# §1.9 Calculus of a Complex Variable

## §1.9(i) Complex Numbers

 1.9.1 $z=x+iy,$ $x,y\in\mathbb{R}$. ⓘ Symbols: $\in$: element of, $\mathbb{R}$: real line and $z$: variable A&S Ref: 3.7.1 Permalink: http://dlmf.nist.gov/1.9.E1 Encodings: TeX, pMML, png See also: Annotations for §1.9(i), §1.9 and Ch.1

### Real and Imaginary Parts

 1.9.2 $\displaystyle\Re z$ $\displaystyle=x,$ $\displaystyle\Im z$ $\displaystyle=y.$ ⓘ Defines: $\Im$: imaginary part and $\Re$: 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 See also: Annotations for §1.9(i), §1.9(i), §1.9 and Ch.1

### Polar Representation

 1.9.3 $\displaystyle x$ $\displaystyle=r\cos\theta,$ $\displaystyle y$ $\displaystyle=r\sin\theta,$ ⓘ Symbols: $\cos\NVar{z}$: cosine function, $\sin\NVar{z}$: 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 See also: Annotations for §1.9(i), §1.9(i), §1.9 and Ch.1

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 See also: Annotations for §1.9(i), §1.9(i), §1.9 and Ch.1

and when $z\neq 0$,

 1.9.5 $\theta=\omega,\;\;\pi-\omega,\;\;-\pi+\omega,\mbox{ or }-\omega,$ ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter and $\theta$: angle Permalink: http://dlmf.nist.gov/1.9.E5 Encodings: TeX, pMML, png See also: Annotations for §1.9(i), §1.9(i), §1.9 and Ch.1

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

 1.9.6 $\omega=\operatorname{arctan}\left(|y/x|\right)\in\left[0,\tfrac{1}{2}\pi\right].$ ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $[\NVar{a},\NVar{b}]$: closed interval, $\in$: element of and $\operatorname{arctan}\NVar{z}$: arctangent function A&S Ref: 3.7.4 Permalink: http://dlmf.nist.gov/1.9.E6 Encodings: TeX, pMML, png See also: Annotations for §1.9(i), §1.9(i), §1.9 and Ch.1

### Modulus and Phase

 1.9.7 $\displaystyle|z|$ $\displaystyle=r,$ $\displaystyle\operatorname{ph}z$ $\displaystyle=\theta+2n\pi,$ $n\in\mathbb{Z}$. ⓘ Defines: $\operatorname{ph}$: phase Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\in$: element of, $\mathbb{Z}$: 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 See also: Annotations for §1.9(i), §1.9(i), §1.9 and Ch.1

The principal value of $\operatorname{ph}z$ corresponds to $n=0$, that is, $-\pi\leq\operatorname{ph}z\leq\pi$. It is single-valued on $\mathbb{C}\setminus\{0\}$, 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 $-\pi<\operatorname{ph}z\leq\pi$.)

 1.9.8 $\displaystyle|\Re z|$ $\displaystyle\leq|z|,$ $\displaystyle|\Im z|$ $\displaystyle\leq|z|,$ ⓘ Symbols: $\Im$: imaginary part, $\Re$: real part and $z$: variable 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 and Ch.1
 1.9.9 $z=re^{i\theta},$ ⓘ Symbols: $\mathrm{e}$: base of natural logarithm, $z$: variable, $r$: radius and $\theta$: angle Permalink: http://dlmf.nist.gov/1.9.E9 Encodings: TeX, pMML, png See also: Annotations for §1.9(i), §1.9(i), §1.9 and Ch.1

where

 1.9.10 $e^{i\theta}=\cos\theta+i\sin\theta;$ ⓘ Symbols: $\cos\NVar{z}$: cosine function, $\mathrm{e}$: base of natural logarithm, $\sin\NVar{z}$: sine function and $\theta$: angle Permalink: http://dlmf.nist.gov/1.9.E10 Encodings: TeX, pMML, png See also: Annotations for §1.9(i), §1.9(i), §1.9 and Ch.1

see §4.14.

