained using differentiation.

Appendix B

A bit of analysis, 2

B.1 The complex exponential function

If z is a complex number, then put

E(z) = ∑_{j=0}^{∞} \frac{z^{j}}{j!} , (B.1.1)

as in (25) on p178 of [309]. One can check that the series on the right converges absolutely, using the ratio test. Alternatively, one can use the comparison test, by comparing this series with a convergent geometri series.

If z ∈ R, then the right side of (B.1.1) is the usual Taylor series expansion for the exponential function. One can also use this to define the exponential function on the real line, and as a complex-valued function on the complex plane.

Of course, we would like to check that this satisfies the usual properties of the exponential function on the real line. We would like to consider its properties on the complex plane as well.

B.1.1 Exponentials of sums

If w is another complex number, then

E(z + w) = ∑_{n=0}^{∞} \frac{(z + w)^{n}}{n!} = ∑_{n=0}^{∞} ( ∑_{j=0}^{n} \frac{z^{j}}{j!} \frac{w^{n−j}}{(n − j)!}), (B.1.2)

using the binomial theorem in the second step, as in Section A.15. The right side is the same as the Cauchy product of the series used to define E(z) and E(w), as in Section A.14. This implies that

E(z + w) = E(z) E(w), (B.1.3) as in (26) on p178 of [309], because these series converge absolutely, as in Subsection A.14.2.

In particular, we can take w = −z, to get that

E(z) E(−z) = E(z − z) = E(0) = 1, (B.1.4)

as in (27) on p178 of [309]. This means that E(z) ≠ 0, with

E(z)^{−1} = E(−z). (B.1.5)

If x ∈ R, then E(x) ∈ R, and one can check that E(x) is positive and strictly increasing for x ≥ 0. One can verify that the same properties hold when x ≤ 0, using (B.1.5).

Similarly, if x ∈ R, then E(x) → +∞ as x → +∞, as on p179 of [309]. It follows that E(x) → 0 as x → −∞, because of (B.1.5), as in [309].

We may use E(z) as the definition of the complex exponential function e^{z} = exp z for z ∈ C, as on p1 of [308]. We shall use this notation from now on here.

B.1.2 Exponentials and complex conjugates

It is easy to see that \overline{(exp z)} = exp(\overline{z}) (B.1.6)

for every z ∈ C. This implies that

(B.1.7)  | exp z|^{2} = (exp z) \overline{(exp z)} = (exp z) (exp \overline{z})
= exp(z + \overline{z}) = exp(2 Re z).

In particular, | exp(i y)| = 1 (B.1.8)

for every y ∈ R. In fact, Euler’s identity states that

(B.1.9) exp(i y) = cos y + i sin y,

as in (46) on p182 of [309], and (5) on p2 of [308].

B.2 Continuous complex-valued functions

Let n be a positive integer, let E be a nonempty subset of R^{n}, and let f be a complex-valued function on E. We say that f is continuous at a point x ∈ E if for every ϵ > 0 there is a δ > 0 such that

|f(x) − f(y)| < ϵ (B.2.1)

for every y ∈ E such that |x − y| < δ. (B.2.2)

Let {x_{j}}_{j=1}^{∞} be a sequence of elements of E that converges to x, as in Subsection A.8.1. One can check that

lim_{j→∞} f(x_{j}) = f(x) (B.2.3)

under these conditions.

It is well known that continuity of f at x is characterized by this property. This can be shown using an argument by contradiction.

If f is continuous at every point in E, then we may simply say that f is continuous on E.

B.2.1 More on continuous functions

Let g be another complex-valued function on E. If f and g are both continuous at x ∈ E, then it is well known that

(B.2.4) f + g and f g are continuous at x,

as complex-valued functions on E. This can be obtained from the characterization of continuity in terms of convergent sequences and the analogous statements for convergent sequences of complex numbers, as in Section A.9.

Suppose for the moment that f(y) ≠ 0 for each y ∈ E, so that 1/f defines a complex-valued function on E. If f is continuous at x, then it is well known that (B.2.5) 1/f is continuous at x

too. This can be obtained from the analogous statement for sequences of complex numbers, as in Subsection A.9.2.

One can use (B.2.4) to get that functions on R^{n} defined by polynomials are continuous. Similarly, a rational function is continuous on any set where the denominator does not take the value 0.

B.2.2 Continuity of compositions

Let A be a nonempty subset of the complex plane, and let ϕ be a complex-valued function on A. If f(y) ∈ A (B.2.6)

for every y ∈ E, then the composition ϕ ∘ f of f with ϕ may be defined as another complex-valued function on E, with

(ϕ ∘ f)(y) = ϕ(f(y)) (B.2.7)

for every y ∈ E, as usual.

If f is continuous at x ∈ E, and ϕ is continuous at f(x), then it is well known that (B.2.8) ϕ ∘ f is continuous at x.

This can be verified directly from the definitions, or using the characterization of continuity in terms of convergent sequences.

Of course, z ↦ 1/z (B.2.9)

defines a complex-valued function on the set C\{0} of nonzero complex numbers. If f(y) ≠ 0 for each y ∈ E, then 1/f is the same as the composition of f with (B.2.9).

If f is any complex-valued function on E, then

|f(y)| (B.2.10)

defines a nonnegative real-valued function on E. This is the same as the composition of f with z ↦ |z|, (B.2.11)

as a nonnegative real-valued function on C. It is well known and not too difficult to show that (B.2.11) is continuous on C, which corresponds to a remark about convergent sequences of complex numbers in Subsection A.8.4. If f is continous at x ∈ E, then it follows that (B.2.10) is continuous at x as well.

Similarly, it is well known and not difficult to show that

w ↦ |w| (B.2.12)

is a continuous real-valued function on R^{n}, which corresponds to a remark about convergent sequences in R^{n} in Subsection A.8.2.

B.2.3 Real and imaginary parts

It is easy to see that f is continuous on E if and only if

(B.2.13) Re f(y) and Im f(y)

are continuous real-valued functions on E. Of course, these two functions are the same as the compositions of f with

(B.2.14) z ↦ Re z and z ↦ Im z,

respectively, as real valued functions on C.

