# §1.9 Calculus of a Complex Variable

## §1.9(i) Complex Numbers

1.9.1.

1.9.2

### ¶ Polar Representation

1.9.3

where

1.9.4

and when ,

1.9.5

according as lies in the 1st, 2nd, 3rd, or 4th quadrants. Here

1.9.6

### ¶ Modulus and Phase

The principal value of corresponds to , that is, . It is single-valued on , except on the interval where it is discontinuous and two-valued. Unless indicated otherwise, these principal values are assumed throughout the DLMF. (However, if we require a principal value to be single-valued, then we can restrict .)

1.9.8

where

see §4.14.

1.9.11
1.9.12
1.9.13

### ¶ Arithmetic Operations

If , , then

1.9.14
1.9.15
1.9.16

provided that . Also,

1.9.17
1.9.18
1.9.19
1.9.20

Equations (1.9.18) and (1.9.20) hold for general values of the phases, but not necessarily for the principal values.

1.9.21.

1.9.23

## §1.9(ii) Continuity, Point Sets, and Differentiation

### ¶ Continuity

A function is continuous at a point if . That is, given any positive number , however small, we can find a positive number such that for all in the open disk .

A function of two complex variables is continuous at if ; compare (1.5.1) and (1.5.2).

### ¶ Point Sets in

A neighborhood of a point is a disk . An open set in is one in which each point has a neighborhood that is contained in the set.

A point is a limit point (limiting point or accumulation point) of a set of points in (or ) if every neighborhood of contains a point of distinct from . ( may or may not belong to .) As a consequence, every neighborhood of a limit point of contains an infinite number of points of . Also, the union of and its limit points is the closure of .

A domain , say, is an open set in that is connected, that is, any two points can be joined by a polygonal arc (a finite chain of straight-line segments) lying in the set. Any point whose neighborhoods always contain members and nonmembers of is a boundary point of . When its boundary points are added the domain is said to be closed, but unless specified otherwise a domain is assumed to be open.

A region is an open domain together with none, some, or all of its boundary points. Points of a region that are not boundary points are called interior points.

A function is continuous on a region if for each point in and any given number () we can find a neighborhood of such that for all points in the intersection of the neighborhood with .

### ¶ Differentiation

A function is differentiable at a point if the following limit exists:

Differentiability automatically implies continuity.

### ¶ Cauchy–Riemann Equations

If exists at and , then

1.9.25

at .

Conversely, if at a given point the partial derivatives , , , and exist, are continuous, and satisfy (1.9.25), then is differentiable at .

### ¶ Analyticity

A function is said to be analytic (holomorphic) at if it is differentiable in a neighborhood of .

A function is analytic in a domain if it is analytic at each point of . A function analytic at every point of is said to be entire.

If is analytic in an open domain , then each of its derivatives , , exists and is analytic in .

### ¶ Harmonic Functions

If is analytic in an open domain , then and are harmonic in , that is,

1.9.26

or in polar form ((1.9.3)) and satisfy

1.9.27

at all points of .

## §1.9(iii) Integration

An arc is given by , , where and are continuously differentiable. If and are continuous and and are piecewise continuous, then defines a contour.

A contour is simple if it contains no multiple points, that is, for every pair of distinct values of , . A simple closed contour is a simple contour, except that .

Next,

1.9.28

for a contour and continuous, . If , , then the integral is defined analogously to the infinite integrals in §1.4(v). Similarly when or .

### ¶ Jordan Curve Theorem

Any simple closed contour divides into two open domains that have as common boundary. One of these domains is bounded and is called the interior domain of ; the other is unbounded and is called the exterior domain of .

### ¶ Cauchy’s Theorem

If is continuous within and on a simple closed contour and analytic within , then

### ¶ Cauchy’s Integral Formula

If is continuous within and on a simple closed contour and analytic within , and if is a point within , then

1.9.30

and

provided that in both cases is described in the positive rotational (anticlockwise) sense.

### ¶ Liouville’s Theorem

Any bounded entire function is a constant.

