§1.10 Functions of a Complex Variable

§1.10(i) Taylor’s Theorem for Complex Variables

Let $f(z)$ be analytic on the disk $|z-z_{0}|. Then

 1.10.1 $f(z)=\sum^{\infty}_{n=0}\frac{f^{(n)}(z_{0})}{n!}(z-z_{0})^{n}.$ ⓘ Symbols: $!$: factorial (as in $n!$), $z$: variable and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.10.E1 Encodings: TeX, pMML, png See also: Annotations for 1.10(i), 1.10 and 1

The right-hand side is the Taylor series for $f(z)$ at $z=z_{0}$, and its radius of convergence is at least $R$.

Examples

 1.10.2 $e^{z}=1+\frac{z}{1!}+\frac{z^{2}}{2!}+\cdots,$ $|z|<\infty$, ⓘ Symbols: $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$) and $z$: variable Permalink: http://dlmf.nist.gov/1.10.E2 Encodings: TeX, pMML, png See also: Annotations for 1.10(i), 1.10(i), 1.10 and 1
 1.10.3 $\ln\left(1+z\right)=z-\frac{z^{2}}{2}+\frac{z^{3}}{3}-\cdots,$ $|z|<1$, ⓘ Symbols: $\ln\NVar{z}$: principal branch of logarithm function and $z$: variable Permalink: http://dlmf.nist.gov/1.10.E3 Encodings: TeX, pMML, png See also: Annotations for 1.10(i), 1.10(i), 1.10 and 1
 1.10.4 $(1-z)^{-\alpha}=1+\alpha z+\frac{\alpha(\alpha+1)}{2!}z^{2}+\frac{\alpha(% \alpha+1)(\alpha+2)}{3!}z^{3}+\cdots,$ $|z|<1$. ⓘ Symbols: $!$: factorial (as in $n!$) and $z$: variable Referenced by: §1.10(i), §1.10(i) Permalink: http://dlmf.nist.gov/1.10.E4 Encodings: TeX, pMML, png See also: Annotations for 1.10(i), 1.10(i), 1.10 and 1

Note that (1.10.4) is a generalization of the binomial expansion (1.2.2) with the binomial coefficients defined in (1.2.6). Again, in these examples $\ln\left(1+z\right)$ and $(1-z)^{-\alpha}$ have their principal values; see §§4.2(i) and 4.2(iv).

Zeros

An analytic function $f(z)$ has a zero of order (or multiplicity) $m$ ($\geq\!1$) at $z_{0}$ if the first nonzero coefficient in its Taylor series at $z_{0}$ is that of $(z-z_{0})^{m}$. When $m=1$ the zero is simple.

§1.10(ii) Analytic Continuation

Let $f_{1}(z)$ be analytic in a domain $D_{1}$. If $f_{2}(z)$, analytic in $D_{2}$, equals $f_{1}(z)$ on an arc in $D=D_{1}\cap D_{2}$, or on just an infinite number of points with a limit point in $D$, then they are equal throughout $D$ and $f_{2}(z)$ is called an analytic continuation of $f_{1}(z)$. We write $(f_{1},D_{1})$, $(f_{2},D_{2})$ to signify this continuation.

Suppose $z(t)=x(t)+iy(t)$, $a\leq t\leq b$, is an arc and $a=t_{0}. Suppose the subarc $z(t)$, $t\in[t_{j-1},t_{j}]$ is contained in a domain $D_{j}$, $j=1,\dots,n$. The function $f_{1}(z)$ on $D_{1}$ is said to be analytically continued along the path $z(t)$, $a\leq t\leq b$, if there is a chain $(f_{1},D_{1})$, $(f_{2},D_{2}),\dots,(f_{n},D_{n})$.

Analytic continuation is a powerful aid in establishing transformations or functional equations for complex variables, because it enables the problem to be reduced to: (a) deriving the transformation (or functional equation) with real variables; followed by (b) finding the domain on which the transformed function is analytic.