Similarly, f is continuous on E if and only if

\overline{f(y)} (B.2.15)

is continuous on E. This is the same as the composition of f with the complexvalued function defined on C by complex conjugation.

B.3 Pointwise and uniform convergence

Let E be a nonempty set, let {f_{j}}_{j=1}^{∞} be a sequence of complex-valued functions on E, and let f be a complex-valued function on E. We say that {f_{j}}_{j=1}^{∞} converges to f pointwise on E if for every y ∈ E,

(B.3.1) {f_{j}(y)}_{j=1}^{∞} converges to f(y),

as a sequence of complex numbers.

We say that {f_{j}}_{j=1}^{∞} converges to f uniformly on E if for every ϵ > 0 there is a positive integer L such that

|f_{j}(y) − f(y)| < ϵ (B.3.2)

for every y ∈ E and j ≥ L. This implies that {f_{j}}_{j=1}^{∞} converges to f pointwise on E. If E has only finitely many elements, and {f_{j}}_{j=1}^{∞} converges to f pointwise on E, then one can verify that {f_{j}}_{j=1}^{∞} converges to f uniformly on E.

B.3.1 An example with E = [0, 1]

Let us take E to be the closed unit interval [0, 1] in the real line, and put

f_{j}(y) = y^{j} (B.3.3)

for each j and 0 ≤ y ≤ 1. This sequence of functions converges pointwise on [0, 1], with

(B.3.4)  lim_{j→∞} f_{j}(y) = 0 when 0 ≤ y < 1
= 1 when y = 1.

If r is a positive real number with r < 1, the one can check that

(B.3.5) {f_{j}}_{j=1}^{∞} converges to 0 uniformly on [0, r].

However, one can also verify that {f_{j}}_{j=1}^{∞} does not converge uniformly on [0, 1]. Indeed, for each j, y^{j} is as close to 1 as we want when y is sufficiently close to 1.

B.3.2 Uniform convergence and continuity

Let n be a posiitve integer, and let E be a nonempty subset of R^{n}. Also let {f_{j}}_{j=1}^{∞} be a sequence of complex-valued functions on E that converges uniformly to a complex-valued function f on E. Suppose that x ∈ E, and that

(B.3.6) f_{j} is continuous at x

for each j. Under these conditions, it is well known and not too difficult to show that (B.3.7) f is continuous at x.

B.4. WEIERSTRASS’ CRITERION FOR UNIFORM CONVERGENCE 43

B.4 Weierstrass’ criterion for uniform convergence

Let E be a nonempty set, and let a_{1}(y), a_{2}(y), a_{3}(y), … be an infinite sequence of complex-valued functions on E. Also let A_{1}, A_{2}, A_{3}, … be an infinite sequence of nonnegative real numbers such that

|a_{j}(y)| ≤ A_{j} (B.4.1)

for every y ∈ E and j ≥ 1. Suppose that

∑_{j=1}^{∞} A_{j} (B.4.2)

converges. This implies that

∑_{j=1}^{∞} a_{j}(y) (B.4.3)

converges absolutely for each y ∈ E, by the comparison test.

In fact, the sequence of partial sums

∑_{j=1}^{l} a_{j}(y) (B.4.4)

converges to (B.4.3) uniformly on E under these conditions. This is a well-known criterion of Weierstrass for uniform convergence.

Indeed, if l is a positive integer, than

|∑_{j=1}^{∞} a_{j}(y) − ∑_{j=1}^{l} a_{j}(y)| = | ∑_{j=l+1}^{∞} a_{j}(y)| ≤ ∑_{j=l+1}^{∞} |a_{j}(y)| ≤ ∑_{j=l+1}^{∞} A_{j} (B.4.5)

for every y ∈ E. The right side tends to 0 as l → ∞, because of the convergence of (B.4.2). This implies that (B.4.4) converges to (B.4.3) uniformly on E, because the right side of (B.4.5) does not depened on y.

B.4.1 Continuity of the sum

Let n be a positive integer, and suppose that E is a subset of R^{n}. If a_{j}(y) is continuous on E for each j, then the partial sums (B.4.4) are continuous on E for each l as well.

If (B.4.2) converges, then the uniform convergence of the partial sums implies that (B.4.3) is continuous on E, as in Subsection B.3.2.

B.5 Continuity of functions defined by power series

Let ∑_{j=0}^{∞} a_{j} z^{j} be a power series with complex coefficients. Suppose for the moment that r is a positive real number such that

∑_{j=0}^{∞} |a_{j}| r^{j} (B.5.1)

converges. This implies that ∑_{j=0}^{∞} a_{j} z^{j} converges absolutely for each z ∈ C with |z| ≤ r, by the comparison test, as in Subsection A.12.1.

Thus

f(z) = ∑_{j=0}^{∞} a_{j} z^{j} (B.5.2)

defines a complex-valued function on the closed disk

{z ∈ C : |z| ≤ r}. (B.5.3)

We can use Weierstrass’ criterion to get that the sequence of partial sums

∑_{j=0}^{l} a_{j} z^{j} (B.5.4)

converges uniformly to f(z) on (B.5.3) as l → ∞. This uses the fact that

|a_{j} z^{j}| = |a_{j}| |z|^{j} ≤ |a_{j}| r^{j} (B.5.5)

on (B.5.3) for each j.

Note that (B.5.4) is continuous as a complex-valued function on C for each nonnegative integer l, as in Subsection B.2.1. In particular, the restriction of (B.5.4) to (B.5.3) is continuous for each l. It follows that f(z) is continuous on (B.5.3), as in Subsection B.4.1.

B.5.1 Continuity on open disks

Suppose now that ρ is a positive real number such that (B.5.1) converges when 0 < r < ρ. It is convenient to include the case where ρ = +∞ here too, so that (B.5.1) converges for all r > 0.

Under these conditions, ∑_{j=0}^{∞} a_{j} z^{j} converges absoutely for every complex number z with |z| < ρ, as before. This means that (B.5.2) defines a complexvalued function on {z ∈ C : |z| < ρ}. (B.5.6)

Of course, (B.5.6) is the same as the complex plane when ρ = +∞.

