§1.9 Calculus of a Complex Variable

1.9.2		$\mathrm{\Re }z$	$=x,$
		$\mathrm{\Im }z$	$=y.$
ⓘ Defines: $\mathrm{\Im }$: imaginary part and $\mathrm{\Re }$: 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

1.9.3		$x$	$=r\cos \theta ,$
		$y$	$=r\sin \theta ,$
ⓘ Symbols: $\cos z$: cosine function, $\sin z$: 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 $z\ne 0$,

1.9.5		$$\theta =\omega ,\pi -\omega ,-\pi +\omega ,\text{ or }-\omega ,$$
ⓘ 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		$$\omega =\arctan \left(\left|y/x\right|\right)\in [0,\frac{1}{2}\pi ].$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $[a,b]$: closed interval, $\in$: element of, $\arctan z$: arctangent function and $\left|x\right|$: absolute value of $x$ 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

1.9.7		$\left|z\right|$	$=r,$
		$\mathrm{ph}z$	$=\theta +2n\pi ,$
	$n\in \mathbb{Z}$.
ⓘ Defines: $\mathrm{ph}$: phase and $\left|x\right|$: absolute value of $x$ (locally) 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 $\mathrm{ph}z$ corresponds to $n=0$, that is, $-\pi \le \mathrm{ph}z\le \pi$. It is single-valued on $ℂ\setminus \{0\}$, except on the interval $(-\mathrm{\infty },0)$ 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 .)

1.9.8		$\left|\mathrm{\Re }z\right|$	$\le \left|z\right|,$
		$\left|\mathrm{\Im }z\right|$	$\le \left|z\right|,$
ⓘ Symbols: $\mathrm{\Im }$: imaginary part, $\mathrm{\Re }$: real part, $z$: variable and $\left|x\right|$: absolute value of $x$ Permalink: http://dlmf.nist.gov/1.9.E8 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §1.9(i), §1.9(i), §1.9 and Ch.1

1.9.9		$$z=r{\mathrm{e}}^{\mathrm{i}\theta },$$
ⓘ Symbols: $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: 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		$${\mathrm{e}}^{\mathrm{i}\theta }=\cos \theta +\mathrm{i}\sin \theta ;$$
ⓘ Symbols: $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\sin z$: 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.

1.9.11		$\overline{z}$	$=x-\mathrm{i}y,$
ⓘ Defines: $\overline{z}$: complex conjugate Symbols: $\mathrm{i}$: imaginary unit and $z$: 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		$\left|\overline{z}\right|$	$=\left|z\right|,$
ⓘ Symbols: $\overline{z}$: complex conjugate, $z$: variable and $\left|x\right|$: absolute value of $x$ 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		$\mathrm{ph}\overline{z}$	$=-\mathrm{ph}z.$
ⓘ Symbols: $\overline{z}$: complex conjugate, $\mathrm{ph}$: 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

If $z_1=x_1+\mathrm{i}{y}_1$, $z_2=x_2+\mathrm{i}{y}_2$, then

1.9.14		$$z_1\pm z_2=x_1\pm x_2+\mathrm{i}(y_1\pm y_2),$$
ⓘ Symbols: $\mathrm{i}$: imaginary unit and $z$: 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		$$z_1{z}_2=x_1{x}_2-y_1{y}_2+\mathrm{i}(x_1{y}_2+x_2{y}_1),$$
ⓘ Symbols: $\mathrm{i}$: imaginary unit and $z$: 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		$$\frac{z_1}{z_2}=\frac{z_1{\overline{z}}_2}{{\left|z_2\right|}^2}=\frac{x_1{x}_2+y_1{y}_2+\mathrm{i}(x_2{y}_1-x_1{y}_2)}{x_2^2+y_2^2},$$
ⓘ Symbols: $\overline{z}$: complex conjugate, $\mathrm{i}$: imaginary unit, $z$: variable and $\left|x\right|$: absolute value of $x$ 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 $z_2\ne 0$. Also,

1.9.17		$$\left|z_1{z}_2\right|=\left|z_1\right|\left|z_2\right|,$$
ⓘ Symbols: $z$: variable and $\left|x\right|$: absolute value of $x$ 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		$$\mathrm{ph}\left(z_1{z}_2\right)=\mathrm{ph}{z}_1+\mathrm{ph}{z}_2,$$
ⓘ Symbols: $\mathrm{ph}$: 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		$$\left|\frac{z_1}{z_2}\right|=\frac{\left|z_1\right|}{\left|z_2\right|},$$
ⓘ Symbols: $z$: variable and $\left|x\right|$: absolute value of $x$ 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		$$\mathrm{ph}\frac{z_1}{z_2}=\mathrm{ph}{z}_1-\mathrm{ph}{z}_2.$$
ⓘ Symbols: $\mathrm{ph}$: 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.

