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§1.9 Calculus of a Complex Variable

Contents

§1.9(i) Complex Numbers

1.9.1 z=x+iy,
x,y.

Real and Imaginary Parts

1.9.2 z =x,
z =y.

Polar Representation

1.9.3 x =rcosθ,
y =rsinθ,

where

1.9.4 r=(x2+y2)1/2,

and when z0,

1.9.5 θ=ω, π-ω, -π+ω, or -ω,

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

1.9.6 ω=arctan(|y/x|)[0,12π].

Modulus and Phase

1.9.7 |z| =r,
phz =θ+2nπ,
n.

The principal value of phz corresponds to n=0, that is, -πphzπ. It is single-valued on \{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<phzπ.)

1.9.8 |z| |z|,
|z| |z|,
1.9.9 z=reiθ,

where

1.9.10 eiθ=cosθ+isinθ;

see §4.14.

Complex Conjugate

1.9.11 z¯ =x-iy,
1.9.12 |z¯| =|z|,
1.9.13 phz¯ =-phz.

Arithmetic Operations

If z1=x1+iy1, z2=x2+iy2, then

1.9.14 z1±z2=x1±x2+i(y1±y2),
1.9.15 z1z2=x1x2-y1y2+i(x1y2+x2y1),
1.9.16 z1z2=z1z¯2|z2|2=x1x2+y1y2+i(x2y1-x1y2)x22+y22,

provided that z20. Also,

1.9.17 |z1z2|=|z1||z2|,
1.9.18 ph(z1z2)=phz1+phz2,
1.9.19 |z1z2|=|z1||z2|,
1.9.20 phz1z2=phz1-phz2.

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

Powers

1.9.21 zn=(xn-(n2)xn-2y2+(n4)xn-4y4-)+i((n1)xn-1y-(n3)xn-3y3+),
n=1,2,.

DeMoivre’s Theorem

1.9.22 cosnθ+isinnθ=(cosθ+isinθ)n,
n.

Triangle Inequality

1.9.23 ||z1|-|z2|||z1+z2||z1|+|z2|.

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

Continuity

A function f(z) is continuous at a point z0 if limzz0f(z)=f(z0). That is, given any positive number ϵ, 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

A neighborhood of a point z0 is a disk |z-z0|<δ. An open set in 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 (or ) if every neighborhood of z0 contains a point of S distinct 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 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 z0 in R and any given number ϵ (>0) we can find a neighborhood of z0 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 differentiable at a point z if the following limit exists:

1.9.24 f(z)=dfdz=limh0f(z+h)-f(z)h.

Differentiability automatically implies continuity.

Cauchy–Riemann Equations

If f(z) exists at z=x+iy and f(z)=u(x,y)+iv(x,y), then

1.9.25 ux =vy,
uy =-vx

at (x,y).

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

Analyticity

A function f(z) is said to be analytic (holomorphic) at z=z0 if it is 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 is said to be entire.

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

Harmonic Functions

If f(z)=u(x,y)+iv(x,y) is analytic in an open domain D, then u and v are harmonic in D, that is,

1.9.26 2ux2+2uy2=2vx2+2vy2=0,

or in polar form ((1.9.3)) u and v satisfy

1.9.27 2ur2+1rur+1r22uθ2=0

at all points of D.

§1.9(iii) Integration

An arc C is given by z(t)=x(t)+iy(t), atb, where x 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,

1.9.28 Cf(z)dz=abf(z(t))(x(t)+iy(t))dt,

for a contour C and f(z(t)) continuous, atb. If f(z(t0))=, at0b, then the integral is defined analogously to the infinite integrals in §1.4(v). Similarly when a=- or b=+.

Jordan Curve Theorem

Any simple closed contour C divides 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

1.9.29 Cf(z)dz=0.

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

1.9.30 f(z0)=12πiCf(z)z-z0dz,

and

1.9.31 f(n)(z0)=n!2πiCf(z)(z-z0)n+1dz,
n=1,2,3,,

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 z0C, then

1.9.32 12πiC1z-z0dz=𝒩(C,z0),

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

Mean Value Property

For u(z) harmonic,

1.9.33 u(z)=12π02πu(z+reiϕ)dϕ.

Poisson Integral

If h(w) is continuous on |w|=R, then with z=reiθ

1.9.34 u(reiθ)=12π02π(R2-r2)h(Reiϕ)dϕR2-2Rrcos(ϕ-θ)+r2

is harmonic in |z|<R. Also with |w|=R, limzwu(z)=h(w) as zw within |z|<R.