Schwarz Reflection Principle

Let $C$ be a simple closed contour consisting of a segment $\mathit{AB}$ of the real axis and a contour in the upper half-plane joining the ends of $\mathit{AB}$. Also, let $f(z)$ be analytic within $C$, continuous within and on $C$, and real on $\mathit{AB}$. Then $f(z)$ can be continued analytically across $\mathit{AB}$ by reflection, that is,

 1.10.5 $f(\overline{z})=\overline{f(z)}.$ ⓘ Symbols: $\overline{\NVar{z}}$: complex conjugate Permalink: http://dlmf.nist.gov/1.10.E5 Encodings: TeX, pMML, png See also: Annotations for 1.10(ii), 1.10(ii), 1.10 and 1

§1.10(iii) Laurent Series

Suppose $f(z)$ is analytic in the annulus $r_{1}<|z-z_{0}|, $0\leq r_{1}, and $r\in(r_{1},r_{2})$. Then

 1.10.6 $f(z)=\sum^{\infty}_{n=-\infty}a_{n}(z-z_{0})^{n},$ ⓘ Symbols: $z$: variable and $n$: nonnegative integer Referenced by: §1.10(iii) Permalink: http://dlmf.nist.gov/1.10.E6 Encodings: TeX, pMML, png See also: Annotations for 1.10(iii), 1.10 and 1

where

 1.10.7 $a_{n}=\frac{1}{2\pi i}\int_{|z-z_{0}|=r}\frac{f(z)}{(z-z_{0})^{n+1}}\mathrm{d}z,$

and the integration contour is described once in the positive sense. The series (1.10.6) converges uniformly and absolutely on compact sets in the annulus.

Let $r_{1}=0$, so that the annulus becomes the punctured neighborhood $N$: $0<|z-z_{0}|, and assume that $f(z)$ is analytic in $N$, but not at $z_{0}$. Then $z=z_{0}$ is an isolated singularity of $f(z)$. This singularity is removable if $a_{n}=0$ for all $n<0$, and in this case the Laurent series becomes the Taylor series. Next, $z_{0}$ is a pole if $a_{n}\not=0$ for at least one, but only finitely many, negative $n$. If $-n$ is the first negative integer (counting from $-\infty$) with $a_{-n}\not=0$, then $z_{0}$ is a pole of order (or multiplicity) $n$. Lastly, if $a_{n}\not=0$ for infinitely many negative $n$, then $z_{0}$ is an isolated essential singularity.

The singularities of $f(z)$ at infinity are classified in the same way as the singularities of $f(1/z)$ at $z=0$.

An isolated singularity $z_{0}$ is always removable when $\lim_{z\to z_{0}}f(z)$ exists, for example $(\sin z)/z$ at $z=0$.

The coefficient $a_{-1}$ of $(z-z_{0})^{-1}$ in the Laurent series for $f(z)$ is called the residue of $f(z)$ at $z_{0}$, and denoted by $\Residue_{z=z_{0}}[f(z)]$, $\Residue\limits_{z=z_{0}}[f(z)]$, or (when there is no ambiguity) $\Residue[f(z)]$.

A function whose only singularities, other than the point at infinity, are poles is called a meromorphic function. If the poles are infinite in number, then the point at infinity is called an essential singularity: it is the limit point of the poles.

Picard’s Theorem

In any neighborhood of an isolated essential singularity, however small, an analytic function assumes every value in $\mathbb{C}$ with at most one exception.

§1.10(iv) Residue Theorem

If $f(z)$ is analytic within a simple closed contour $C$, and continuous within and on $C$—except in both instances for a finite number of singularities within $C$—then

 1.10.8 $\frac{1}{2\pi i}\int_{C}f(z)\mathrm{d}z=\mbox{sum of the residues of f(z) % within C}.$

Here and elsewhere in this subsection the path $C$ is described in the positive sense.

Phase (or Argument) Principle

If the singularities within $C$ are poles and $f(z)$ is analytic and nonvanishing on $C$, then

 1.10.9 $N-P=\frac{1}{2\pi i}\int_{C}\frac{f^{\prime}(z)}{f(z)}\mathrm{d}z=\frac{1}{2% \pi}\Delta_{C}(\operatorname{ph}f(z)),$

where $N$ and $P$ are respectively the numbers of zeros and poles, counting multiplicity, of $f$ within $C$, and $\Delta_{C}(\operatorname{ph}f(z))$ is the change in any continuous branch of $\operatorname{ph}\left(f(z)\right)$ as $z$ passes once around $C$ in the positive sense. For examples of applications see Olver (1997b, pp. 252–254).