### Complex Conjugate

 1.9.11 $\displaystyle\overline{z}$ $\displaystyle=x-iy,$ ⓘ Defines: $\overline{\NVar{z}}$: complex conjugate Symbols: $z$: variable A&S Ref: 3.7.7 Permalink: http://dlmf.nist.gov/1.9.E11 Encodings: TeX, pMML, png See also: Annotations for §1.9(i), §1.9(i), §1.9 and Ch.1 1.9.12 $\displaystyle|\overline{z}|$ $\displaystyle=|z|,$ ⓘ Symbols: $\overline{\NVar{z}}$: complex conjugate and $z$: variable A&S Ref: 3.7.8 Permalink: http://dlmf.nist.gov/1.9.E12 Encodings: TeX, pMML, png See also: Annotations for §1.9(i), §1.9(i), §1.9 and Ch.1 1.9.13 $\displaystyle\operatorname{ph}\overline{z}$ $\displaystyle=-\operatorname{ph}z.$ ⓘ Symbols: $\overline{\NVar{z}}$: complex conjugate, $\operatorname{ph}$: phase and $z$: variable A&S Ref: 3.7.9 Permalink: http://dlmf.nist.gov/1.9.E13 Encodings: TeX, pMML, png See also: Annotations for §1.9(i), §1.9(i), §1.9 and Ch.1

### Arithmetic Operations

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

 1.9.14 $z_{1}\pm z_{2}=x_{1}\pm x_{2}+\mathrm{i}(y_{1}\pm y_{2}),$ ⓘ Symbols: $z$: variable Permalink: http://dlmf.nist.gov/1.9.E14 Encodings: TeX, pMML, png See also: Annotations for §1.9(i), §1.9(i), §1.9 and Ch.1
 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 See also: Annotations for §1.9(i), §1.9(i), §1.9 and Ch.1
 1.9.16 $\frac{z_{1}}{z_{2}}=\frac{z_{1}\overline{z}_{2}}{|z_{2}|^{2}}=\frac{x_{1}x_{2}% +y_{1}y_{2}+i(x_{2}y_{1}-x_{1}y_{2})}{x_{2}^{2}+y_{2}^{2}},$ ⓘ Symbols: $\overline{\NVar{z}}$: complex conjugate and $z$: variable A&S Ref: 3.7.13 Permalink: http://dlmf.nist.gov/1.9.E16 Encodings: TeX, pMML, png See also: Annotations for §1.9(i), §1.9(i), §1.9 and Ch.1

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 See also: Annotations for §1.9(i), §1.9(i), §1.9 and Ch.1
 1.9.18 $\operatorname{ph}\left(z_{1}z_{2}\right)=\operatorname{ph}z_{1}+\operatorname{% ph}z_{2},$ ⓘ Symbols: $\operatorname{ph}$: phase and $z$: variable A&S Ref: 3.7.12 Referenced by: §1.9(i), (25.10.2) Permalink: http://dlmf.nist.gov/1.9.E18 Encodings: TeX, pMML, png See also: Annotations for §1.9(i), §1.9(i), §1.9 and Ch.1
 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 See also: Annotations for §1.9(i), §1.9(i), §1.9 and Ch.1
 1.9.20 $\operatorname{ph}\frac{z_{1}}{z_{2}}=\operatorname{ph}z_{1}-\operatorname{ph}z% _{2}.$ ⓘ Symbols: $\operatorname{ph}$: phase and $z$: variable A&S Ref: 3.7.15 Referenced by: §1.9(i), (25.10.2) Permalink: http://dlmf.nist.gov/1.9.E20 Encodings: TeX, pMML, png See also: Annotations for §1.9(i), §1.9(i), §1.9 and Ch.1

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 $z^{n}=\left(x^{n}-\genfrac{(}{)}{0.0pt}{}{n}{2}x^{n-2}y^{2}+\genfrac{(}{)}{0.0% pt}{}{n}{4}x^{n-4}y^{4}-\cdots\right)+i\left(\genfrac{(}{)}{0.0pt}{}{n}{1}x^{n% -1}y-\genfrac{(}{)}{0.0pt}{}{n}{3}x^{n-3}y^{3}+\cdots\right),$ $n=1,2,\dots$. ⓘ Symbols: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$: 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 See also: Annotations for §1.9(i), §1.9(i), §1.9 and Ch.1

### DeMoivre’s Theorem

 1.9.22 $\cos n\theta+i\sin n\theta=(\cos\theta+i\sin\theta)^{n},$ $n\in\mathbb{Z}$.