In fact, f(z) is continuous as a complex-valued function on (B.5.6). To see this, let z_{0} ∈ C with |z_{0}| < ρ be given. It suffices to check that f is continuous at z_{0}, as a function on (B.5.6).

Let r be a positive real number such that

|z_{0}| < r < ρ. (B.5.7)

We have already seen that f is continuous at z_{0} as a function defined on (B.5.3). One can use this to get that f is continuous at z_{0} as a function on (B.5.6), because |z_{0}| < r.

B.6 Bounded functions

Let E be a nonempty set, and let f be a complex-valued function on E. We say that f is bounded on E if there is a nonnegative real number C such that

|f(y)| ≤ C (B.6.1)

for every y ∈ E.

Suppose that f satisfies (B.6.1), and that g is another bounded complexvalued function on E, with |g(y)| ≤ C′ (B.6.2)

for some C′ ≥ 0 and all y ∈ E. In this case, f + g is bounded on E, with

|f(y) + g(y)| ≤ |f(y)| + |g(y)| ≤ C + C′ (B.6.3)

for every y ∈ E. Similarly, f g is bounded on E, with

|f(y) g(y)| = |f(y)| |g(y)| ≤ C C′ (B.6.4)

for every y ∈ E.

B.6.1 A version of the extreme value theorem

Let n be a positive integer, let w be an element of R^{n}, and let r be a positive real number. Remember that \overline{B}(w, r) is the closed ball in R^{n} centered at w with radius r, as in Subsection A.2.2. Let f be a continuous real-valued function on \overline{B}(w, r). Under these conditions, the extreme value theorem states that f attains its maximum and minimum on \overline{B}(w, r). This means that there are points u, v in \overline{B}(w, r) such that f(u) ≤ f(y) ≤ f(v) (B.6.5)

for every y ∈ E.

If f is a continuous complex-valued function on \overline{B}(w, r), then |f| is a continuous real-valued function on \overline{B}(w, r), as in Subsection B.2.2. This implies that |f| attains its maximum and minimum on \overline{B}(w, r), as in the preceding paragraph. In particular, this means that f is bounded on \overline{B}(w, r).

B.7 Uniformly bounded sequences of functions

Let E be a nonempty set again, and let {f_{j}}_{j=1}^{∞} be a sequence of complexvalued functions on E. We say that {f_{j}}_{j=1}^{∞} is uniformly bounded if there is a nonnegative real number C such that

|f_{j}(y)| ≤ C (B.7.1)

for every y ∈ E and positive integer j. Of course, this implies in particular that f_{j} is bounded nn E for each j.

Suppose that {f_{j}}_{j=1}^{∞} satisfies (B.7.1) on E. Suppose also for the moment that {f_{j}}_{j=1}^{∞} converges pointwise to a complex-valued function f on E. In this case, one can check that (B.6.1) holds for every y ∈ E. In particular, this means that f is bounded on E.

Let {g_{j}}_{j=1}^{∞} be another uniformly bounded sequence of complex-valued functions on E, with |g_{j}(y)| ≤ C′ (B.7.2)

for some C′ ≥ 0 and all y ∈ E and j ≥ 1. Observe that {f_{j} +g_{j}}_{j=1}^{∞} is uniformly bounded on E, with |f_{j}(y) + g_{j}(y)| ≤ C + C′ (B.7.3)

for every y ∈ E and j ≥ 1. Similarly, {f_{j} g_{j}}_{j=1}^{∞} is uniformly bounded on E, with |f_{j}(y) g_{j}(y)| ≤ C C′ (B.7.4)

for each y ∈ E and j ≥ 1.

B.7.1 More on uniform convergence

Suppose now that {f_{j}}_{j=1}^{∞} and {g_{j}}_{j=1}^{∞} are sequences of complex-valued functions on E that converge uniformly to complex-valued functions f and g on E, respectively. One can check that

(B.7.5) {f_{j} + g_{j}}_{j=1}^{∞} converges uniformly to f + g on E

under these conditions. If a is a complex number, then one can verify that

(B.7.6) {a f_{j}}_{j=1}^{∞} converges uniformly to a f on E.

Suppose in addition that {f_{j}}_{j=1}^{∞} and {g_{j}}_{j=1}^{∞} are uniformly bounded on E. This implies that f and g are bounded on E too, as before. In this case, it is well known and not too difficult to show that

(B.7.7) {f_{j} g_{j}}_{j=1}^{∞} converges uniformly to f g on E.

To see this, observe that

f_{j}(y) g_{j}(y) − f(y) g(y) = (f_{j}(y) − f(y)) g_{j}(y) + f(y) (g_{j}(y) − g(y)) (B.7.8)

B.8. UNIFORM CONVERGENCE AND INTEGRATION 47

for every y ∈ E and j ≥ 1. This implies that |f_{j}(y) g_{j}(y) − f(y) g(y)| ≤ |f_{j}(y) − f(y)| |g_{j}(y)| + |f(y)| |g_{j}(y) − g(y)| (B.7.9) for all y ∈ E and j ≥ 1. If C, C′ ≥ 0 are as in (B.6.1) and (B.7.2), respectively, then we get that |f_{j}(y) g_{j}(y) − f(y) g(y)| ≤ C′ |f_{j}(y) − f(y)| + C |g_{j}(y) − g(y)| (B.7.10) for each y ∈ E and j ≥ 1. One can use this to get (B.7.7).

B.7.2 Boundedness and uniform convergence

Let {f_{j}}_{j=1}^{∞} be a sequence of complex-valued functions on E that converges uniformly to a complex-valued function f on E again. If (B.7.11) f_{j} is bounded on E for each j, then one can check that (B.7.12) f is bounded on E.

Using this, one can also verify that (B.7.13) {f_{j}}_{j=1}^{∞} is uniformly bounded on E under these conditions.