 1.10.10 $\frac{1}{2\pi i}\int_{C}\frac{zf^{\prime}(z)}{f(z)}\mathrm{d}z=\mbox{(sum of % locations of zeros)}-\mbox{(sum of locations of poles)},$

each location again being counted with multiplicity equal to that of the corresponding zero or pole.

Rouché’s Theorem

If $f(z)$ and $g(z)$ are analytic on and inside a simple closed contour $C$, and $|g(z)|<|f(z)|$ on $C$, then $f(z)$ and $f(z)+g(z)$ have the same number of zeros inside $C$.

§1.10(v) Maximum-Modulus Principle

Analytic Functions

If $f(z)$ is analytic in a domain $D$, $z_{0}\in D$ and $|f(z)|\leq|f(z_{0})|$ for all $z\in D$, then $f(z)$ is a constant in $D$.

Let $D$ be a bounded domain with boundary $\partial D$ and let $\overline{D}=D\cup\partial D$. If $f(z)$ is continuous on $\overline{D}$ and analytic in $D$, then $|f(z)|$ attains its maximum on $\partial D$.

Harmonic Functions

If $u(z)$ is harmonic in $D$, $z_{0}\in D$, and $u(z)\leq u(z_{0})$ for all $z\in D$, then $u(z)$ is constant in $D$. Moreover, if $D$ is bounded and $u(z)$ is continuous on $\overline{D}$ and harmonic in $D$, then $u(z)$ is maximum at some point on $\partial D$.

Schwarz’s Lemma

In $|z|, if $f(z)$ is analytic, $|f(z)|\leq M$, and $f(0)=0$, then

 1.10.11 $|f(z)|\leq\frac{M|z|}{R}\;\mbox{ and }\;|f^{\prime}(0)|\leq\frac{M}{R}.$ ⓘ Symbols: $z$: variable and $R$: radius Permalink: http://dlmf.nist.gov/1.10.E11 Encodings: TeX, pMML, png See also: Annotations for 1.10(v), 1.10(v), 1.10 and 1

Equalities hold iff $f(z)=Az$, where $A$ is a constant such that $\left|A\right|=M/R$.

§1.10(vi) Multivalued Functions

Functions which have more than one value at a given point $z$ are called multivalued (or many-valued) functions. Let $F(z)$ be a multivalued function and $D$ be a domain. If we can assign a unique value $f(z)$ to $F(z)$ at each point of $D$, and $f(z)$ is analytic on $D$, then $f(z)$ is a branch of $F(z)$.

Example

$F(z)=\sqrt{z}$ is two-valued for $z\not=0$. If $D=\mathbb{C}\setminus(-\infty,0]$ and $z=re^{i\theta}$, then one branch is $\sqrt{r}e^{i\theta/2}$, the other branch is $-\sqrt{r}e^{i\theta/2}$, with $-\pi<\theta<\pi$ in both cases. Similarly if $D=\mathbb{C}\setminus[0,\infty)$, then one branch is $\sqrt{r}e^{i\theta/2}$, the other branch is $-\sqrt{r}e^{i\theta/2}$, with $0<\theta<2\pi$ in both cases.

A cut domain is one from which the points on finitely many nonintersecting simple contours (§1.9(iii)) have been removed. Each contour is called a cut. A cut neighborhood is formed by deleting a ray emanating from the center. (Or more generally, a simple contour that starts at the center and terminates on the boundary.)

Suppose $F(z)$ is multivalued and $a$ is a point such that there exists a branch of $F(z)$ in a cut neighborhood of $a$, but there does not exist a branch of $F(z)$ in any punctured neighborhood of $a$. Then $a$ is a branch point of $F(z)$. For example, $z=0$ is a branch point of $\sqrt{z}$.

Branches can be constructed in two ways:

(a) By introducing appropriate cuts from the branch points and restricting $F(z)$ to be single-valued in the cut plane (or domain).

(b) By specifying the value of $F(z)$ at a point $z_{0}$ (not a branch point), and requiring $F(z)$ to be continuous on any path that begins at $z_{0}$ and does not pass through any branch points or other singularities of $F(z)$.

If the path circles a branch point at $z=a$, $k$ times in the positive sense, and returns to $z_{0}$ without encircling any other branch point, then its value is denoted conventionally as $F((z_{0}-a)e^{2k\pi i}+a)$.