1.9.21		$$z^n=\left(x^n-\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)x^{n-2}{y}^2+\left(\genfrac{}{}{0.0pt}{}{n}{4}\right)x^{n-4}{y}^4-\mathrm{\cdots }\right)+\mathrm{i}\left(\left(\genfrac{}{}{0.0pt}{}{n}{1}\right)x^{n-1}y-\left(\genfrac{}{}{0.0pt}{}{n}{3}\right)x^{n-3}{y}^3+\mathrm{\cdots }\right),$$
$n=1,2,\mathrm{\dots }$.
ⓘ Symbols: $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\mathrm{i}$: imaginary unit, $z$: 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

1.9.22		$$\cos n\theta +\mathrm{i}\sin n\theta ={(\cos \theta +\mathrm{i}\sin \theta )}^n,$$
$n\in \mathbb{Z}$.
ⓘ Symbols: $\cos z$: cosine function, $\in$: element of, $\mathrm{i}$: imaginary unit, $\mathbb{Z}$: set of all integers, $\sin z$: sine function, $n$: nonnegative integer and $\theta$: angle Permalink: http://dlmf.nist.gov/1.9.E22 Encodings: TeX, pMML, png See also: Annotations for §1.9(i), §1.9(i), §1.9 and Ch.1

1.9.23		$$\left|\left|z_1\right|-\left|z_2\right|\right|\le \left|z_1+z_2\right|\le \left|z_1\right|+\left|z_2\right|.$$
ⓘ Symbols: $z$: variable and $\left|x\right|$: absolute value of $x$ 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

A function $f(z)$ is continuous at a point $z_0$ if $\underset{z\to z_0}{lim}f(z)=f(z_0)$. That is, given any positive number $ϵ$, however small, we can find a positive number $\delta$ such that for all $z$ in the open disk .

A function of two complex variables $f(z,w)$ is continuous at $(z_0,w_0)$ if $\underset{(z,w)\to (z_0,w_0)}{lim}f(z,w)=f(z_0,w_0)$; compare (1.5.1) and (1.5.2).

A neighborhood of a point $z_0$ is a disk . An open set in $ℂ$ 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 $ℂ$ (or $ℂ\cup \mathrm{\infty }$) 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 $ℂ$ 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 $ϵ$ ($>0$) we can find a neighborhood of $z_0$ such that for all points $z$ in the intersection of the neighborhood with $R$.

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

1.9.24		$$f^{\prime }(z)=\frac{df}{dz}=\underset{h\to 0}{lim}\frac{f(z+h)-f(z)}{h}.$$
ⓘ Symbols: $ℂ$: complex plane, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ and $z$: variable Referenced by: §1.9(ii) Permalink: http://dlmf.nist.gov/1.9.E24 Encodings: TeX, pMML, png Modification (effective with 1.2.0): We now emphasise below that $h\to 0$ in $ℂ$. See also: Annotations for §1.9(ii), §1.9(ii), §1.9 and Ch.1

The limit is taken for $h\to 0$ in $ℂ$.

Differentiability automatically implies continuity.

If $f^{\prime }(z)$ exists at $z=x+\mathrm{i}y$ and $f(z)=u(x,y)+\mathrm{i}v(x,y)$, 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: $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$, $\partial x$: 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 $\partial u/\partial x$, $\partial u/\partial y$, $\partial v/\partial x$, and $\partial v/\partial y$ exist, are continuous, and satisfy (1.9.25), then $f(z)$ is differentiable at $z=x+\mathrm{i}y$.

A function $f(z)$ is said to be analytic (holomorphic) at $z=z_0$ if it is complex 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 $ℂ$ is said to be entire.

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

If $f(z)=u(x,y)+\mathrm{i}v(x,y)$ 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 $x$, $\partial x$: partial differential of $x$, $u(x,y)$: function and $v(x,y)$: function A&S Ref: 3.7.32 Permalink: http://dlmf.nist.gov/1.9.E26 Encodings: TeX, pMML, png See also: Annotations for §1.9(ii), §1.9(ii), §1.9 and Ch.1

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 $x$, $\partial x$: partial differential of $x$, $r$: 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 See also: Annotations for §1.9(ii), §1.9(ii), §1.9 and Ch.1

at all points of $D$.

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

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

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		$$f(z_0)=\frac{1}{2\pi \mathrm{i}}{\int }_C\frac{f(z)}{z-z_0}dz,$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{i}$: imaginary unit, $\int$: 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		$$f^{(n)}(z_0)=\frac{n!}{2\pi \mathrm{i}}{\int }_C\frac{f(z)}{{(z-z_0)}^{n+1}}dz,$$
$n=1,2,3,\mathrm{\dots }$,
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\int$: integral, $n$: nonnegative integer and $C$: closed contour Permalink: http://dlmf.nist.gov/1.9.E31 Encodings: TeX, pMML, png See also: Annotations for §1.9(iii), §1.9(iii), §1.9 and Ch.1

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

Any bounded entire function is a constant.

