§1.9 Calculus of a Complex Variable ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods (original) (raw)

Contents

1. §1.9(i) Complex Numbers
2. §1.9(ii) Continuity, Point Sets, and Differentiation
3. §1.9(iii) Integration
4. §1.9(iv) Conformal Mapping
5. §1.9(v) Infinite Sequences and Series
6. §1.9(vi) Power Series
7. §1.9(vii) Inversion of Limits

§1.9(i) Complex Numbers

Real and Imaginary Parts

Polar Representation

where

and when z≠0,

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

Modulus and Phase

| 1.9.7 | |z| | =r, | | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ | ---------- | ---- | | ph⁡z | =θ+2⁢n⁢π, | | | n∈ℤ. | | | | ⓘ Defines: ph: phase and|x|: absolute value of x (locally) Symbols: π: the ratio of the circumference of a circle to its diameter,∈: element of,ℤ: set of all integers,z: variable,n: nonnegative integer,r: radius andθ: angle Permalink: http://dlmf.nist.gov/1.9.E7 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §1.9(i),§1.9(i),§1.9 andCh.1 | | |

The _principal value_of ph⁡z corresponds to n=0, that is, −π≤ph⁡z≤π. It is single-valued on ℂ∖{0}, except on the interval(−∞,0) 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−π<ph⁡z≤π.)

| 1.9.8 | |ℜ⁡z| | ≤|z|, | | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ | ------ | ------ | | |ℑ⁡z| | ≤|z|, | | | ⓘ Symbols: ℑ⁡: imaginary part,ℜ⁡: real part,z: variable and|x|: absolute value of x Permalink: http://dlmf.nist.gov/1.9.E8 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §1.9(i),§1.9(i),§1.9 andCh.1 | | |

where

see §4.14.

Complex Conjugate

Arithmetic Operations

If z1=x1+i⁢y1, z2=x2+i⁢y2, then

provided that z2≠0. Also,

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

Powers

DeMoivre’s Theorem

Triangle Inequality

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

Continuity

A function f⁡(z) is continuous at a point z0 iflimz→z0f⁡(z)=f⁡(z0). That is, given any positive numberϵ, however small, we can find a positive number δ such that|f⁡(z)−f⁡(z0)|<ϵ for all z in the open disk |z−z0|<δ.

A function of two complex variables f⁡(z,w) is continuous at(z0,w0) if lim(z,w)→(z0,w0)f⁡(z,w)=f⁡(z0,w0); compare (1.5.1) and (1.5.2).

Point Sets in ℂ

A neighborhood of a point z0 is a disk |z−z0|<δ. An_open set_ in ℂ is one in which each point has a neighborhood that is contained in the set.

A point z0 is a limit point (limiting point or_accumulation point_) of a set of points S in ℂ (orℂ∪∞) if every neighborhood of z0 contains a point of Sdistinct from z0. (z0 may or may not belong to S.) As a consequence, every neighborhood of a limit point of S contains an infinite number of points of S. Also, the union of S and its limit points is the closure of S.

A domain D, say, is an open set in ℂ 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 D is a boundary point_of D. 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 f⁡(z) is continuous on a region R if for each point z0in R and any given number ϵ (>0) we can find a neighborhood ofz0 such that |f⁡(z)−f⁡(z0)|<ϵ for all points z in the intersection of the neighborhood with R.

Differentiation

A function f⁡(z) is complex differentiable at a point z if the following limit exists:

The limit is taken for h→0 in ℂ.

Differentiability automatically implies continuity.

Cauchy–Riemann Equations

If f′⁡(z) exists at z=x+i⁢y and f⁡(z)=u⁡(x,y)+i⁢v⁡(x,y), then

at (x,y).

Conversely, if at a given point (x,y) the partial derivatives∂u/∂x, ∂u/∂y, ∂v/∂x, and ∂v/∂yexist, are continuous, and satisfy (1.9.25), then f⁡(z) is differentiable at z=x+i⁢y.

Analyticity

A function f⁡(z) is said to be analytic (holomorphic) atz=z0 if it is complex differentiable in a neighborhood of z0.

A function f⁡(z) is analytic in a domain D if it is analytic at each point of D. A function analytic at every point of ℂ is said to be entire.

If f⁡(z) is analytic in an open domain D, then each of its derivativesf′⁡(z), f′′⁡(z), … exists and is analytic in D.

Harmonic Functions

If f⁡(z)=u⁡(x,y)+i⁢v⁡(x,y) is analytic in an open domain D, then u andv are_harmonic_ in D, that is,

or in polar form (1.9.3) u and v satisfy

at all points of D.

§1.9(iii) Integration

An arc C is given by z⁡(t)=x⁡(t)+i⁢y⁡(t), a≤t≤b, wherex and y are continuously differentiable. If x⁡(t) and y⁡(t) are continuous and x′⁡(t) and y′⁡(t) are piecewise continuous, then z⁡(t)defines a contour.

A contour is simple if it contains no multiple points, that is, for every pair of distinct values t1,t2 of t, z⁡(t1)≠z⁡(t2). A_simple closed contour_ is a simple contour, except that z⁡(a)=z⁡(b).

Next,

for a contour C and f⁡(z⁡(t)) continuous, a≤t≤b. If f⁡(z⁡(t0))=∞, a≤t0≤b, then the integral is defined analogously to the infinite integrals in §1.4(v). Similarly when a=−∞ orb=+∞.