Example

Let $\alpha$ and $\beta$ be real or complex numbers that are not integers. The function $F(z)=(1-z)^{\alpha}(1+z)^{\beta}$ is many-valued with branch points at $\pm 1$. Branches of $F(z)$ can be defined, for example, in the cut plane $D$ obtained from $\mathbb{C}$ by removing the real axis from $1$ to $\infty$ and from $-1$ to $-\infty$; see Figure 1.10.1. One such branch is obtained by assigning $(1-z)^{\alpha}$ and $(1+z)^{\beta}$ their principal values (§4.2(iv)).

Alternatively, take $z_{0}$ to be any point in $D$ and set $F(z_{0})=e^{\alpha\ln\left(1-z_{0}\right)}e^{\beta\ln\left(1+z_{0}\right)}$ where the logarithms assume their principal values. (Thus if $z_{0}$ is in the interval $(-1,1)$, then the logarithms are real.) Then the value of $F(z)$ at any other point is obtained by analytic continuation.

Thus if $F(z)$ is continued along a path that circles $z=1$ $m$ times in the positive sense and returns to $z_{0}$ without circling $z=-1$, then $F((z_{0}-1)e^{2m\pi i}+1)=e^{\alpha\ln\left(1-z_{0}\right)}e^{\beta\ln\left(1+% z_{0}\right)}e^{2\pi im\alpha}$. If the path also circles $z=-1$ $n$ times in the clockwise or negative sense before returning to $z_{0}$, then the value of $F(z_{0})$ becomes $e^{\alpha\ln\left(1-z_{0}\right)}e^{\beta\ln\left(1+z_{0}\right)}e^{2\pi im% \alpha}e^{-2\pi in\beta}$.

§1.10(vii) Inverse Functions

Lagrange Inversion Theorem

Suppose $f(z)$ is analytic at $z=z_{0}$, $f^{\prime}(z_{0})\not=0$, and $f(z_{0})=w_{0}$. Then the equation

 1.10.12 $f(z)=w$ ⓘ Symbols: $z$: variable and $w$: variable Referenced by: §1.10(vii), §1.10(vii) Permalink: http://dlmf.nist.gov/1.10.E12 Encodings: TeX, pMML, png See also: Annotations for 1.10(vii), 1.10(vii), 1.10 and 1

has a unique solution $z=F(w)$ analytic at $w=w_{0}$, and

 1.10.13 $F(w)=z_{0}+\sum^{\infty}_{n=1}F_{n}(w-w_{0})^{n}$ ⓘ Defines: $F(w)$: inverse function of $f(z)$ (locally) Symbols: $z$: variable, $w$: variable, $n$: nonnegative integer and $F_{n}$: residue Referenced by: §1.10(vii) Permalink: http://dlmf.nist.gov/1.10.E13 Encodings: TeX, pMML, png See also: Annotations for 1.10(vii), 1.10(vii), 1.10 and 1

in a neighborhood of $w_{0}$, where $nF_{n}$ is the residue of $1/(f(z)-f(z_{0}))^{n}$ at $z=z_{0}$. (In other words $nF_{n}$ is the coefficient of $(z-z_{0})^{-1}$ in the Laurent expansion of $1/(f(z)-f(z_{0}))^{n}$ in powers of $(z-z_{0})$; compare §1.10(iii).)

Furthermore, if $g(z)$ is analytic at $z_{0}$, then

 1.10.14 $g(F(w))=g(z_{0})+\sum^{\infty}_{n=1}G_{n}(w-w_{0})^{n},$ ⓘ Symbols: $z$: variable, $w$: variable, $n$: nonnegative integer, $F(w)$: inverse function of $f(z)$ and $G_{n}$: residue Referenced by: §1.10(vii) Permalink: http://dlmf.nist.gov/1.10.E14 Encodings: TeX, pMML, png See also: Annotations for 1.10(vii), 1.10(vii), 1.10 and 1

where $nG_{n}$ is the residue of $g^{\prime}(z)/(f(z)-f(z_{0}))^{n}$ at $z=z_{0}$.

