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).

An analytic function $f(z)$ has a zero of order (or multiplicity) $m$ ($\ge 1$) 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.

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

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

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

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

In addition,

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

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

If $f(z)$ is analytic in a domain $D$, $z_0\in D$ and $\left|f(z)\right|\le \left|f(z_0)\right|$ 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 $\left|f(z)\right|$ attains its maximum on $\partial D$.

If $u(z)$ is harmonic in $D$, $z_0\in D$, and $u(z)\le u(z_0)$ 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 $\partial D$.

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

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

$F(z)=\sqrt{z}$ is two-valued for $z\ne 0$. If $D=ℂ\setminus (-\mathrm{\infty },0]$ and $z=r{\mathrm{e}}^{\mathrm{i}\theta }$, then one branch is $\sqrt{r}{\mathrm{e}}^{\mathrm{i}\theta /2}$, the other branch is $-\sqrt{r}{\mathrm{e}}^{\mathrm{i}\theta /2}$, with in both cases. Similarly if $D=ℂ\setminus [0,\mathrm{\infty })$, then one branch is $\sqrt{r}{\mathrm{e}}^{\mathrm{i}\theta /2}$, the other branch is $-\sqrt{r}{\mathrm{e}}^{\mathrm{i}\theta /2}$, with 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 $F((z_0-a){\mathrm{e}}^{2k\pi \mathrm{i}}+a)$.

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 $ℂ$ by removing the real axis from $1$ to $\mathrm{\infty }$ and from $-1$ to $-\mathrm{\infty }$; 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 $F(z_0)={\mathrm{e}}^{\alpha \ln \left(1-z_0\right)}{\mathrm{e}}^{\beta \ln \left(1+z_0\right)}$ 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 $F((z_0-1){\mathrm{e}}^{2m\pi \mathrm{i}}+1)={\mathrm{e}}^{\alpha \ln \left(1-z_0\right)}{\mathrm{e}}^{\beta \ln \left(1+z_0\right)}{\mathrm{e}}^{2\pi \mathrm{i}m\alpha }$. 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 ${\mathrm{e}}^{\alpha \ln \left(1-z_0\right)}{\mathrm{e}}^{\beta \ln \left(1+z_0\right)}{\mathrm{e}}^{2\pi \mathrm{i}m\alpha }{\mathrm{e}}^{-2\pi \mathrm{i}n\beta }$.

Suppose $f(z)$ is analytic at $z=z_0$, $f^{\prime }(z_0)\ne 0$, and $f(z_0)=w_0$. Then the equation

has a unique solution $z=F(w)$ analytic at $w=w_0$, and

in a neighborhood of $w_0$, where $n{F}_n$ is the residue of $1/{(f(z)-f(z_0))}^n$ at $z=z_0$. (In other words $n{F}_n$ 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

where $n{G}_n$ is the residue of $g^{\prime }(z)/{(f(z)-f(z_0))}^n$ at $z=z_0$.

Suppose that

where $\mu >0$, $f_0\ne 0$, 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

in a neighborhood of $w_0$, $n{F}_n$ 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

where $n{G}_n$ is the residue of $g^{\prime }(z)/{(f(z)-f(z_0))}^{n/\mu }$ at $z=z_0$.

If $\left|f(z,t)\right|\le M(t)$ for $z\in S$ and ${\int }_a^{b}M(t)dt$ converges, then the integral (1.10.18) converges uniformly and absolutely in $S$.

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

and

then the product ${\prod }_{n=1}^{\mathrm{\infty }}(1+a_n(z))$ 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), $\left|a_n(z)\right|\le M_n$ for all $n$, and again .

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

is an entire function with zeros at $z_n$.

If $\{a_n\}$ and $\{z_n\}$ are sequences such that $z_m\ne z_n$ ($m\ne n$) and ${\sum }_{n=1}^{\mathrm{\infty }}\left|a_n{z}_n^{-2}\right|$ is convergent, then

is analytic in $ℂ$, except for simple poles at $z=z_n$ of residue $a_n$.

Ultraspherical polynomials have generating function

Many properties are a direct consequence of this representation: Taking the $x$-derivative gives us

and hence $\frac{d}{dx}{C}_n^{(\lambda )}\left(x\right)=2\lambda C_{n-1}^{(\lambda +1)}\left(x\right)$, that is (18.9.19). The recurrence relation for $C_n^{(\lambda )}\left(x\right)$ in §18.9(i) follows from $(1-2xz+z^2)\frac{\partial }{\partial z}F(x,\lambda ;z)=2\lambda (x-z)F(x,\lambda ;z)$, and the contour integral representation for $C_n^{(\lambda )}\left(x\right)$ in §18.10(iii) is just (1.10.27).