### ¶ Winding Number

If is a closed contour, and , then

1.9.32

where is an integer called the winding number of with respect to . If is simple and oriented in the positive rotational sense, then is 1 or 0 depending whether is inside or outside .

### ¶ Poisson Integral

If is continuous on , then with

is harmonic in . Also with , as within .

## §1.9(iv) Conformal Mapping

The extended complex plane, , consists of the points of the complex plane together with an ideal point called the point at infinity. A system of open disks around infinity is given by

Each is a neighborhood of . Also,

1.9.36
1.9.37,
1.9.38
1.9.39.

A function is analytic at if is analytic at , and we set .

### ¶ Conformal Transformation

Suppose is analytic in a domain and are two arcs in passing through . Let be the images of and under the mapping . The angle between and at is the angle between the tangents to the two arcs at , that is, the difference of the signed angles that the tangents make with the positive direction of the real axis. If , then the angle between and equals the angle between and both in magnitude and sense. We then say that the mapping is conformal (angle-preserving) at .

The linear transformation , , has and maps conformally onto .

### ¶ Bilinear Transformation

1.9.40, .
1.9.41
1.9.42.
1.9.43
1.9.44

The transformation (1.9.40) is a one-to-one conformal mapping of onto itself.

The cross ratio of is defined by

1.9.45

or its limiting form, and is invariant under bilinear transformations.

Other names for the bilinear transformation are fractional linear transformation, homographic transformation, and Möbius transformation.

## §1.9(v) Infinite Sequences and Series

A sequence converges to if . For , the sequence converges iff the sequences and separately converge. A series converges if the sequence converges. The series is divergent if does not converge. The series converges absolutely if converges. A series converges (diverges) absolutely when (), or when (). Absolutely convergent series are also convergent.

Let be a sequence of functions defined on a set . This sequence converges pointwise to a function if

1.9.46

for each . The sequence converges uniformly on , if for every there exists an integer , independent of , such that

for all and .

A series converges uniformly on , if the sequence converges uniformly on .

### ¶ Weierstrass -test

Suppose is a sequence of real numbers such that converges and for all and all . Then the series converges uniformly on .

A doubly-infinite series converges (uniformly) on iff each of the series and converges (uniformly) on .

## §1.9(vi) Power Series

For a series there is a number , , such that the series converges for all in and diverges for in . The circle is called the circle of convergence of the series, and is the radius of convergence. Inside the circle the sum of the series is an analytic function . For in (), the convergence is absolute and uniform. Moreover,

and

1.9.49

For the converse of this result see §1.10(i).

### ¶ Operations

When and both converge

1.9.50

and

1.9.51

where

1.9.52

Next, let

1.9.53.

Then the expansions (1.9.54), (1.9.57), and (1.9.60) hold for all sufficiently small .

1.9.54

where

1.9.55
1.9.56.

With ,

(principal value), where

1.9.58

and

1.9.59.

Also,

1.9.60

(principal value), where ,

1.9.61

and

For the definitions of the principal values of and see §§4.2(i) and 4.2(iv).

Lastly, a power series can be differentiated any number of times within its circle of convergence:

## §1.9(vii) Inversion of Limits

### ¶ Double Sequences and Series

A set of complex numbers where and take all positive integer values is called a double sequence. It converges to if for every , there is an integer such that

for all . Suppose converges to and the repeated limits

exist. Then both repeated limits equal .

A double series is the limit of the double sequence

If the limit exists, then the double series is convergent; otherwise it is divergent. The double series is absolutely convergent if it is convergent when is replaced by .

If a double series is absolutely convergent, then it is also convergent and its sum is given by either of the repeated sums

### ¶ Term-by-Term Integration

Suppose the series , where is continuous, converges uniformly on every compact set of a domain , that is, every closed and bounded set in . Then

for any finite contour in .

### ¶ Dominated Convergence Theorem

Let be a finite or infinite interval, and be real or complex continuous functions, . Suppose converges uniformly in any compact interval in , and at least one of the following two conditions is satisfied:

1.9.69

Then

1.9.71