§1.9 Calculus of a Complex Variable

Keywords:: calculus
Referenced by:: §1.1
Permalink:: http://dlmf.nist.gov/1.9

§1.9(i) Complex Numbers
§1.9(ii) Continuity, Point Sets, and Differentiation
§1.9(iii) Integration
§1.9(iv) Conformal Mapping
§1.9(v) Infinite Sequences and Series
§1.9(vi) Power Series
§1.9(vii) Inversion of Limits

§1.9(i) Complex Numbers

Notes:: See Copson (1935, Chapter 1), Levinson and Redheffer (1970, Chapter 1), or Markushevich (1983, pp. 14–18).
Referenced by:: §1.11(iv), §4.2(i), §4.45(ii)
Permalink:: http://dlmf.nist.gov/1.9.i

1.9.1 $z=x+iy,$ $x,y\in\Real$ .

Symbols:: $\in$ : element of, $\Real$ : real line (excluding infinity) and $z$ : variable
A&S Ref:: 3.7.1
Permalink:: http://dlmf.nist.gov/1.9.E1
Encodings:: TeX, pMML, png

¶ Real and Imaginary Parts

Keywords:: complex numbers

1.9.2

$\realpart{z}=x,$

$\imagpart{z}=y.$

Defines:: $\imagpart{}$ : imaginary part and $\realpart{}$ : real part
Symbols:: $z$ : variable
A&S Ref:: 3.7.5 3.7.6
Permalink:: http://dlmf.nist.gov/1.9.E2
Encodings:: TeX, TeX, pMML, pMML, png, png

¶ Polar Representation

Keywords:: complex numbers, polar representation

1.9.3

$x=r\mathop{\cos\/}\nolimits\theta,$

$y=r\mathop{\sin\/}\nolimits\theta,$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, $r$ : radius and $\theta$ : angle
A&S Ref:: 3.7.2
Referenced by:: ¶ ‣ §1.9(ii)
Permalink:: http://dlmf.nist.gov/1.9.E3
Encodings:: TeX, TeX, pMML, pMML, png, png

where

1.9.4 $r=(x^{2}+y^{2})^{{1/2}},$

Symbols:: $r$ : radius
A&S Ref:: 3.7.3
Permalink:: http://dlmf.nist.gov/1.9.E4
Encodings:: TeX, pMML, png

and when $z\neq 0$ ,

1.9.5 $\theta=\omega,\;\;\pi-\omega,\;\;-\pi+\omega,\mbox{ or }-\omega,$

Symbols:: $\theta$ : angle
Permalink:: http://dlmf.nist.gov/1.9.E5
Encodings:: TeX, pMML, png

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

1.9.6 $\omega=\mathop{\mathrm{arctan}\/}\nolimits\!\left(|y/x|\right)\in\left[0,\tfrac{1}{2}\pi\right].$

Symbols:: $[a,b]$ : closed interval, $\in$ : element of and $\mathop{\mathrm{arctan}\/}\nolimits z$ : arctangent function
A&S Ref:: 3.7.4
Permalink:: http://dlmf.nist.gov/1.9.E6
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¶ Modulus and Phase

Keywords:: complex numbers

1.9.7

$|z|=r,$

$\mathop{\mathrm{ph}\/}\nolimits z=\theta+2n\pi,$ $n\in\Integer$ .

Defines:: $\mathop{\mathrm{ph}\/}\nolimits$ : phase
Symbols:: $\in$ : element of, $\Integer$ : set of all integers, $z$ : variable, $n$ : nonnegative integer, $r$ : radius and $\theta$ : angle
Permalink:: http://dlmf.nist.gov/1.9.E7
Encodings:: TeX, TeX, pMML, pMML, png, png

The principal value of $\mathop{\mathrm{ph}\/}\nolimits z$ corresponds to $n=0$ , that is, $-\pi\leq\mathop{\mathrm{ph}\/}\nolimits z\leq\pi$ . It is single-valued on $\Complex\setminus\{ 0\}$ , except on the interval $(-\infty,0)$ 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 $-\pi<\mathop{\mathrm{ph}\/}\nolimits z\leq\pi$ .)

1.9.8

$|\realpart{z}|\leq|z|,$

$|\imagpart{z}|\leq|z|,$

Symbols:: $\imagpart{}$ : imaginary part, $\realpart{}$ : real part and $z$ : variable
Permalink:: http://dlmf.nist.gov/1.9.E8
Encodings:: TeX, TeX, pMML, pMML, png, png

1.9.9 $z=re^{{i\theta}},$

Symbols:: $e$ : base of exponential function, $z$ : variable, $r$ : radius and $\theta$ : angle
Permalink:: http://dlmf.nist.gov/1.9.E9
Encodings:: TeX, pMML, png

where

1.9.10 $e^{{i\theta}}=\mathop{\cos\/}\nolimits\theta+i\mathop{\sin\/}\nolimits\theta;$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $e$ : base of exponential function, $\mathop{\sin\/}\nolimits z$ : sine function and $\theta$ : angle
Permalink:: http://dlmf.nist.gov/1.9.E10
Encodings:: TeX, pMML, png

see §4.14.