### 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 See also: Annotations for §1.9(i), §1.9(i), §1.9 and Ch.1

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

### Continuity

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 $\mathbb{C}$

A neighborhood of a point $z_{0}$ is a disk $\left|z-z_{0}\right|<\delta$. An open set in $\mathbb{C}$ 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 $\mathbb{C}$ (or $\mathbb{C}\cup\infty$) 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 $\mathbb{C}$ 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 $z_{0}$ 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

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

 1.9.24 $f^{\prime}(z)=\frac{\mathrm{d}f}{\mathrm{d}z}=\lim_{h\to 0}\frac{f(z+h)-f(z)}{% h}.$ ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative of $f$ with respect to $x$ and $z$: variable Permalink: http://dlmf.nist.gov/1.9.E24 Encodings: TeX, pMML, png See also: Annotations for §1.9(ii), §1.9(ii), §1.9 and Ch.1

Differentiability automatically implies continuity.

### Cauchy–Riemann Equations

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

 1.9.25 $\displaystyle\frac{\partial u}{\partial x}$ $\displaystyle=\frac{\partial v}{\partial y},$ $\displaystyle\frac{\partial u}{\partial y}$ $\displaystyle=-\frac{\partial v}{\partial x}$ ⓘ Symbols: $\frac{\partial\NVar{f}}{\partial\NVar{x}}$: partial derivative of $f$ with respect to $x$, $\partial\NVar{x}$: partial differential of $x$, $u(x,y)$: 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 See also: Annotations for §1.9(ii), §1.9(ii), §1.9 and Ch.1

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

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 $\mathbb{C}$ 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

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

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

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

at all points of $D$.

## §1.9(iii) Integration

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 $\int_{C}f(z)\mathrm{d}z=\int_{a}^{b}f(z(t))(x^{\prime}(t)+iy^{\prime}(t))% \mathrm{d}t,$ ⓘ Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral and $C$: closed contour Permalink: http://dlmf.nist.gov/1.9.E28 Encodings: TeX, pMML, png See also: Annotations for §1.9(iii), §1.9 and Ch.1

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

Any simple closed contour $C$ divides $\mathbb{C}$ 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

 1.9.29 $\int_{C}f(z)\mathrm{d}z=0.$ ⓘ Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral and $C$: closed contour Permalink: http://dlmf.nist.gov/1.9.E29 Encodings: TeX, pMML, png See also: Annotations for §1.9(iii), §1.9(iii), §1.9 and Ch.1

### 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 $f(z_{0})=\frac{1}{2\pi i}\int_{C}\frac{f(z)}{z-z_{0}}\mathrm{d}z,$ ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral and $C$: closed contour Referenced by: §2.3(iii) Permalink: http://dlmf.nist.gov/1.9.E30 Encodings: TeX, pMML, png See also: Annotations for §1.9(iii), §1.9(iii), §1.9 and Ch.1

and

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

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 $z_{0}\not\in C$, then

 1.9.32 $\frac{1}{2\pi i}\int_{C}\frac{1}{z-z_{0}}\mathrm{d}z=\mathcal{N}(C,z_{0}),$ ⓘ Defines: $\mathcal{N}(C,z_{0})$: winding number of $C$ (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral and $C$: closed contour Permalink: http://dlmf.nist.gov/1.9.E32 Encodings: TeX, pMML, png See also: Annotations for §1.9(iii), §1.9(iii), §1.9 and Ch.1