Jordan Curve Theorem

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

Cauchy’s Theorem

If f⁡(z) is continuous within and on a simple closed contour C and analytic within C, then

Cauchy’s Integral Formula

If f⁡(z) is continuous within and on a simple closed contour C and analytic within C, and if z0 is a point within C, then

and

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

Liouville’s Theorem

Any bounded entire function is a constant.

Winding Number

If C is a closed contour, and z0∉C, then

where 𝒩⁡(C,z0) is an integer called the winding number of Cwith respect to z0. If C is simple and oriented in the positive rotational sense, then 𝒩⁡(C,z0) is 1 or 0 depending whetherz0 is inside or outside C.

Mean Value Property

Poisson Integral

If h⁡(w) is continuous on |w|=R, then with z=r⁢ei⁢θ

is harmonic in |z|<R. Also with |w|=R,limz→wu⁡(z)=h⁡(w) as z→w within |z|<R.

§1.9(iv) Conformal Mapping

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

| 1.9.35 | Sr={z∣|z|>1/r}∪{∞}, | | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ | ---------------------- | | 0<r<∞. | | | ⓘ Symbols: ∪: union,z: variable,r: radius,Sr: neighborhood and|x|: absolute value of x Permalink: http://dlmf.nist.gov/1.9.E35 Encodings: TeX, pMML, png See also: Annotations for §1.9(iv),§1.9 andCh.1 | |

Each Sr is a _neighborhood_of ∞. Also,

A function f⁡(z) is analytic at ∞ if g⁡(z)=f⁡(1/z) is analytic at z=0, and we set f′⁡(∞)=g′⁡(0).

Conformal Transformation

Suppose f⁡(z) is analytic in a domain D and C1,C2 are two arcs in Dpassing through z0. Let C1′,C2′ be the images of C1 and C2 under the mapping w=f⁡(z). The angle between C1 and C2 atz0 is the angle between the tangents to the two arcs at z0, that is, the difference of the signed angles that the tangents make with the positive direction of the real axis. If f′⁡(z0)≠0, then the angle between C1and C2 equals the angle between C1′ and C2′ both in magnitude and sense. We then say that the mapping w=f⁡(z) is conformal(angle-preserving) at z0.

The linear transformation f⁡(z)=a⁢z+b, a≠0, has f′⁡(z)=aand w=f⁡(z) maps ℂ conformally onto ℂ.

Bilinear Transformation

The transformation (1.9.40) is a one-to-one conformal mapping of ℂ∪{∞} onto itself.

The _cross ratio_of z1,z2,z3,z4∈ℂ∪{∞} is defined by

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 {zn} converges_to z if limn→∞zn=z. For zn=xn+i⁢yn, the sequence {zn} converges iff the sequences {xn} and {yn}separately converge. A series ∑n=0∞zn converges_if the sequence sn=∑k=0nzk converges. The series is_divergent if sn does not converge. The series converges_absolutely if ∑n=0∞|zn| converges. A series∑n=0∞zn converges (diverges) absolutely whenlimn→∞|zn|1/n<1 (>1), or whenlimn→∞|zn+1/zn|<1 (>1). Absolutely convergent series are also convergent.

Let {fn⁡(z)} be a sequence of functions defined on a set S. This sequence _converges pointwise_to a function f⁡(z) if

for each z∈S. The sequence _converges uniformly_on S, if for every ϵ>0 there exists an integer N, independent ofz, such that

for all z∈S and n≥N.

A series∑n=0∞fn⁡(z) converges uniformly on S, if the sequencesn⁡(z)=∑k=0nfk⁡(z) converges uniformly on S.

Weierstrass M-test

Suppose {Mn} is a sequence of real numbers such that∑n=0∞Mn converges and |fn⁡(z)|≤Mn for all z∈Sand all n≥0. Then the series ∑n=0∞fn⁡(z) converges uniformly on S.

A doubly-infinite series∑n=−∞∞fn⁡(z) converges (uniformly) on S iff each of the series ∑n=0∞fn⁡(z) and ∑n=1∞f−n⁡(z) converges (uniformly) on S.

§1.9(vi) Power Series

For a series ∑n=0∞an⁢(z−z0)n there is a number R, 0≤R≤∞, such that the series converges for all z in |z−z0|<R and diverges for z in |z−z0|>R. The circle |z−z0|=R is called the_circle of convergence_of the series, and R is the radius of convergence. Inside the circle the sum of the series is an analytic function f⁡(z). For z in|z−z0|≤ρ (<R), the convergence is absolute and uniform. Moreover,

and

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

Operations

When ∑an⁢zn and ∑bn⁢zn both converge

and

where

Next, let

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

where

With a0=1,

(principal value), where

and

Also,

(principal value), where ν∈ℂ,

and

For the definitions of the principal values of ln⁡f⁡(z) and (f⁡(z))νsee §§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 {zm,n} where m and n take all positive integer values is called a double sequence. It converges to zif for every ϵ>0, there is an integer N such that

for all m,n≥N. Suppose {zm,n} converges to z and the repeated limits

exist. Then both repeated limits equal z.

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 ζm,n is replaced by |ζm,n|.

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 ∑n=0∞fn⁡(z), where fn⁡(z) is continuous, converges uniformly on every _compact set_of a domain D, that is, every closed and bounded set in D. Then

for any finite contour C in D.

Dominated Convergence Theorem

Let (a,b) be a finite or infinite interval, and f0⁡(t),f1⁡(t),…be real or complex continuous functions, t∈(a,b). Suppose∑n=0∞fn⁡(t) converges uniformly in any compact interval in(a,b), and at least one of the following two conditions is satisfied:

Then