¶ Complex Conjugate

Keywords:: complex numbers

1.9.11 $\conj{z}=x-iy,$

Defines:: $\conj{z}$ : complex conjugate
Symbols:: $z$ : variable
A&S Ref:: 3.7.7
Permalink:: http://dlmf.nist.gov/1.9.E11
Encodings:: TeX, pMML, png

1.9.12 $|\conj{z}|=|z|,$

Symbols:: $\conj{z}$ : complex conjugate and $z$ : variable
A&S Ref:: 3.7.8
Permalink:: http://dlmf.nist.gov/1.9.E12
Encodings:: TeX, pMML, png

1.9.13 $\mathop{\mathrm{ph}\/}\nolimits\conj{z}=-\mathop{\mathrm{ph}\/}\nolimits z.$

Symbols:: $\conj{z}$ : complex conjugate, $\mathop{\mathrm{ph}\/}\nolimits$ : phase and $z$ : variable
A&S Ref:: 3.7.9
Permalink:: http://dlmf.nist.gov/1.9.E13
Encodings:: TeX, pMML, png

¶ Arithmetic Operations

Keywords:: complex numbers

If $z_{1}=x_{1}+iy_{1}$ , $z_{2}=x_{2}+iy_{2}$ , then

1.9.14 $z_{1}\pm z_{2}=x_{1}\pm x_{2}+i(y_{1}\pm y_{2}),$

Symbols:: $z$ : variable
Permalink:: http://dlmf.nist.gov/1.9.E14
Encodings:: TeX, pMML, png

1.9.15 $z_{1}z_{2}=x_{1}x_{2}-y_{1}y_{2}+i(x_{1}y_{2}+x_{2}y_{1}),$

Symbols:: $z$ : variable
A&S Ref:: 3.7.10
Permalink:: http://dlmf.nist.gov/1.9.E15
Encodings:: TeX, pMML, png

1.9.16 $\frac{z_{1}}{z_{2}}=\frac{z_{1}\conj{z}_{2}}{|z_{2}|^{2}}=\frac{x_{1}x_{2}+y_{1}y_{2}+i(x_{2}y_{1}-x_{1}y_{2})}{x_{2}^{2}+y_{2}^{2}},$

Symbols:: $\conj{z}$ : complex conjugate and $z$ : variable
A&S Ref:: 3.7.13
Permalink:: http://dlmf.nist.gov/1.9.E16
Encodings:: TeX, pMML, png

provided that $z_{2}\neq 0$ . Also,

1.9.17 $|z_{1}z_{2}|=|z_{1}|\;|z_{2}|,$

Symbols:: $z$ : variable
A&S Ref:: 3.7.11
Permalink:: http://dlmf.nist.gov/1.9.E17
Encodings:: TeX, pMML, png

1.9.18 $\mathop{\mathrm{ph}\/}\nolimits\!\left(z_{1}z_{2}\right)=\mathop{\mathrm{ph}\/}\nolimits z_{1}+\mathop{\mathrm{ph}\/}\nolimits z_{2},$

Symbols:: $\mathop{\mathrm{ph}\/}\nolimits$ : phase and $z$ : variable
A&S Ref:: 3.7.12
Referenced by:: ¶ ‣ §1.9(i)
Permalink:: http://dlmf.nist.gov/1.9.E18
Encodings:: TeX, pMML, png

1.9.19 $\left|\frac{z_{1}}{z_{2}}\right|=\frac{|z_{1}|}{|z_{2}|},$

Symbols:: $z$ : variable
A&S Ref:: 3.7.14
Permalink:: http://dlmf.nist.gov/1.9.E19
Encodings:: TeX, pMML, png

1.9.20 $\mathop{\mathrm{ph}\/}\nolimits\frac{z_{1}}{z_{2}}=\mathop{\mathrm{ph}\/}\nolimits z_{1}-\mathop{\mathrm{ph}\/}\nolimits z_{2}.$

Symbols:: $\mathop{\mathrm{ph}\/}\nolimits$ : phase and $z$ : variable
A&S Ref:: 3.7.15
Referenced by:: ¶ ‣ §1.9(i)
Permalink:: http://dlmf.nist.gov/1.9.E20
Encodings:: TeX, pMML, png

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

¶ Powers

Keywords:: complex numbers

1.9.21 $z^{n}=\left(x^{n}-\binom{n}{2}x^{{n-2}}y^{2}+\binom{n}{4}x^{{n-4}}y^{4}-\cdots\right)+i\left(\binom{n}{1}x^{{n-1}}y-\binom{n}{3}x^{{n-3}}y^{3}+\cdots\right),$ $n=1,2,\dots$ .