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

For $u(z)$ harmonic,

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

### Poisson Integral

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

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

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

## §1.9(iv) Conformal Mapping

The extended complex plane, $\mathbb{C}\,\cup\,\{\infty\}$, consists of the points of the complex plane $\mathbb{C}$ 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. ⓘ Symbols: $\cup$: union, $z$: variable, $r$: radius and $S_{r}$: neighborhood Permalink: http://dlmf.nist.gov/1.9.E35 Encodings: TeX, pMML, png See also: Annotations for §1.9(iv), §1.9 and Ch.1

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 See also: Annotations for §1.9(iv), §1.9 and Ch.1
 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 See also: Annotations for §1.9(iv), §1.9 and Ch.1
 1.9.38 $z/\infty=0,$ ⓘ Symbols: $z$: variable Permalink: http://dlmf.nist.gov/1.9.E38 Encodings: TeX, pMML, png See also: Annotations for §1.9(iv), §1.9 and Ch.1
 1.9.39 $z/0=\infty,$ $z\neq 0$. ⓘ Symbols: $z$: variable Permalink: http://dlmf.nist.gov/1.9.E39 Encodings: TeX, pMML, png See also: Annotations for §1.9(iv), §1.9 and Ch.1

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

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 $\mathbb{C}$ conformally onto $\mathbb{C}$.

### Bilinear 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 See also: Annotations for §1.9(iv), §1.9(iv), §1.9 and Ch.1
 1.9.41 $\displaystyle f(-d/c)$ $\displaystyle=\infty,$ $\displaystyle f(\infty)$ $\displaystyle=a/c.$ ⓘ Permalink: http://dlmf.nist.gov/1.9.E41 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §1.9(iv), §1.9(iv), §1.9 and Ch.1
 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 See also: Annotations for §1.9(iv), §1.9(iv), §1.9 and Ch.1
 1.9.43 $f^{\prime}(\infty)=\frac{bc-ad}{c^{2}}.$ ⓘ Permalink: http://dlmf.nist.gov/1.9.E43 Encodings: TeX, pMML, png See also: Annotations for §1.9(iv), §1.9(iv), §1.9 and Ch.1
 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 See also: Annotations for §1.9(iv), §1.9(iv), §1.9 and Ch.1

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

The cross ratio of $z_{1},z_{2},z_{3},z_{4}\in\mathbb{C}\cup\{\infty\}$ 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 See also: Annotations for §1.9(iv), §1.9(iv), §1.9 and Ch.1

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 $\{z_{n}\}$ 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 See also: Annotations for §1.9(v), §1.9 and Ch.1

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 See also: Annotations for §1.9(v), §1.9 and Ch.1

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

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

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}| 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 $|z-z_{0}|\leq\rho$ ($), the convergence is absolute and uniform. Moreover,

 1.9.48 $a_{n}=\frac{f^{(n)}(z_{0})}{n!},$ ⓘ Symbols: $!$: factorial (as in $n!$), $z$: variable and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.9.E48 Encodings: TeX, pMML, png See also: Annotations for §1.9(vi), §1.9 and Ch.1

and

 1.9.49 $R=\liminf_{n\to\infty}|a_{n}|^{-1/n}.$ ⓘ Defines: $R$: radius of convergence (locally) Symbols: $\liminf$: least limit point and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.9.E49 Encodings: TeX, pMML, png See also: Annotations for §1.9(vi), §1.9 and Ch.1

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

### Operations

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 See also: Annotations for §1.9(vi), §1.9(vi), §1.9 and Ch.1

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 See also: Annotations for §1.9(vi), §1.9(vi), §1.9 and Ch.1

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 See also: Annotations for §1.9(vi), §1.9(vi), §1.9 and Ch.1

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 See also: Annotations for §1.9(vi), §1.9(vi), §1.9 and Ch.1