If $C$ is a closed contour, and $z_0\notin C$, then

1.9.32		$$\frac{1}{2\pi \mathrm{i}}{\int }_C\frac{1}{z-z_0}dz=𝒩(C,z_0),$$
ⓘ Defines: $𝒩(C,z_0)$: winding number of $C$ (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{i}$: imaginary unit, $\int$: 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 $𝒩(C,z_0)$ 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 $𝒩(C,z_0)$ is $1$ or $0$ depending whether $z_0$ is inside or outside $C$.

For $u(z)$ harmonic,

1.9.33		$$u(z)=\frac{1}{2\pi }{\int }_0^{2\pi }u(z+r{\mathrm{e}}^{\mathrm{i}\varphi })d\varphi .$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $r$: radius and $u(z)$: harmonic function Permalink: http://dlmf.nist.gov/1.9.E33 Encodings: TeX, pMML, png See also: Annotations for §1.9(iii), §1.9(iii), §1.9 and Ch.1

If $h(w)$ is continuous on $\left|w\right|=R$, then with $z=r{\mathrm{e}}^{\mathrm{i}\theta }$

1.9.34		$$u(r{\mathrm{e}}^{\mathrm{i}\theta })=\frac{1}{2\pi }{\int }_0^{2\pi }\frac{(R^2-r^2)h(R{\mathrm{e}}^{\mathrm{i}\varphi })d\varphi }{R^2-2Rr\cos \left(\varphi -\theta \right)+r^2}$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: 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 See also: Annotations for §1.9(iii), §1.9(iii), §1.9 and Ch.1

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

Suppose $f(z)$ is analytic in a domain $D$ and $C_1,C_2$ are two arcs in $D$ passing through $z_0$. Let $C_1^{\prime },C_2^{\prime }$ 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 $f^{\prime }(z_0)\ne 0$, then the angle between $C_1$ and $C_2$ equals the angle between $C_1^{\prime }$ and $C_2^{\prime }$ both in magnitude and sense. We then say that the mapping $w=f(z)$ is conformal (angle-preserving) at $z_0$.

The linear transformation $f(z)=az+b$, $a\ne 0$, has $f^{\prime }(z)=a$ and $w=f(z)$ maps $ℂ$ conformally onto $ℂ$.

1.9.40		$$w=f(z)=\frac{az+b}{cz+d},$$
$ad-bc\ne 0$, $c\ne 0$.
ⓘ 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		$f(-d/c)$	$=\mathrm{\infty },$
		$f(\mathrm{\infty })$	$=a/c.$
ⓘ 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		$$f^{\prime }(z)=\frac{ad-bc}{{(cz+d)}^2},$$
$z\ne -d/c$.
ⓘ 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		$$f^{\prime }(\mathrm{\infty })=\frac{bc-ad}{c^2}.$$
ⓘ 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 $ℂ\cup \{\mathrm{\infty }\}$ onto itself.

The cross ratio of $z_1,z_2,z_3,z_4\in ℂ\cup \{\mathrm{\infty }\}$ 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.

Suppose $\{M_n\}$ is a sequence of real numbers such that ${\sum }_{n=0}^{\mathrm{\infty }}{M}_n$ converges and $\left|f_n(z)\right|\le M_n$ for all $z\in S$ and all $n\ge 0$. Then the series ${\sum }_{n=0}^{\mathrm{\infty }}{f}_n(z)$ converges uniformly on $S$.

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

When $\sum a_n{z}^n$ and $\sum b_n{z}^n$ both converge

1.9.50		$$\sum _{n=0}^{\mathrm{\infty }}(a_n\pm b_n)z^n=\sum _{n=0}^{\mathrm{\infty }}{a}_n{z}^n\pm \sum _{n=0}^{\mathrm{\infty }}{b}_n{z}^n,$$
ⓘ 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		$$\left(\sum _{n=0}^{\mathrm{\infty }}{a}_n{z}^n\right)\left(\sum _{n=0}^{\mathrm{\infty }}{b}_n{z}^n\right)=\sum _{n=0}^{\mathrm{\infty }}{c}_n{z}^n,$$
ⓘ 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		$$c_n=\sum _{k=0}^n{a}_k{b}_{n-k}.$$
ⓘ 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		$$f(z)=a_0+a_{1}z+a_2{z}^2+\mathrm{\cdots },$$
$a_0\ne 0$.
ⓘ 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 $\left|z\right|$.

1.9.54		$$\frac{1}{f(z)}=b_0+b_{1}z+b_2{z}^2+\mathrm{\cdots },$$
ⓘ 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		$b_0$	$=1/a_0,$
		$b_1$	$=-a_1/a_0^2,$
		$b_2$	$=(a_1^2-a_0{a}_2)/a_0^3,$
ⓘ Permalink: http://dlmf.nist.gov/1.9.E55 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §1.9(vi), §1.9(vi), §1.9 and Ch.1

1.9.56		$$b_n=-(a_1{b}_{n-1}+a_2{b}_{n-2}+\mathrm{\cdots }+a_n{b}_0)/a_0,$$
$n\ge 1$.
ⓘ 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