B.8 Uniform convergence and integration

Let a and b be real numbers with a < b, and let {f_{j}}_{j=1}^{∞} be a sequence of continuous real-valued functions on [a, b] that converges uniformly to a realvalued function f on [a, b]. Remember that f is also continuous on [a, b], as in Subsection B.3.2. It is well known and not too difficult to show that

lim_{j→∞} ∫_{a}^{b} f_{j}(x) dx = ∫_{a}^{b} f(x) dx. (B.8.1)

More precisely,

(B.8.2)  |∫_{a}^{b} f_{j}(x) dx − ∫_{a}^{b} f(x) dx| = |∫_{a}^{b}(f_{j}(x) − f(x)) dx|
≤ ∫_{a}^{b} |f_{j}(x) − f(x)| dx

for each j. One can use uniform convergence to get that

lim_{j→∞} ∫_{a}^{b} |f_{j}(x) − f(x)| dx = 0. (B.8.3)

There are analogous statements for multiple integrals over suitable regions in R^{n}.

One may also consider the case where the functions f_{j} are Riemann integrable, instead of continuous. It is well known that uniform convergence implies that f is Riemann integrable as well.

One may consider integrals of complex-valued functions too. The integral of a complex-valued function may be defined under suitable conditions as the complex number whose real and imaginary parts are the corresponding integrals of the real and imaginary parts of the function.

B.8.1 The bounded convergence theorem

Let {f_{j}}_{j=1}^{∞} be a sequence of continuous real-valued functions on [a, b] again. Suppose now that {f_{j}}_{j=1}^{∞} converges to a continuous real-valued function f pointwise on [a, b], and that {f_{j}}_{j=1}^{∞} is uniformly bounded on [a, b]. Under these conditions, a classical theorem of Arzelà and Osgood impies that (B.8.1) holds, as in [83, 133, 235, 242, 348]. This also works when the f_{j}’s and f are Riemann integrable on [a, b], instead of continuous.

More results like these can be obtained using Lebesgue integrals.

B.9 Uniform convergence and differentiation

Let a and b be real numbers with a < b again, and let {g_{j}}_{j=1}^{∞} be a sequence of continuously-differentiable real-valued functions on [a, b]. This means that for each positive integer j, the derivative g′_{j}(x) of g_{j} exists at every x ∈ [a, b], using the appropriate one-sided derivative when x = a or b, and that g′_{j} is continuous on [a, b].

Suppose that {g_{j}(a)}_{j=1}^{∞} converges to a real number g(a), and that {g′_{j}}_{j=1}^{∞} converges uniformly to a real-valued function f on [a, b]. Note that f is continuous on [a, b], as in Subsection B.3.2.

Of course,

g_{j}(x) = g_{j}(a) + ∫_{a}^{x} g′_{j}(t) dt (B.9.1)

for each j ≥ 1 and x ∈ [a, b], by the fundamental theorem of calculus. Let g be the real-valued function defined on [a, b] by

g(x) = g(a) + ∫_{a}^{x} f(t) dt (B.9.2)

for each x ∈ [a, b].

One can check that

(B.9.3) {g_{j}}_{j=1}^{∞} converges uniformly to g on [a, b],

using the same type of arguments as mentioned at the beginning of the previous section. We also have that g is uniformly continuous on [a, b], with g′ = f.

There are analogous statements for complex-valued functions, as before. The derivative of a complex-valued function at a point may be defined as the complex number whose real and imaginary parts are the derivatives of the real and imaginary parts of the function at the same point, when they exist.

B.9.1 Differentiating power series

Let ∑_{j=0}^{∞} a_{j} x^{j} be a power series with real or complex coefficients, and suppose that

∑_{j=0}^{∞} j |a_{j}| r^{j} (B.9.4)

converges for some positive real number r. It is easy to see that this implies that

∑_{j=0}^{∞} |a_{j}| r^{j} (B.9.5)

converges.

Put

f(x) = ∑_{j=0}^{∞} a_{j} x^{j} (B.9.6)

and

ϕ(x) = ∑_{j=1}^{∞} j a_{j} x^{j−1} (B.9.7)

for each x ∈ R with |x| ≤ r. The partial sums of these series converge uniformly on [−r, r], as in Section B.5. One can use the remarks at the beginning of the section to get that f is differentiable on [−r, r], with

f′ = ϕ (B.9.8)

on [−r, r].

B.9.2 Differentiating exp(a t)

If a is a complex number, then

exp(a t) = ∑_{j=1}^{∞} \frac{a^{j} t^{j}}{j!} (B.9.9)

defines a complex-valued function of t on the real line, as in Section B.1. One can use the remarks in the previous subsection to get that this function is differentiable on R, with

\frac{d}{dt} exp(a t) = a exp(a t). (B.9.10)

Alternatively, one can check more directly that this holds at t = 0. In order to take the derivative at other points in R, one can use the fact that the exponential of a sum if equal to the product of the exponentials, as on p179 of [309], and p2 of [308].

B.9.3 More on differentiating power series

Suppose that 0 < ρ ≤ +∞ has the property that (B.9.5) converges for all positive real numbers r with r < ρ. In this case, it is well known and not difficult to show that this implies that (B.9.4) converges when r < ρ too. More precisely, if r < t < ρ, then one can use the convergence of

∑_{j=0}^{∞} |a_{j}| t^{j} (B.9.11)

to get that (B.9.4) converges.

It follows that the series on the right sides of (B.9.6) and (B.9.7) converge absolutely for every x ∈ R with |x| < ρ, so that f(x) and ϕ(x) may be defined on (−ρ, ρ) as before. We also get that f is differentiable on (−ρ, ρ), with derivative as in (B.9.8).

B.10 Open and closed sets

Let n be a positive integer, and let U be a subset of R^{n}. We say that U is an open set in R^{n} if for every x ∈ U there is a positive real number r such that

B(x, r) ⊆ U. (B.10.1)

Remember that B(x, r) is the open ball in R^{n} centered at x with radius r with respect to the standard Euclidean metric, as in Subsection A.2.2.

If w ∈ R^{n} and t is a positive real number, then it is well known and not difficult to show that (B.10.2) B(w, t) is an open set.

Indeed, if x ∈ B(w, t), then one can check that

B(x, r) ⊆ B(w, t) (B.10.3)

when r = t − d(w, x) > 0, using the triangle inequality.