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 See also: Annotations for §1.9(vi), §1.9(vi), §1.9 and Ch.1

where

 1.9.55 $\displaystyle b_{0}$ $\displaystyle=1/a_{0},$ $\displaystyle b_{1}$ $\displaystyle=-a_{1}/a_{0}^{2},$ $\displaystyle b_{2}$ $\displaystyle=(a_{1}^{2}-a_{0}a_{2})/a_{0}^{3},$ ⓘ Permalink: http://dlmf.nist.gov/1.9.E55 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §1.9(vi), §1.9(vi), §1.9 and Ch.1
 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 See also: Annotations for §1.9(vi), §1.9(vi), §1.9 and Ch.1

With $a_{0}=1$,

 1.9.57 $\ln f(z)=q_{1}z+q_{2}z^{2}+q_{3}z^{3}+\cdots,$ ⓘ Symbols: $\ln\NVar{z}$: 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 See also: Annotations for §1.9(vi), §1.9(vi), §1.9 and Ch.1

(principal value), where

 1.9.58 $\displaystyle q_{1}$ $\displaystyle=a_{1},$ $\displaystyle q_{2}$ $\displaystyle=(2a_{2}-a_{1}^{2})/2,$ $\displaystyle q_{3}$ $\displaystyle=(3a_{3}-3a_{1}a_{2}+a_{1}^{3})/3,$ ⓘ Symbols: $q_{j}$: coefficients Permalink: http://dlmf.nist.gov/1.9.E58 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §1.9(vi), §1.9(vi), §1.9 and Ch.1

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 See also: Annotations for §1.9(vi), §1.9(vi), §1.9 and Ch.1

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 See also: Annotations for §1.9(vi), §1.9(vi), §1.9 and Ch.1

(principal value), where $\nu\in\mathbb{C}$,

 1.9.61 $\displaystyle p_{0}$ $\displaystyle=1,$ $\displaystyle p_{1}$ $\displaystyle=\nu a_{1},$ $\displaystyle p_{2}$ $\displaystyle=\nu((\nu-1)a_{1}^{2}+2a_{2})/2,$ ⓘ 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 See also: Annotations for §1.9(vi), §1.9(vi), §1.9 and Ch.1

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 See also: Annotations for §1.9(vi), §1.9(vi), §1.9 and Ch.1

For the definitions of the principal values of $\ln f(z)$ 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|, $m=0,1,2,\dots$.

## §1.9(vii) Inversion of Limits

### Double Sequences and Series

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 See also: Annotations for §1.9(vii), §1.9(vii), §1.9 and Ch.1

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 See also: Annotations for §1.9(vii), §1.9(vii), §1.9 and Ch.1

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 See also: Annotations for §1.9(vii), §1.9(vii), §1.9 and Ch.1

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 See also: Annotations for §1.9(vii), §1.9(vii), §1.9 and Ch.1

### 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 $\int_{C}\sum^{\infty}_{n=0}f_{n}(z)\mathrm{d}z=\sum^{\infty}_{n=0}\int_{C}f_{n% }(z)\mathrm{d}z$

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

### Dominated Convergence Theorem

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:

 1.9.69 $\int^{b}_{a}\sum^{\infty}_{n=0}|f_{n}(t)|\mathrm{d}t<\infty,$ ⓘ Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral and $n$: nonnegative integer Referenced by: §1.9(vii) Permalink: http://dlmf.nist.gov/1.9.E69 Encodings: TeX, pMML, png See also: Annotations for §1.9(vii), §1.9(vii), §1.9 and Ch.1
 1.9.70 $\sum^{\infty}_{n=0}\int^{b}_{a}|f_{n}(t)|\mathrm{d}t<\infty.$ ⓘ Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.9.E70 Encodings: TeX, pMML, png See also: Annotations for §1.9(vii), §1.9(vii), §1.9 and Ch.1

Then

 1.9.71 $\int^{b}_{a}\sum^{\infty}_{n=0}f_{n}(t)\mathrm{d}t=\sum^{\infty}_{n=0}\int^{b}% _{a}f_{n}(t)\mathrm{d}t.$ ⓘ Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral and $n$: nonnegative integer Referenced by: §1.9(vii) Permalink: http://dlmf.nist.gov/1.9.E71 Encodings: TeX, pMML, png See also: Annotations for §1.9(vii), §1.9(vii), §1.9 and Ch.1