Symbols:: $\binom{m}{n}$ : binomial coefficient, $z$ : variable and $n$ : nonnegative integer
A&S Ref:: 3.7.22
Permalink:: http://dlmf.nist.gov/1.9.E21
Encodings:: TeX, pMML, png

¶ DeMoivre’s Theorem

Keywords:: DeMoivre’s theorem, complex numbers

1.9.22 $\mathop{\cos\/}\nolimits n\theta+i\mathop{\sin\/}\nolimits n\theta=(\mathop{\cos\/}\nolimits\theta+i\mathop{\sin\/}\nolimits\theta)^{n},$ $n\in\Integer$ .

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\in$ : element of, $\Integer$ : set of all integers, $\mathop{\sin\/}\nolimits z$ : sine function, $n$ : nonnegative integer and $\theta$ : angle
Permalink:: http://dlmf.nist.gov/1.9.E22
Encodings:: TeX, pMML, png

¶ Triangle Inequality

Keywords:: complex numbers, triangle inequality

1.9.23 $\left|\left|z_{1}\right|-\left|z_{2}\right|\right|\leq\left|z_{1}+z_{2}\right|\leq\left|z_{1}\right|+\left|z_{2}\right|.$

Symbols:: $z$ : variable
A&S Ref:: 3.7.29
Permalink:: http://dlmf.nist.gov/1.9.E23
Encodings:: TeX, pMML, png

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

Notes:: See Levinson and Redheffer (1970, Chapters 1,2 and pp. 133–138) and Copson (1935, Chapters 2,3).
Referenced by:: §1.6(iv)
Permalink:: http://dlmf.nist.gov/1.9.ii

¶ Continuity

Keywords:: continuous function, disk, limits of functions

A function $f(z)$ is continuous at a point $z_{0}$ if $\lim\limits _{{z\to z_{0}}}f(z)=f(z_{0})$ . That is, given any positive number $\epsilon$ , however small, we can find a positive number $\delta$ such that $|f(z)-f(z_{0})|<\epsilon$ for all $z$ in the open disk $|z-z_{0}|<\delta$ .

A function of two complex variables $f(z,w)$ is continuous at $(z_{0},w_{0})$ if $\lim\limits _{{(z,w)\to(z_{0},w_{0})}}f(z,w)=f(z_{0},w_{0})$ ; compare (1.5.1) and (1.5.2).

¶ Point Sets in $\Complex$

Keywords:: accumulation point, boundary points, closed point set, closure, connected point set, continuous function, domain, interior points, limit points (or limiting points), neighborhood, open point set, point sets in complex plane, points in complex plane, region

A neighborhood of a point $z_{0}$ is a disk $\left|z-z_{0}\right|<\delta$ . An open set in $\Complex$ is one in which each point has a neighborhood that is contained in the set.

A point $z_{0}$ is a limit point (limiting point or accumulation point) of a set of points $S$ in $\Complex$ (or $\Complex\cup\infty$ ) if every neighborhood of $z_{0}$ contains a point of $S$ distinct from $z_{0}$ . ( $z_{0}$ 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 $\Complex$ 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 $z_{0}$ in $R$ and any given number $\epsilon$ ( $>0$ ) we can find a neighborhood of $z_{0}$ such that $\left|f(z)-f(z_{0})\right|<\epsilon$ for all points $z$ in the intersection of the neighborhood with $R$ .

¶ Differentiation

Keywords:: differentiable functions

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

1.9.24 $f^{{\prime}}(z)=\frac{df}{dz}=\lim _{{h\to 0}}\frac{f(z+h)-f(z)}{h}.$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ and $z$ : variable
Permalink:: http://dlmf.nist.gov/1.9.E24
Encodings:: TeX, pMML, png

Differentiability automatically implies continuity.

¶ Cauchy–Riemann Equations

Keywords:: Cauchy–Riemann equations, differentiation

If $f^{{\prime}}(z)$ exists at $z=x+iy$ and $f(z)=u(x,y)+iv(x,y)$ , then

1.9.25

$\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y},$

$\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$

Symbols:: $\frac{\partial f}{\partial x}$ : partial derivative of $f$ with respect to $x$ , $u(x,y)$ : function and $v(x,y)$ : function
A&S Ref:: 3.7.30
Referenced by:: ¶ ‣ §1.9(ii)
Permalink:: http://dlmf.nist.gov/1.9.E25
Encodings:: TeX, TeX, pMML, pMML, png, png

at $(x,y)$ .

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

¶ Analyticity

Keywords:: analytic function, entire functions, functions, holomorphic function

A function $f(z)$ is said to be analytic (holomorphic) at $z=z_{0}$ if it is differentiable in a neighborhood of $z_{0}$ .

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 $\Complex$ is said to be entire.

If $f(z)$ is analytic in an open domain $D$ , then each of its derivatives $f^{{\prime}}(z)$ , $f^{{\prime\prime}}(z)$ , $\dots$ exists and is analytic in $D$ .

