# §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)

### Real and Imaginary Parts

 1.9.2 $\displaystyle\Re{z}=x,$ $\displaystyle\Im{z}=y.$ 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)

### Polar Representation

 1.9.3 $\displaystyle x$ $\displaystyle=r\mathop{\cos\/}\nolimits\theta,$ $\displaystyle y$ $\displaystyle=r\mathop{\sin\/}\nolimits\theta,$ Symbols: $\mathop{\cos\/}\nolimits\NVar{z}$: cosine function, $\mathop{\sin\/}\nolimits\NVar{z}$: 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)

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)

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)

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: $\pi$: the ratio of the circumference of a circle to its diameter, $[\NVar{a},\NVar{b}]$: closed interval, $\in$: element of and $\mathop{\mathrm{arctan}\/}\nolimits\NVar{z}$: 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)

### Modulus and Phase

 1.9.7 $\displaystyle|z|$ $\displaystyle=r,$ $\displaystyle\mathop{\mathrm{ph}\/}\nolimits z$ $\displaystyle=\theta+2n\pi,$ $n\in\mathbb{Z}$. Defines: $\mathop{\mathrm{ph}\/}\nolimits$: 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)

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 $\mathbb{C}\setminus\{0\}$, 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 $-\pi<\mathop{\mathrm{ph}\/}\nolimits z\leq\pi$.)

 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)
 1.9.9 $z=re^{i\theta},$ Symbols: $\mathrm{e}$: base of exponential function, $z$: 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)

where

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

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) 1.9.12 $\displaystyle|\overline{z}|$ $\displaystyle=|z|,$ 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) 1.9.13 $\displaystyle\mathop{\mathrm{ph}\/}\nolimits\overline{z}$ $\displaystyle=-\mathop{\mathrm{ph}\/}\nolimits z.$ Symbols: $\overline{\NVar{z}}$: 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 See also: Annotations for 1.9(i)

### 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)
 1.9.15 $z_{1}z_{2}=x_{1}x_{2}-y_{1}y_{2}+i(x_{1}y_{2}+x_{2}y_{1}),$ 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)
 1.9.16 $\frac{z_{1}}{z_{2}}=\frac{z_{1}\overline{z}_{2}}{|z_{2}|^{2}}=\frac{x_{1}x_{2}% +y_{1}y_{2}+i(x_{2}y_{1}-x_{1}y_{2})}{x_{2}^{2}+y_{2}^{2}},$ 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)

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)
 1.9.18 $\mathop{\mathrm{ph}\/}\nolimits\!\left(z_{1}z_{2}\right)=\mathop{\mathrm{ph}\/% }\nolimits z_{1}+\mathop{\mathrm{ph}\/}\nolimits z_{2},$ 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 See also: Annotations for 1.9(i)
 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)
 1.9.20 $\mathop{\mathrm{ph}\/}\nolimits\frac{z_{1}}{z_{2}}=\mathop{\mathrm{ph}\/}% \nolimits z_{1}-\mathop{\mathrm{ph}\/}\nolimits z_{2}.$ 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 See also: Annotations for 1.9(i)

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

### DeMoivre’s Theorem

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

### 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(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)

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

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)

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)

### 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)

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)

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

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)

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)
 1.9.37 $\infty\cdot z=z\cdot\infty=\infty,$ $z\not=0$, Symbols: $z$: variable Permalink: http://dlmf.nist.gov/1.9.E37 Encodings: TeX, pMML, png See also: Annotations for 1.9(iv)
 1.9.38 $z/\infty=0,$ Symbols: $z$: variable Permalink: http://dlmf.nist.gov/1.9.E38 Encodings: TeX, pMML, png See also: Annotations for 1.9(iv)
 1.9.39 $z/0=\infty,$ $z\neq 0$. Symbols: $z$: variable Permalink: http://dlmf.nist.gov/1.9.E39 Encodings: TeX, pMML, png See also: Annotations for 1.9(iv)

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)
 1.9.41 $\displaystyle f(-d/c)$ $\displaystyle=\infty,$ $\displaystyle f(\infty)$ $\displaystyle=a/c.$ Permalink: http://dlmf.nist.gov/1.9.E41 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 1.9(iv)
 1.9.42 $f^{\prime}(z)=\frac{ad-bc}{(cz+d)^{2}},$ $z\not=-d/c$. Symbols: $z$: variable Permalink: http://dlmf.nist.gov/1.9.E42 Encodings: TeX, pMML, png See also: Annotations for 1.9(iv)
 1.9.43 $f^{\prime}(\infty)=\frac{bc-ad}{c^{2}}.$ Permalink: http://dlmf.nist.gov/1.9.E43 Encodings: TeX, pMML, png See also: Annotations for 1.9(iv)
 1.9.44 $z=\frac{dw-b}{-cw+a}.$ 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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)
 1.9.56 $b_{n}=-(a_{1}b_{n-1}+a_{2}b_{n-2}+\dots+a_{n}b_{0})/a_{0},$ $n\geq 1$. Symbols: $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.9.E56 Encodings: TeX, pMML, png See also: Annotations for 1.9(vi)

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: $\mathop{\ln\/}\nolimits\NVar{z}$: 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)

(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)

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)

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)

(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)

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)

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|, $m=0,1,2,\dots$.

## §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)

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)

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)

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)

### 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)
 1.9.70 $\sum^{\infty}_{n=0}\int^{b}_{a}|f_{n}(t)|\mathrm{d}t<\infty.$ 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)

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)