If U_{1}, …, U_{l} are finitely many open sets in R^{n}, then one can verify that

(B.10.4) U_{1} ∩ ⋯ ∩ U_{n} is an open set.

It is easy to see that the union of any family of open sets in R^{n} is an open set too.

B.10. OPEN AND CLOSED SETS 51

B.10.1 Limit points and closed sets

Let E be a subset of R^{n}, and let x be an element of R^{n}. We say that x is a limit point of E if for every positive real number r there is a y ∈ E such that x ≠ y and |x − y| < r. (B.10.5) Note that x need not be an element of E, and that elements of E may not be limit points of E.

If x is a limit point of E and t is a positive real number, then it is well known and not difficult to show that (B.10.6) there are infinitely many elements of E in B(x, t). In particular, if E has only finitely many elements, then E has no limit points in R^{n}.

We say that E is a closed set in R^{n} if E contains all of its limit points. Equivalently, this means that if x ∈ R^{n} is a limit point of E, then x ∈ E. (B.10.7)

It is well known that (B.10.8) closed balls in R^{n} are closed sets. This can be verified directly from the definitions, or using another characterization of closed sets in the next subsection.

B.10.2 Complements of subsets of R^{n}

If A and B are sets, then put A \ B = {x ∈ A : x ∉ B}. (B.10.9) If B is a subset of A, then this may be called the complement of B in A. Observe that A \ (A \ B) = B (B.10.10) when B ⊆ A.

If E is any subset of R^{n}, then it is well known and not too difficult to show that E is a closed set if and only if (B.10.11) R^{n} \ E is an open set. Equivalently, this means that a subset U of R^{n} is an open set if and only if (B.10.12) R^{n} \ U is a closed set, because of (B.10.10).

If E_{1}, …, E_{l} are finitely many closed sets in R^{n}, then it is well known that

(B.10.13) E_{1} ∪ ⋯ ∪ E_{l} is a closed set in R^{n}.

Indeed, if E_{1}, …, E_{l} are any subsets of R^{n}, then one can check that

R^{n} \ ( ⋃_{j=1}^{l} E_{j}) = ⋂_{j=1}^{l}(R^{n} \ E_{j}). (B.10.14)

If E_{1}, …, E_{l} are closed sets, then the right side is an open set, because of (B.10.4).

It is also well known that the intersection of any family of closed sets in R^{n} is a closed set as well. This can be obtained from the definition of a closed set, or using the fact that the complement of an intersection is equal to the union of the complements of the sets.

B.10.3 Bounded sets and the extreme value theorem

A subset E of R^{n} is said to be bounded if there is a nonnegative real number C such that |x| ≤ C. (B.10.15)

One can check that open and closed balls in R^{n} are bounded sets, using the triangle inequality.

Suppose that E is a nonempty subset of R^{n} that is both closed and bounded. Also let f be a continuous real-valued continuous function on E. Another version of the extreme value theorem states that f attains its maximum and minimum on E. This was mentioned in Subsection B.6.1 when E is a cosed ball in R^{n}.

If f is a continuous complex-valued function on E, then one can apply the previous statement to |f|, to get that f is bounded on E, as before.

B.10.4 Closures and boundaries

If E is any subset of R^{n}, then the closure of E in R^{n} is the subset \overline{E} of R^{n} consisting of the elements of E and the limit points of E in R^{n}. In particular,

E ⊆ \overline{E}, (B.10.16)

by definition. One can check that E = \overline{E} if and only if E is a closed set, directly from the definitions.

If E is any subset of R^{n} again, then it is well known and not too difficult to show that (B.10.17) \overline{E} is a closed set.

This is the same as saying that

(B.10.18) R^{n} \ \overline{E} is an open set, as before.

The boundary of E in R^{n} is defined to be the set

∂E = \overline{E} ∩ \overline{(R^{n} \ E)}. (B.10.19)

This is a closed set in R^{n}, because it is the intersection of two closed sets.

If U is an open set in R^{n} then the boundary of U may be expressed more simply as ∂U = \overline{U} \ U. (B.10.20)

This is because the complement of U in R^{n} is a closed set, and thus equal to its own closure.

B.11 Partial derivatives

Let n be a positive integer, let U be an open subset of R^{n}, and let f be a real-valued function on U. If x is an element of U and l is a positive integer less than or equal to n, then the partial derivative of f at x in the lth variable may be denoted

∂_{l}f(x) = D_{l}f(x) = \frac{∂f}{∂x_{l}} (x), (B.11.1)

when it exists. Sometimes subscripts are used to indicate partial derivatives, so that (B.11.1) may also be denoted f_{x_{l}}(x).

If n is not too large, then we may use different letters for the coordinates of a point in R^{n}, and similar notation for partial derivatives. A point in R^{2} may be expressed as (x, y), for instance, and the partial derivatives of f with respect to x and y may be expressed as

f_{x} = \frac{∂f}{∂x} and f_{y} = \frac{∂f}{∂y} , (B.11.2)

respectively.

Sometimes one of the variables may be denoted t, and the other variables may be expressed as in either of the previous two paragraphs. The partial derivative of f with respect to t may be expressed as

f_{t} = \frac{∂f}{∂t} , (B.11.3)

and the other partial derivatives may be expressed as in (B.11.1) or (B.11.2), as appropriate.

One may also consider complex-valued functions f on U. In this case, the real and imaginary parts of a partial derivative of f are the same as the corresponding partial derivative of the real and imaginary parts of f, respectively, when they exist.

B.11.1 Continuous differentiability

Suppose that f is a real or complex-valued function on U such that the partial derivatives of f in each variable exist at every point in U. If each of the partial derivatives of f is continuous on U, then f is said to be continuously differentiable on U.

If f is continuously differentiable on U, then it is well known that

(B.11.4) f is continuous on U.

More precisely, this holds when the partial derivatives of f are locally bounded on U, in the sense that for each x ∈ U there is a positive real number r such that B(x, r) ⊆ U and the partial derivatives of f are bounded on B(x, r). If n = 1, then a function is continuous at any point at which the derivative exists.