¶ Harmonic Functions

Keywords:: functions, 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 $\frac{{\partial}^{2}u}{{\partial x}^{2}}+\frac{{\partial}^{2}u}{{\partial y}^{2}}=\frac{{\partial}^{2}v}{{\partial x}^{2}}+\frac{{\partial}^{2}v}{{\partial y}^{2}}=0,$

Symbols:: $\frac{\partial f}{\partial x}$ : partial derivative of $f$ with respect to $x$ , $u(x,y)$ : function and $v(x,y)$ : function
A&S Ref:: 3.7.32
Permalink:: http://dlmf.nist.gov/1.9.E26
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or in polar form ((1.9.3)) $u$ and $v$ satisfy

1.9.27 $\frac{{\partial}^{2}u}{{\partial r}^{2}}+\frac{1}{r}\frac{\partial u}{\partial r}+\frac{1}{r^{2}}\frac{{\partial}^{2}u}{{\partial\theta}^{2}}=0$

Symbols:: $\frac{\partial f}{\partial x}$ : partial derivative of $f$ with respect to $x$ , $r$ : radius, $\theta$ : angle and $u(x,y)$ : function
A&S Ref:: 3.7.33
Referenced by:: ¶ ‣ §1.15(iii)
Permalink:: http://dlmf.nist.gov/1.9.E27
Encodings:: TeX, pMML, png

at all points of $D$ .

§1.9(iii) Integration

Notes:: See Levinson and Redheffer (1970, Chapter 3 and p. 360), Copson (1935, pp. 56–69), and Ahlfors (1966, pp. 168–169). For a proof of the Jordan Curve Theorem see, for example, Dienes (1931, pp. 177–197). The theorem is valid with less restrictive conditions than those assumed here.
Keywords:: arc(s), contour, integrals, integration, simple closed contour
Referenced by:: ¶ ‣ §1.10(vi), §15.16, §3.4(ii)
Permalink:: http://dlmf.nist.gov/1.9.iii

An arc $C$ is given by $z(t)=x(t)+iy(t)$ , $a\leq t\leq b$ , where $x$ and $y$ are continuously differentiable. If $x(t)$ and $y(t)$ are continuous and $x^{{\prime}}(t)$ and $y^{{\prime}}(t)$ 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 $t_{1},t_{2}$ of $t$ , $z(t_{1})\neq z(t_{2})$ . A simple closed contour is a simple contour, except that $z(a)=z(b)$ .

Next,

1.9.28 $\int _{C}f(z)dz=\int _{a}^{b}f(z(t))(x^{{\prime}}(t)+iy^{{\prime}}(t))dt,$

Symbols:: $dx$ : differential of $x$ , $\int$ : integral and $C$ : arc
Permalink:: http://dlmf.nist.gov/1.9.E28
Encodings:: TeX, pMML, png

for a contour $C$ and $f(z(t))$ continuous, $a\leq t\leq b$ . If $f(z(t_{0}))=\infty$ , $a\leq t_{0}\leq b$ , then the integral is defined analogously to the infinite integrals in §1.4(v). Similarly when $a=-\infty$ or $b=+\infty$ .

¶ Jordan Curve Theorem

Keywords:: Jordan curve theorem, domain, point sets in complex plane

Any simple closed contour $C$ divides $\Complex$ 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

Keywords:: Cauchy’s theorem

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

1.9.29 $\int _{C}f(z)dz=0.$

Symbols:: $dx$ : differential of $x$ , $\int$ : integral and $C$ : closed contour
Permalink:: http://dlmf.nist.gov/1.9.E29
Encodings:: TeX, pMML, png

¶ Cauchy’s Integral Formula

Keywords:: Cauchy’s integral formula

If $f(z)$ is continuous within and on a simple closed contour $C$ and analytic within $C$ , and if $z_{0}$ is a point within $C$ , then

1.9.30 $f(z_{0})=\frac{1}{2\pi i}\int _{C}\frac{f(z)}{z-z_{0}}dz,$

Symbols:: $dx$ : differential of $x$ , $\int$ : integral and $C$ : closed contour
Referenced by:: §2.3(iii)
Permalink:: http://dlmf.nist.gov/1.9.E30
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and

1.9.31 $f^{{(n)}}(z_{0})=\frac{n!}{2\pi i}\int _{C}\frac{f(z)}{(z-z_{0})^{{n+1}}}dz,$ $n=1,2,3,\dots$ ,

Symbols:: $dx$ : differential of $x$ , $!$ : $n!$ : factorial, $\int$ : integral, $n$ : nonnegative integer and $C$ : closed contour
Permalink:: http://dlmf.nist.gov/1.9.E31
Encodings:: TeX, pMML, png

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

¶ Liouville’s Theorem

Keywords:: Liouville’s theorem for entire functions, entire functions

Any bounded entire function is a constant.