§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

1.9.35 Sr={z|z|>1/r}{},
0<r<.

Each Sr is a neighborhood of . Also,

1.9.36 ±z=z±=,
1.9.37 z=z=,
z0,
1.9.38 z/=0,
1.9.39 z/0=,
z0.

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 D passing through z0. Let C1,C2 be the images of C1 and C2 under the mapping w=f(z). The angle between C1 and C2 at z0 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 C1 and 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)=az+b, a0, has f(z)=a and w=f(z) maps conformally onto .

Bilinear Transformation

1.9.40 w=f(z)=az+bcz+d,
ad-bc0, c0.
1.9.41 f(-d/c) =,
f() =a/c.
1.9.42 f(z)=ad-bc(cz+d)2,
z-d/c.
1.9.43 f()=bc-adc2.
1.9.44 z=dw-b-cw+a.

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

The cross ratio of z1,z2,z3,z4{} is defined by

1.9.45 (z1-z2)(z3-z4)(z1-z4)(z3-z2),

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 limnzn=z. For zn=xn+iyn, the sequence {zn} converges iff the sequences {xn} and {yn} separately converge. A series n=0zn 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=0zn converges (diverges) absolutely when limn|zn|1/n<1 (>1), or when limn|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

1.9.46 f(z)=limnfn(z)

for each zS. The sequence converges uniformly on S, if for every ϵ>0 there exists an integer N, independent of z, such that

1.9.47 |fn(z)-f(z)|<ϵ

for all zS and nN.

A series n=0fn(z) converges uniformly on S, if the sequence sn(z)=k=0nfk(z) converges uniformly on S.

Weierstrass M-test

Suppose {Mn} is a sequence of real numbers such that n=0Mn converges and |fn(z)|Mn for all zS and all n0. Then the series n=0fn(z) converges uniformly on S.

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

§1.9(vi) Power Series

For a series n=0an(z-z0)n there is a number R, 0R, 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,

1.9.48 an=f(n)(z0)n!,

and

1.9.49 R=lim infn|an|-1/n.

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

Operations

When anzn and bnzn both converge

1.9.50 n=0(an±bn)zn=n=0anzn±n=0bnzn,

and

1.9.51 (n=0anzn)(n=0bnzn)=n=0cnzn,

where

1.9.52 cn=k=0nakbn-k.

Next, let

1.9.53 f(z)=a0+a1z+a2z2+,
a00.

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

1.9.54 1f(z)=b0+b1z+b2z2+,

where

1.9.55 b0 =1/a0,
b1 =-a1/a02,
b2 =(a12-a0a2)/a03,
1.9.56 bn=-(a1bn-1+a2bn-2++anb0)/a0,
n1.

With a0=1,

1.9.57 lnf(z)=q1z+q2z2+q3z3+,

(principal value), where

1.9.58 q1 =a1,
q2 =(2a2-a12)/2,
q3 =(3a3-3a1a2+a13)/3,

and

1.9.59 qn=(nan-(n-1)a1qn-1-(n-2)a2qn-2--an-1q1)/n,
n2.

Also,

1.9.60 (f(z))ν=p0+p1z+p2z2+,

(principal value), where ν,

1.9.61 p0 =1,
p1 =νa1,
p2 =ν((ν-1)a12+2a2)/2,

and

1.9.62 pn=((ν-n+1)a1pn-1+(2ν-n+2)a2pn-2++((n-1)ν-1)an-1p1+nνan)/n,
n1.

For the definitions of the principal values of lnf(z) and (f(z))ν see §§4.2(i) and 4.2(iv).

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

1.9.63 f(m)(z)=n=0(n+1)man+m(z-z0)n,
|z-z0|<R, m=0,1,2,.

§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 z if for every ϵ>0, there is an integer N such that

1.9.64 |zm,n-z|<ϵ

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

1.9.65 limm(limnzm,n),
limn(limmzm,n)

exist. Then both repeated limits equal z.

A double series is the limit of the double sequence

1.9.66 zp,q=m=0pn=0qζm,n.

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

1.9.67 m=0(n=0ζm,n),
n=0(m=0ζm,n).

Term-by-Term Integration

Suppose the series n=0fn(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

1.9.68 Cn=0fn(z)dz=n=0Cfn(z)dz

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=0fn(t) converges uniformly in any compact interval in (a,b), and at least one of the following two conditions is satisfied:

1.9.69 abn=0|fn(t)|dt<,
1.9.70 n=0ab|fn(t)|dt<.

Then

1.9.71 abn=0fn(t)dt=n=0abfn(t)dt.