Let g be another real or complex-valued function on U, and suppose that f and g are both continuously differentiable on U. It is easy to see that f + g and f g are continuously differentiable on U as well.

B.11.2 Directional derivatives

Let f be a real or complex-valued function on U again, and let x ∈ U and w ∈ R^{n} be given. Also let U(x, w) be the of real numbers t such that

x + t w ∈ U. (B.11.5)

One can check that this is an open set in the real line that contains 0.

The directional derivative of f at x in the direction of w is defined to be the derivative of f(x + t w) (B.11.6)

as a function of t ∈ U(x, w) at 0, when it exists. This may be denoted

D_{w}f(x). (B.11.7)

If f is continuously differentiable on U, then it is well known that the directional derivative exists, with

D_{w}f(x) = ∑_{j=1}^{n} w_{j} \frac{∂f}{∂x_{j}} (x). (B.11.8)

Note that the right side is linear in w.

B.11.3 Twice continuous differentiability

Let f be a real or complex-valued function on U, and suppose for the moment that the partial derivative of f in the lth variable exists at every point in U for some l ≤ n. Consider the partial derivative

\frac{∂}{∂x_{j}} (\frac{∂f}{∂x_{l}} ) (B.11.9)

B.12. CONNECTED SETS IN R^{N} 55

of ∂f/∂x_{l} in the jth variable, 1 ≤ j ≤ n, when it exists.

Suppose now that ∂f/∂x_{l} exists at every point in U for each l = 1, …, n, and that its partial derivatives (B.11.9) exist at every point in U for each j = 1, …, n. If each of these second derivatives of f is continuous on U, then f is said to be twice continuously differentiable on U.

Equivalently, this means that

(B.11.10) \frac{∂f}{∂x_{l}} is continuously differentiable on U

for each l. This implies that

(B.11.11) \frac{∂f}{∂x_{l}} is continuous on U

for each l, as before, so that

(B.11.12) f is continuously differentiable on U.

It follows that f is continuous on U, as before.

If f is twice continuously differentiable on U, then it is well known that

\frac{∂}{∂x_{j}} (\frac{∂f}{∂x_{l}} ) = \frac{∂}{∂x_{l}} (\frac{∂f}{∂x_{j}} ) (B.11.13)

on U. These second derivatives may be denoted

\frac{∂^{2}f}{∂x_{j} ∂x_{l}} = \frac{∂^{2}f}{∂x_{l} ∂x_{j}} (B.11.14)

or f_{x_{j} x_{l}} = f_{x_{l} x_{j}}. (B.11.15)

Let g be another real or complex-valued function on U, and suppose that f and g are both twice continuously differentiable on U. One can check that f +g and f g are also twice continuously differentiable on U.

B.12 Connected sets in R^{n}

Let n be a positive integer, and let A, B be subsets of R^{n}. We say that A and B are separated if \overline{A} ∩ B = A ∩ \overline{B} = ∅, (B.12.1)

where \overline{A}, \overline{B} are the closures of A, B in R^{n}, respectively, as in Subsection B.10.4.

Note that disjoint closed subsets of R^{n} are separated. It is well known and not difficult to show that disjoint open subsets of R^{n} are separated too.

If a subset E of R^{n} can be expressed as

E = A ∪ B, (B.12.2) where A, B are separated and A, B ≠ ∅, then E is said to be not connected in R^{n}. Otherwise, E is said to be connected.

If E is a closed set in R^{n}, and E is expressed as in (B.12.2), where A, B are separated sets in R^{n}, then one can check that A and B are closed sets in R^{n}. In this case, we get that E is connected if and only if E cannot be expressed as the union of two nonempty disjoint closed subsets of R^{n}.

Similarly, if E is an open set in R^{n}, and E is expressed as in (B.12.2), where A, B are separated sets in R^{n}, then one can verify that A and B are open sets in R^{n}. This means that E is connected if and only if E cannot be expressed as the union of two nonempty disjoint open subsets of R^{n}.

B.12.1 Connectedness in the real line

It is well known that a subset E of the real line is connected if and only if for every x, y ∈ E with x < y and every w ∈ R with

x < w < y, (B.12.3)

we have that w ∈ E. (B.12.4)

The necessity of this condition can be obtained directly from the definitions. More precisely, if w ∉ E, then one can check that

A = {t ∈ E : t < w} (B.12.5)

and B = {t ∈ E : t > w} (B.12.6)

are nonempty separated subsets of R whose union is equal to E.

One can use this to show that

(B.12.7) convex subsets of R^{n} are connected.

The converse holds when n = 1, as in the preceding paragraph.

B.12.2 Locally constant functions on subsets of R^{n}

Let E be a nonempty subset of R^{n}, and let f be a function on E with values in any set. We say that

(B.12.8) f is locally constant at a point x ∈ E

if there is a positive real number r such that

f(x) = f(y) (B.12.9)

for every y ∈ E with |x − y| < r. (B.12.10)

Supose for the moment that f is locally constant on E, in the sense that f is locally constant at every point in E. Let w be an element of E, and put

A = {x ∈ E : f(x) = f(w)} (B.12.11)

and B = {x ∈ E : f(x) ≠ w}. (B.12.12)

Clearly w ∈ A, and E is equal to the union of A and B.

One can check that A and B are separated in R^{n}, because f is locally constant on E. If E is connected, then it follows that B = ∅, so that f is constant on E.

Conversely, if E is not connected, then E may be expressed as the union of two nonempty separated sets A and B in R^{n}. In this case, one can get a locally constant function f on E that is not constant by taking f to be constant on each of A and b, with different constant values on these sets.

B.12.3 Locally constant functions on open sets

Let U be a nonempty open set in R^{n}, and let f be a real-valued function on U. Suppose that the first partial derivatives of f exist at every point in U, with

\frac{∂f}{∂x_{l}} = 0 (B.12.13)

on U for each l = 1, …, n. Under these cnditions, it is well known and not difficult to show that f is locally constant on U. If U is connected, then it follows that f is constant on U. This also works for complex-valued functions on U.