¶ Winding Number

Keywords:: winding number

If $C$ is a closed contour, and $z_{0}\not\in C$ , then

1.9.32 $\frac{1}{2\pi i}\int _{C}\frac{1}{z-z_{0}}dz=\mathcal{N}(C,z_{0}),$

Defines:: $\mathcal{N}$ : winding number
Symbols:: $dx$ : differential of $x$ , $\int$ : integral and $C$ : closed contour
Permalink:: http://dlmf.nist.gov/1.9.E32
Encodings:: TeX, pMML, png

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

¶ Mean Value Property

Keywords:: harmonic functions, mean value property for harmonic functions

For $u(z)$ harmonic,

1.9.33 $u(z)=\frac{1}{2\pi}\int^{{2\pi}}_{0}u(z+re^{{i\phi}})d\phi.$

Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $r$ : radius and $u(z)$ : harmonic function
Permalink:: http://dlmf.nist.gov/1.9.E33
Encodings:: TeX, pMML, png

¶ Poisson Integral

Keywords:: Poisson integral, harmonic functions

If $h(w)$ is continuous on $|w|=R$ , then with $z=re^{{i\theta}}$

1.9.34 $u(re^{{i\theta}})=\frac{1}{2\pi}\int^{{2\pi}}_{0}\frac{(R^{2}-r^{2})h(Re^{{i\phi}})d\phi}{R^{2}-2Rr\mathop{\cos\/}\nolimits\!\left(\phi-\theta\right)+r^{2}}$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $r$ : radius, $u(z)$ : harmonic function and $R$ : radius
Referenced by:: ¶ ‣ §1.15(v)
Permalink:: http://dlmf.nist.gov/1.9.E34
Encodings:: TeX, pMML, png

is harmonic in $|z|<R$ . Also with $\left|w\right|=R$ , $\lim\limits _{{z\to w}}u(z)=h(w)$ as $z\to w$ within $|z|<R$ .

§1.9(iv) Conformal Mapping

Notes:: See Markushevich (1983, pp. 41–46), Markushevich (1985, vol. 1, §34), and Levinson and Redheffer (1970, pp. 259–277).
Keywords:: analytic function, conformal mapping, disk, extended complex plane, neighborhood, open disks around infinity, points in complex plane
Permalink:: http://dlmf.nist.gov/1.9.iv

The extended complex plane, $\Complex\,\cup\,\{\infty\}$ , consists of the points of the complex plane $\Complex$ together with an ideal point $\infty$ called the point at infinity. A system of open disks around infinity is given by

1.9.35 $S_{r}=\{ z\mid|z|>1/r\}\cup\{\infty\},$ $0<r<\infty$ .

Symbols:: $\{\ldots\}$ : sequence, asymptotic sequence (or scale), or enumerable set, $\cup$ : union, $z$ : variable, $r$ : radius and $S_{r}$ : neighborhood
Permalink:: http://dlmf.nist.gov/1.9.E35
Encodings:: TeX, pMML, png

Each $S_{r}$ is a neighborhood of $\infty$ . Also,

1.9.36 $\infty\pm z=z\pm\infty=\infty,$

Symbols:: $z$ : variable
Permalink:: http://dlmf.nist.gov/1.9.E36
Encodings:: TeX, pMML, png

1.9.37 $\infty\cdot z=z\cdot\infty=\infty,$ $z\not=0$ ,

Symbols:: $z$ : variable
Permalink:: http://dlmf.nist.gov/1.9.E37
Encodings:: TeX, pMML, png

1.9.38 $z/\infty=0,$

Symbols:: $z$ : variable
Permalink:: http://dlmf.nist.gov/1.9.E38
Encodings:: TeX, pMML, png

1.9.39 $z/0=\infty,$ $z\neq 0$ .

Symbols:: $z$ : variable
Permalink:: http://dlmf.nist.gov/1.9.E39
Encodings:: TeX, pMML, png

A function $f(z)$ is analytic at $\infty$ if $g(z)=f(1/z)$ is analytic at $z=0$ , and we set $f^{{\prime}}(\infty)=g^{{\prime}}(0)$ .

¶ Conformal Transformation

Keywords:: angle between arcs, arc(s), conformal mapping, linear transformation

Suppose $f(z)$ is analytic in a domain $D$ and $C_{1},C_{2}$ are two arcs in $D$ passing through $z_{0}$ . Let $C^{{\prime}}_{1},C^{{\prime}}_{2}$ be the images of $C_{1}$ and $C_{2}$ under the mapping $w=f(z)$ . The angle between $C_{1}$ and $C_{2}$ at $z_{0}$ is the angle between the tangents to the two arcs at $z_{0}$ , that is, the difference of the signed angles that the tangents make with the positive direction of the real axis. If $f^{{\prime}}(z_{0})\not=0$ , then the angle between $C_{1}$ and $C_{2}$ equals the angle between $C^{{\prime}}_{1}$ and $C^{{\prime}}_{2}$ both in magnitude and sense. We then say that the mapping $w=f(z)$ is conformal (angle-preserving) at $z_{0}$ .

The linear transformation $f(z)=az+b$ , $a\not=0$ , has $f^{{\prime}}(z)=a$ and $w=f(z)$ maps $\Complex$ conformally onto $\Complex$ .

¶ Bilinear Transformation

Keywords:: Möbius transformation, bilinear transformation, cross ratio, fractional linear transformation, homographic transformation

1.9.40 $w=f(z)=\frac{az+b}{cz+d},$ $ad-bc\not=0$ , $c\not=0$ .