Extended Inversion Theorem

Suppose that

 1.10.15 $f(z)=f(z_{0})+\sum^{\infty}_{n=0}f_{n}(z-z_{0})^{\mu+n},$ ⓘ Symbols: $z$: variable, $n$: nonnegative integer, $f_{n}$: residue and $\mu$: parameter Referenced by: §2.3(iii) Permalink: http://dlmf.nist.gov/1.10.E15 Encodings: TeX, pMML, png See also: Annotations for 1.10(vii), 1.10(vii), 1.10 and 1

where $\mu>0$, $f_{0}\not=0$, and the series converges in a neighborhood of $z_{0}$. (For example, when $\mu$ is an integer $f(z)-f(z_{0})$ has a zero of order $\mu$ at $z_{0}$.) Let $w_{0}=f(z_{0})$. Then (1.10.12) has a solution $z=F(w)$, where

 1.10.16 $F(w)=z_{0}+\sum^{\infty}_{n=1}F_{n}(w-w_{0})^{n/\mu}$

in a neighborhood of $w_{0}$, $nF_{n}$ being the residue of $1/(f(z)-f(z_{0}))^{n/\mu}$ at $z=z_{0}$.

It should be noted that different branches of $(w-w_{0})^{1/\mu}$ used in forming $(w-w_{0})^{n/\mu}$ in (1.10.16) give rise to different solutions of (1.10.12). Also, if in addition $g(z)$ is analytic at $z_{0}$, then

 1.10.17 $g(F(w))=g(z_{0})+\sum^{\infty}_{n=1}G_{n}(w-w_{0})^{n/\mu},$

where $nG_{n}$ is the residue of $g^{\prime}(z)/(f(z)-f(z_{0}))^{n/\mu}$ at $z=z_{0}$.

§1.10(viii) Functions Defined by Contour Integrals

Let $D$ be a domain and $[a,b]$ be a closed finite segment of the real axis. Assume that for each $t\in[a,b]$, $f(z,t)$ is an analytic function of $z$ in $D$, and also that $f(z,t)$ is a continuous function of both variables. Then

 1.10.18 $F(z)=\int^{b}_{a}f(z,t)\mathrm{d}t$ ⓘ Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral and $z$: variable Referenced by: §1.10(viii), §1.10(viii) Permalink: http://dlmf.nist.gov/1.10.E18 Encodings: TeX, pMML, png See also: Annotations for 1.10(viii), 1.10 and 1

is analytic in $D$ and its derivatives of all orders can be found by differentiating under the sign of integration.

This result is also true when $b=\infty$, or when $f(z,t)$ has a singularity at $t=b$, with the following conditions. For each $t\in[a,b)$, $f(z,t)$ is analytic in $D$; $f(z,t)$ is a continuous function of both variables when $z\in D$ and $t\in[a,b)$; the integral (1.10.18) converges at $b$, and this convergence is uniform with respect to $z$ in every compact subset $S$ of $D$.

The last condition means that given $\epsilon$ ($>0$) there exists a number $a_{0}\in[a,b)$ that is independent of $z$ and is such that

 1.10.19 $\left|\int_{a_{1}}^{b}f(z,t)\mathrm{d}t\right|<\epsilon,$ ⓘ Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $z$: variable and $\epsilon$: positive number Permalink: http://dlmf.nist.gov/1.10.E19 Encodings: TeX, pMML, png See also: Annotations for 1.10(viii), 1.10 and 1

for all $a_{1}\in[a_{0},b)$ and all $z\in S$; compare §1.5(iv).

$M$-test

If $|f(z,t)|\leq M(t)$ for $z\in S$ and $\int^{b}_{a}M(t)\mathrm{d}t$ converges, then the integral (1.10.18) converges uniformly and absolutely in $S$.

§1.10(ix) Infinite Products

Let $p_{k,m}=\prod_{n=k}^{m}(1+a_{n})$. If for some $k\geq 1$, $p_{k,m}\to p_{k}\not=0$ as $m\to\infty$, then we say that the infinite product $\prod^{\infty}_{n=1}(1+a_{n})$ converges. (The integer $k$ may be greater than one to allow for a finite number of zero factors.) The convergence of the product is absolute if $\prod^{\infty}_{n=1}(1+|a_{n}|)$ converges. The product $\prod^{\infty}_{n=1}(1+a_{n})$, with $a_{n}\not=-1$ for all $n$, converges iff $\sum^{\infty}_{n=1}\ln\left(1+a_{n}\right)$ converges; and it converges absolutely iff $\sum^{\infty}_{n=1}|a_{n}|$ converges.