B.13 Vector-valued functions

Let m and n be positive integers, let E be a nonempty subset of R^{n}, and let f be a function on E with values in R^{m}. Thus, if x ∈ E, then

f(x) = (f_{1}(x), …, f_{m}(x)) (B.13.1)

is an element of R^{m}. This means that f_{1}, …, f_{m} are real-valued functions on E.

One can define what it means for f to be continuous at a point x ∈ E in essentially the same way as in Section B.2, using the standard Euclidean metrics on R^{n} and R^{m}. One can check that f is continuous at x if and only if f_{1}, …, f_{m} are all continuous at x, as real-valued functions on E.

Continuity of f at x can be characterized in terms of convergence of sequences as before. This can be shown using the same type of arguments as for real or complex-valued functions, or by reducing to the case of real-valued functions.

If f is continuous at every point in E, then we may simply sat that f is continuous on E, as before.

B.13.1 More on continuity of compositions

Let A be a nonempty subset of R^{m}, let p be another positive integer, and let ϕ be a function on A with values in R^{p}. If

f(y) ∈ A (B.13.2)

for every y ∈ E, then the composition ϕ ∘ f of f with ϕ may be defined as a function on E with values in R^{p} by

(ϕ ∘ f)(y) = ϕ(f(y)), (B.13.3)

as usual.

If f is continuous at x ∈ E, and ϕ is continuous at f(x), then ϕ ∘ f is continuous at x, as in Subsection B.2.2.

Remember that v ↦ |v| is continuous as a real-valued function on R^{m}, as in Subsection B.2.2. If f is continuous at x ∈ E, then |f| is continuous at x as a real-valued function on E, as before.

B.13.2 More on partial derivatives

Suppose now that E is an open set in R^{n}. If x ∈ E and l is a positive integer less than or equal to n, then the partial derivative of f at x in the lth variable may be denoted as in (B.11.1), when it exists. Of course, the partial derivative is an element of R^{m}, when it exists.

This partial derivative exists exactly when the partial derivatives of the components of f at x in the lth variable exist. In this case, the components of the partial derivative of f at x in the lth variable are equal to the corresponding partial derivatives of the components of f at x.

We say that f is continuously differentiable on E if the partial derivatives of f in each variable exist at every point in E, and are continuous as R^{m}- valued functions on E. This is the same as saying that the components of f are continuously differentiable as real-valued functions on E, as in Subsection B.11.1. This implies that f is continuous as an R^{m}-valued function on E, as before.

If x ∈ E and w ∈ R^{n}, then the directional derivative D_{w}f(x) of f at x in the direction of w may be defined as in Subsection B.11.2, when it exists. Note that D_{w}f(x) is an element of R^{m}, when it exists. This happens exactly when the directional derivatives of the components of f at x in the direction of w exist, in which case the componentis of D_{w}f(x) are the same as the corresponding directional derivatives of the components of f.

If f is continuously differentiable on E, then D_{w}f(x) exists and may be expressed as in (B.11.8), as before.

B.13.3 Compositions and continuous differentiability

Suppose that A is an open subset of R^{m} too. If f is continuously differentiable on E, and ϕ is continuously differentiable on A, then it is well known that

(B.13.4) ϕ ∘ f is continuously differentiable on E.

In fact, if x ∈ U and w ∈ R^{n}, then the directional derivative of ϕ ∘ f at x in the direction of w may be expressed as

D_{w}(ϕ ∘ f)(x) = (D_{D_{w}f(x)}ϕ)(f(x)), (B.13.5)

which is to say that it is equal to the directional derivative of ϕ at f(x) in the direction of D_{w}f(x). This is the version of the chain rule for vector-valued functions of several variables.

Let e_{1}, …, e_{n} be the standard basis vectors in R^{n}. This means that the jth component of e_{l} is equal to 1 when j = l and to 0 otherwise, 1 ≤ j, l ≤ n. The partial derivative at x of a function on E in the lth variable is the same as the directional derivative at x of the function in the direction e_{l}.

Thus (B.13.5) may be used to express the partial derivatives of ϕ ∘ f at x in terms of the partial derivatives of ϕ at f(x) and the partial derivatives of f at x. More precisely,

∂_{l}(ϕ ∘ f)(x) = (D_{∂_{l}f(x)}ϕ)(f(x)) = ∑_{k=1}^{m}(∂_{l}f_{k}(x)) (∂_{k}ϕ(f(x))) (B.13.6)

for each l = 1, …, n, where f_{k} is the kth component of f for each k = 1, …, m.

It is easy to reduce to the case where p = 1, by considering the components of ϕ separately, as real-valued functions on A. If we also have that m = 1, then this version of the chain rule reduces to the usual formulation for real-valued functions of one variable. Indeed, (B.13.5) is the same as saying that

D_{w}(ϕ ∘ f)(x) = ϕ′(f(x)) (D_{w}f(x)) (B.13.7)

in this case. In particular, (B.13.6) simplifies to

∂_{l}(ϕ ∘ f)(x) = ϕ′(f(x)) (∂_{l}f(x)). (B.13.8)

The continuity of the partial derivatives of ϕ ∘ f may be obtained more directly from the continuity of f, its first partial derivatives, and ϕ′ under these conditions.

B.13.4 More on twice continuous differentiability

If ∂f/∂x_{l} exists at every point in E, then we may consider its partial derivatives, as in (B.11.9), when they exist. If ∂f/∂x_{l} is continuously differentiable on E for each l = 1, …, n, then f is said to be twice continuously differentiable on E, as in Subsection B.11.3. This happens exactly when the components of f are continuously differentiable as real-valued functions on E, as usual.

If f is twice continuously differentiable on E, then f is continuously differentiable on E, and thus continuous on E, as before. We also have that (B.11.13) holds in this case, as before.

If f is twice continuously differentiable on E, and ϕ is twice continuously differentiable on A, then

(B.13.9) ϕ ∘ f is twice continuously differentiable on E.

Equivalently, one can check that the partial derivatives of ϕ∘f are continuously differentiable on E, using (B.13.6). This also uses the fact that

(B.13.10) (∂_{k}ϕ) ∘ f is continuously differentiable on E,

because ∂_{k}ϕ is continuously differentiable on A and f is continuously differentiable on E.