Symbols:: $z$ : variable and $w$ : variable
Referenced by:: ¶ ‣ §1.9(iv)
Permalink:: http://dlmf.nist.gov/1.9.E40
Encodings:: TeX, pMML, png

1.9.41

$f(-d/c)=\infty,$

$f(\infty)=a/c.$

Permalink:: http://dlmf.nist.gov/1.9.E41
Encodings:: TeX, TeX, pMML, pMML, png, png

1.9.42 $f^{{\prime}}(z)=\frac{ad-bc}{(cz+d)^{2}},$ $z\not=-d/c$ .

Symbols:: $z$ : variable
Permalink:: http://dlmf.nist.gov/1.9.E42
Encodings:: TeX, pMML, png

1.9.43 $f^{{\prime}}(\infty)=\frac{bc-ad}{c^{2}}.$

Permalink:: http://dlmf.nist.gov/1.9.E43
Encodings:: TeX, pMML, png

1.9.44 $z=\frac{dw-b}{-cw+a}.$

Symbols:: $z$ : variable and $w$ : variable
Permalink:: http://dlmf.nist.gov/1.9.E44
Encodings:: TeX, pMML, png

The transformation (1.9.40) is a one-to-one conformal mapping of $\Complex\,\cup\,\{\infty\}$ onto itself.

The cross ratio of $z_{1},z_{2},z_{3},z_{4}\in\Complex\cup\{\infty\}$ is defined by

1.9.45 $\frac{(z_{1}-z_{2})(z_{3}-z_{4})}{(z_{1}-z_{4})(z_{3}-z_{2})},$

Symbols:: $z$ : variable
Permalink:: http://dlmf.nist.gov/1.9.E45
Encodings:: TeX, pMML, png

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

Notes:: See Copson (1935, pp. 19–24, 92–98).
Keywords:: convergence, infinite sequences, infinite series
Permalink:: http://dlmf.nist.gov/1.9.v

A sequence $\{ z_{n}\}$ converges to $z$ if $\lim\limits _{{n\to\infty}}z_{n}=z$ . For $z_{n}=x_{n}+iy_{n}$ , the sequence $\{ z_{n}\}$ converges iff the sequences $\{ x_{n}\}$ and $\{ y_{n}\}$ separately converge. A series $\sum^{\infty}_{{n=0}}z_{n}$ converges if the sequence $s_{n}=\sum^{n}_{{k=0}}z_{k}$ converges. The series is divergent if $s_{n}$ does not converge. The series converges absolutely if $\sum^{\infty}_{{n=0}}|z_{n}|$ converges. A series $\sum^{\infty}_{{n=0}}z_{n}$ converges (diverges) absolutely when $\lim\limits _{{n\to\infty}}|z_{n}|^{{1/n}}<1$ ( $>1$ ), or when $\lim\limits _{{n\to\infty}}\left|\ifrac{z_{{n+1}}}{z_{n}}\right|<1$ ( $>1$ ). Absolutely convergent series are also convergent.

Let $\{ f_{n}(z)\}$ be a sequence of functions defined on a set $S$ . This sequence converges pointwise to a function $f(z)$ if

1.9.46 $f(z)=\lim _{{n\to\infty}}f_{n}(z)$

Symbols:: $z$ : variable and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.9.E46
Encodings:: TeX, pMML, png

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

1.9.47 $|f_{n}(z)-f(z)|<\epsilon$

Symbols:: $z$ : variable, $n$ : nonnegative integer and $\epsilon$ : positive number
Permalink:: http://dlmf.nist.gov/1.9.E47
Encodings:: TeX, pMML, png

for all $z\in S$ and $n\geq N$ .

A series $\sum^{\infty}_{{n=0}}f_{n}(z)$ converges uniformly on $S$ , if the sequence $s_{n}(z)=\sum^{n}_{{k=0}}f_{k}(z)$ converges uniformly on $S$ .

¶ Weierstrass $M$ -test

Keywords:: $M$ -test for uniform convergence, Weierstrass $M$ -test, convergence, infinite series

Suppose $\{ M_{n}\}$ is a sequence of real numbers such that $\sum^{\infty}_{{n=0}}M_{n}$ converges and $|f_{n}(z)|\leq M_{n}$ for all $z\in S$ and all $n\geq 0$ . Then the series $\sum^{\infty}_{{n=0}}f_{n}(z)$ converges uniformly on $S$ .

A doubly-infinite series $\sum^{\infty}_{{n=-\infty}}f_{n}(z)$ converges (uniformly) on $S$ iff each of the series $\sum^{\infty}_{{n=0}}f_{n}(z)$ and $\sum^{\infty}_{{n=1}}f_{{-n}}(z)$ converges (uniformly) on $S$ .