Suppose $a_{n}=a_{n}(z)$, $z\in D$, a domain. The convergence of the infinite product is uniform if the sequence of partial products converges uniformly.

$M$-test

Suppose that $a_{n}(z)$ are analytic functions in $D$. If there is an $N$, independent of $z\in D$, such that

 1.10.20 $|\ln\left(1+a_{n}(z)\right)|\leq M_{n},$ $n\geq N$, ⓘ Symbols: $\ln\NVar{z}$: principal branch of logarithm function, $z$: variable and $n$: nonnegative integer Referenced by: §1.10(ix) Permalink: http://dlmf.nist.gov/1.10.E20 Encodings: TeX, pMML, png See also: Annotations for 1.10(ix), 1.10(ix), 1.10 and 1

and

 1.10.21 $\sum^{\infty}_{n=1}M_{n}<\infty,$ ⓘ Symbols: $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.10.E21 Encodings: TeX, pMML, png See also: Annotations for 1.10(ix), 1.10(ix), 1.10 and 1

then the product $\prod^{\infty}_{n=1}(1+a_{n}(z))$ converges uniformly to an analytic function $p(z)$ in $D$, and $p(z)=0$ only when at least one of the factors $1+a_{n}(z)$ is zero in $D$. This conclusion remains true if, in place of (1.10.20), $|a_{n}(z)|\leq M_{n}$ for all $n$, and again $\sum^{\infty}_{n=1}M_{n}<\infty$.

Weierstrass Product

If $\{z_{n}\}$ is a sequence such that $\sum^{\infty}_{n=1}|z_{n}^{-2}|$ is convergent, then

 1.10.22 $P(z)=\prod^{\infty}_{n=1}\left(1-\frac{z}{z_{n}}\right)e^{z/z_{n}}$ ⓘ Symbols: $\mathrm{e}$: base of natural logarithm, $z$: variable and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.10.E22 Encodings: TeX, pMML, png See also: Annotations for 1.10(ix), 1.10(ix), 1.10 and 1

is an entire function with zeros at $z_{n}$.

§1.10(x) Infinite Partial Fractions

Suppose $D$ is a domain, and

 1.10.23 $F(z)=\prod^{\infty}_{n=1}a_{n}(z),$ $z\in D$, ⓘ Defines: $F$: function (locally) Symbols: $\in$: element of, $z$: variable, $n$: nonnegative integer and $D$: domain Permalink: http://dlmf.nist.gov/1.10.E23 Encodings: TeX, pMML, png See also: Annotations for 1.10(x), 1.10 and 1

where $a_{n}(z)$ is analytic for all $n\geq 1$, and the convergence of the product is uniform in any compact subset of $D$. Then $F(z)$ is analytic in $D$.

If, also, $a_{n}(z)\neq 0$ when $n\geq 1$ and $z\in D$, then $F(z)\neq 0$ on $D$ and

 1.10.24 $\frac{F^{\prime}(z)}{F(z)}=\sum^{\infty}_{n=1}\frac{a_{n}^{\prime}(z)}{a_{n}(z% )}.$ ⓘ Symbols: $z$: variable, $n$: nonnegative integer and $F$: function Permalink: http://dlmf.nist.gov/1.10.E24 Encodings: TeX, pMML, png See also: Annotations for 1.10(x), 1.10 and 1

Mittag-Leffler’s Expansion

If $\{a_{n}\}$ and $\{z_{n}\}$ are sequences such that $z_{m}\neq z_{n}$ ($m\neq n$) and $\sum^{\infty}_{n=1}|a_{n}z_{n}^{-2}|$ is convergent, then

 1.10.25 $f(z)=\sum^{\infty}_{n=1}a_{n}\left(\frac{1}{z-z_{n}}+\frac{1}{z_{n}}\right)$ ⓘ Symbols: $z$: variable and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.10.E25 Encodings: TeX, pMML, png See also: Annotations for 1.10(x), 1.10(x), 1.10 and 1

is analytic in $\mathbb{C}$, except for simple poles at $z=z_{n}$ of residue $a_{n}$.