B.14 More on connectedness

Let m and n be positive integers, let E be a nonempty subset of R^{n}, and let f be a continuous mapping from E into R^{m}. Put

f(E) = {f(y) : y ∈ E}, (B.14.1)

so that f(E) is a subset of R^{m}. If E is connected as a subset of R^{n}, then it is well known that (B.14.2) f(E) is connected as a subset of R^{m}.

More precisely, in order to obtain this from standard results about continuity and connectedness, one also uses the fact that E is connected when it is considered as a subset of itself, with respect to the restriction of the standard Euclidean metric on R^{n} to E. This is not needed when f is the restriction to E of a continuous mapping from R^{n} into R^{m}.

The intermediate value theorem basically corresponds to the case where m = n = 1.

Suppose for the moment that n = 1, and that E = [a, b] for some a, b ∈ R with a ≤ b. If f is a continuous mapping from [a, b] into R^{m}, then f can be extended to a continuous mapping fom R into R^{m}, by putting

f(t) = f(a) (B.14.3)

when t < a, and f(t) = f(b) (B.14.4)

when t > b.

B.14.1 Path connectedness

A subset E of R^{n} is said to be path connected if for every x, y ∈ E there is a continuous path in E from x to y. This means that there is a continuous mapping p from the closed unit interval [0, 1] in the real line into R^{m} such that

p(0) = x, p(1) = y, and p([0, 1]) ⊆ E. (B.14.5)

It is easy to see that

(B.14.6) convex sets in R^{n} are path connected.

It is well known and not too difficult to show that

(B.14.7) path-connected sets in R^{n} are connected.

This uses the fact that if p is a continuous mapping from [0, 1] into R^{n}, then p([0, 1]) is a connected set in R^{n}.

It is also well known that

(B.14.8) connected open sets in R^{n} are path connected.

Suppose that E is a path-connected subset of R^{n}, and that f is a continuous mapping from E into R^{m}. Under these conditions, one can check that

(B.14.9) f(E) is path connected in R^{m}.

B.14.2 Nonemptiness of the boundary of an open set

Let U be a nonempty open subset of R^{n} that is a proper subset of R^{n}, so that U ≠ R^{n}. One can verify that ∂U ≠ ∅. (B.14.10)

This is the same as saying that U is not a closed set in R^{n}, because of the description of the boundary of an open set in Subsection B.10.4. Otherwise, one could check that R^{n} would not be connected.

Alternatively, let x be an element of U, and let z be an element of R^{n} \ U. One can show that there is a real number t_{0} such that 0 < t_{0} ≤ 1 and

(1 − t_{0}) x + t_{0} z ∈ ∂U. (B.14.11)

In fact, one can take t_{0} to be the infimum or greatest lower bound of the set of positive real numbers t such that

(1 − t) x + t z ∈ R^{n} \ U. (B.14.12)

Appendix C

A bit of analysis, 3 Bibliography

[1] S. Abian and A. Brown, On the solution of the differential equation f(x, y, y′) = 0, American Mathematical Monthly 66 (1959), 192–199.

[2] W. Adkins and M. Davison, Putzer’s algorithm for e^{A t} via the Laplace transform, Mathematics Magazine 83 (2010), 267–275.

[3] F. Adler, E. Allman, and L. de Pillis, A letter from the guest editors, American Mathematical Monthly 121 (2014), 751–753.

[4] R. Agnew, Differential Equations, 2nd edition, McGraw-Hill, 1960.

[5] R. Agnew, Views and approximations of differential equations, American Mathematical Monthly 60 (1953), 1–6.

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Index

absolute value of a complex number, 23 of a real number, 13

alternating series test, 33

binomial coefficients, 37 binomial theorem, 37 boundary of a set, 53 bounded functions, 45

uniformly bounded sequences, 46 bounded sequences, 25 bounded sets, 52

C, 22 Cauchy products, 36 Cauchy–Schwarz inequality, 17, 24 chain rule, 59 closed balls, 16, 21 closed half-lines, 14 closed intervals, 14 closed sets, 51 closure of a set, 52 comparison test, 32 complement of a set, 51 complex conjugates, 22 complex exponential function, 39 complex numbers, 22 compositions of functions, 40, 58 connected sets, 56 constant coefficients, 8, 11 continuous differentiability, 8, 54, 58

twice, 11, 55, 59 continuous functions, 39, 40, 57 convergent sequences, 24, 26 convex sets, 22

differential operators, 5, 9 directional derivatives, 54, 58 dot product, 16

Euclidean metric on R, 13 on R^{n}, 15

Euclidean norm on R^{n}, 15, 19 Euler’s identity, 39 extended real numbers, 14 extreme value theorem, 45, 52

geometric series, 31 greatest lower bound, 26

homogeneous equations, 5, 9

imaginary parts, 22

implicit theorem theorem, 3

infimum, 26 infinite series

absolute convergence, 32 convergent, 30 initial value problems, 2 inner products, 16 intervals, 1 invariance under translations

differential equations, 6, 10 metrics, 16, 20

least upper bound, 25 limit of a sequence, 24 limit points of sets, 51 linear differential equations, 4, 9 linear mappings, 5 locally constant functions, 56, 57 lower bounds, 25

metrics associated to norms, 20 modulus of a complex number, 23 monotone sequences, 25

norms, 19

open balls, 15, 21 open half-lines, 14 open intervals, 13

unbounded, 14 open sets, 8, 50

partial derivatives, 8, 53, 58 partial sums, 30 path-connected sets, 61 pointwise convergence, 42 power series, 33

R, 1, 13 R^{2}, 8 R^{n}, 14 radius of convergence, 34 real parts, 22 rearrangements, 34

separated sets, 55 standard basis vectors, 59 supremum, 25

triangle inequality absolute values, 13, 23 arbitrary norms, 19 Euclidean metrics, 13, 15 Euclidean norm, 15

uniform convergence, 42 upper bounds, 25

vector spaces, 6, 15

Weierstrass’ criterion, 43

Z_{+}, 34