§1.9(vi) Power Series

Notes:: See Copson (1935, pp. 37–40), Levinson and Redheffer (1970, pp. 349–351), or Markushevich (1983, pp. 131–135). For the operations on series, see Henrici (1974, Chapter 1) or Olver (1997b, pp. 19–22).
Keywords:: convergence, power series
Referenced by:: §22.10(ii)
Permalink:: http://dlmf.nist.gov/1.9.vi

For a series $\sum^{\infty}_{{n=0}}a_{n}(z-z_{0})^{n}$ there is a number $R$ , $0\leq R\leq\infty$ , such that the series converges for all $z$ in $|z-z_{0}|<R$ and diverges for $z$ in $|z-z_{0}|>R$ . The circle $|z-z_{0}|=R$ 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 $|z-z_{0}|\leq\rho$ ( $<R$ ), the convergence is absolute and uniform. Moreover,

1.9.48 $a_{n}=\frac{f^{{(n)}}(z_{0})}{n!},$

Symbols:: $!$ : $n!$ : factorial, $z$ : variable and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.9.E48
Encodings:: TeX, pMML, png

and

1.9.49 $R=\liminf _{{n\to\infty}}|a_{n}|^{{-1/n}}.$

Symbols:: $\liminf$ : least limit point, $n$ : nonnegative integer and $R$ : radius of convergence
Permalink:: http://dlmf.nist.gov/1.9.E49
Encodings:: TeX, pMML, png

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

¶ Operations

Keywords:: differentiation, power series

When $\sum a_{n}z^{n}$ and $\sum b_{n}z^{n}$ both converge

1.9.50 $\sum^{\infty}_{{n=0}}(a_{n}\pm b_{n})z^{n}=\sum^{\infty}_{{n=0}}a_{n}z^{n}\pm\sum^{\infty}_{{n=0}}b_{n}z^{n},$

Symbols:: $z$ : variable and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.9.E50
Encodings:: TeX, pMML, png

and

1.9.51 $\left(\sum^{\infty}_{{n=0}}a_{n}z^{n}\right)\left(\sum^{\infty}_{{n=0}}b_{n}z^{n}\right)=\sum^{\infty}_{{n=0}}c_{n}z^{n},$

Symbols:: $z$ : variable and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.9.E51
Encodings:: TeX, pMML, png

where

1.9.52 $c_{n}=\sum^{n}_{{k=0}}a_{k}b_{{n-k}}.$

Symbols:: $k$ : integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.9.E52
Encodings:: TeX, pMML, png

Next, let

1.9.53 $f(z)=a_{0}+a_{1}z+a_{2}z^{2}+\cdots,$ $a_{0}\not=0$ .

Symbols:: $z$ : variable
Permalink:: http://dlmf.nist.gov/1.9.E53
Encodings:: TeX, pMML, png

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

1.9.54 $\frac{1}{f(z)}=b_{0}+b_{1}z+b_{2}z^{2}+\cdots,$

Symbols:: $z$ : variable
Referenced by:: ¶ ‣ §1.9(vi)
Permalink:: http://dlmf.nist.gov/1.9.E54
Encodings:: TeX, pMML, png

where

1.9.55

$b_{0}=1/a_{0},$

$b_{1}=-a_{1}/a_{0}^{2},$

$b_{2}=(a_{1}^{2}-a_{0}a_{2})/a_{0}^{3},$

Permalink:: http://dlmf.nist.gov/1.9.E55
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

1.9.56 $b_{n}=-(a_{1}b_{{n-1}}+a_{2}b_{{n-2}}+\dots+a_{n}b_{0})/a_{0},$ $n\geq 1$ .

Symbols:: $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.9.E56
Encodings:: TeX, pMML, png

With $a_{0}=1$ ,

1.9.57 $\mathop{\ln\/}\nolimits f(z)=q_{1}z+q_{2}z^{2}+q_{3}z^{3}+\cdots,$

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $z$ : variable and $q_{j}$ : coefficients
Referenced by:: ¶ ‣ §1.9(vi)
Permalink:: http://dlmf.nist.gov/1.9.E57
Encodings:: TeX, pMML, png

(principal value), where

1.9.58

$q_{1}=a_{1},$

$q_{2}=(2a_{2}-a_{1}^{2})/2,$

$q_{3}=(3a_{3}-3a_{1}a_{2}+a_{1}^{3})/3,$

Symbols:: $q_{j}$ : coefficients
Permalink:: http://dlmf.nist.gov/1.9.E58
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

and

1.9.59 $q_{n}=(na_{n}-(n-1)a_{1}q_{{n-1}}-(n-2)a_{2}q_{{n-2}}-\cdots-a_{{n-1}}q_{1})/n,$ $n\geq 2$ .

Symbols:: $n$ : nonnegative integer and $q_{j}$ : coefficients
Permalink:: http://dlmf.nist.gov/1.9.E59
Encodings:: TeX, pMML, png

Also,

1.9.60 $(f(z))^{\nu}=p_{0}+p_{1}z+p_{2}z^{2}+\cdots,$

Symbols:: $z$ : variable, $\nu$ : complex and $p_{j}$ : coefficients
Referenced by:: ¶ ‣ §1.9(vi)
Permalink:: http://dlmf.nist.gov/1.9.E60
Encodings:: TeX, pMML, png

(principal value), where $\nu\in\Complex$ ,

1.9.61

$p_{0}=1,$

$p_{1}=\nu a_{1},$

$p_{2}=\nu((\nu-1)a_{1}^{2}+2a_{2})/2,$

Symbols:: $\nu$ : complex and $p_{j}$ : coefficients
Permalink:: http://dlmf.nist.gov/1.9.E61
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

and

1.9.62 $p_{n}=((\nu-n+1)a_{1}p_{{n-1}}+(2\nu-n+2)a_{2}p_{{n-2}}+\dots+((n-1)\nu-1)a_{{n-1}}p_{1}+n\nu a_{n})/n,$ $n\geq 1$ .

Symbols:: $n$ : nonnegative integer, $\nu$ : complex and $p_{j}$ : coefficients
Permalink:: http://dlmf.nist.gov/1.9.E62
Encodings:: TeX, pMML, png

For the definitions of the principal values of $\mathop{\ln\/}\nolimits f(z)$ and $(f(z))^{\nu}$ 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 $f^{{(m)}}(z)=\sum _{{n=0}}^{\infty}\left(n+1\right)_{{m}}a_{{n+m}}(z-z_{0})^{n},$ $\left|z-z_{0}\right|<R$ , $m=0,1,2,\dots$ .

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $z$ : variable, $m$ : nonnegative integer, $n$ : nonnegative integer and $R$ : radius of convergence
Permalink:: http://dlmf.nist.gov/1.9.E63
Encodings:: TeX, pMML, png

§1.9(vii) Inversion of Limits

Notes:: See Copson (1935, pp. 27–30, 95–97). For (1.9.69)–(1.9.71), see Titchmarsh (1962, §1.77).
Keywords:: convergence, infinite sequences, infinite series
Referenced by:: §1.16(i), §1.3(iii)
Permalink:: http://dlmf.nist.gov/1.9.vii

¶ Double Sequences and Series

Keywords:: convergence, double sequence, double series, infinite sequences

A set of complex numbers $\{ z_{{m,n}}\}$ where $m$ and $n$ take all positive integer values is called a double sequence. It converges to $z$ if for every $\epsilon>0$ , there is an integer $N$ such that

1.9.64 $|z_{{m,n}}-z|<\epsilon$

Symbols:: $z$ : variable, $m$ : nonnegative integer, $n$ : nonnegative integer and $\epsilon$ : positive number
Permalink:: http://dlmf.nist.gov/1.9.E64
Encodings:: TeX, pMML, png

for all $m,n\geq N$ . Suppose $\{ z_{{m,n}}\}$ converges to $z$ and the repeated limits

1.9.65

$\lim _{{m\to\infty}}\left(\lim _{{n\to\infty}}z_{{m,n}}\right),$

$\lim _{{n\to\infty}}\left(\lim _{{m\to\infty}}z_{{m,n}}\right)$

Symbols:: $z$ : variable, $m$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.9.E65
Encodings:: TeX, TeX, pMML, pMML, png, png

exist. Then both repeated limits equal $z$ .

A double series is the limit of the double sequence

1.9.66 $z_{{p,q}}=\sum^{p}_{{m=0}}\sum^{q}_{{n=0}}\zeta _{{m,n}}.$

Symbols:: $z$ : variable, $m$ : nonnegative integer, $n$ : nonnegative integer and $\zeta _{{p,q}}$ : sum
Permalink:: http://dlmf.nist.gov/1.9.E66
Encodings:: TeX, pMML, png

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 $\zeta _{{m,n}}$ is replaced by $|\zeta _{{m,n}}|$ .

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

$\sum^{\infty}_{{m=0}}\left(\sum^{\infty}_{{n=0}}\zeta _{{m,n}}\right),$

$\sum^{\infty}_{{n=0}}\left(\sum^{\infty}_{{m=0}}\zeta _{{m,n}}\right).$

Symbols:: $m$ : nonnegative integer, $n$ : nonnegative integer and $\zeta _{{p,q}}$ : sum
Permalink:: http://dlmf.nist.gov/1.9.E67
Encodings:: TeX, TeX, pMML, pMML, png, png

¶ Term-by-Term Integration

Keywords:: compact set, infinite series, integration, point sets in complex plane, term-by-term integration

Suppose the series $\sum^{\infty}_{{n=0}}f_{n}(z)$ , where $f_{n}(z)$ 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 $\int _{C}\sum^{\infty}_{{n=0}}f_{n}(z)dz=\sum^{\infty}_{{n=0}}\int _{C}f_{n}(z)dz$

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $z$ : variable, $n$ : nonnegative integer and $C$ : finite contour in $D$
Permalink:: http://dlmf.nist.gov/1.9.E68
Encodings:: TeX, pMML, png

for any finite contour $C$ in $D$ .

¶ Dominated Convergence Theorem

Keywords:: calculus, dominated convergence theorem, infinite series

Let $(a,b)$ be a finite or infinite interval, and $f_{0}(t),f_{1}(t),\dots$ be real or complex continuous functions, $t\in(a,b)$ . Suppose $\sum^{\infty}_{{n=0}}f_{n}(t)$ converges uniformly in any compact interval in $(a,b)$ , and at least one of the following two conditions is satisfied: