1 Algebraic and Analytic Methods Areas 1.9 Calculus of a Complex Variable 1.11 Zeros of Polynomials

§1.10 Functions of a Complex Variable

Keywords:: functions
Permalink:: http://dlmf.nist.gov/1.10

§1.10(i) Taylor’s Theorem for Complex Variables
§1.10(ii) Analytic Continuation
§1.10(iii) Laurent Series
§1.10(iv) Residue Theorem
§1.10(v) Maximum-Modulus Principle
§1.10(vi) Multivalued Functions
§1.10(vii) Inverse Functions
§1.10(viii) Functions Defined by Contour Integrals
§1.10(ix) Infinite Products
§1.10(x) Infinite Partial Fractions

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

Notes:: See Copson (1935, pp. 72–75) and Levinson and Redheffer (1970, pp. 140–143).
Keywords:: Taylor series, Taylor’s theorem
Referenced by:: §1.9(vi), §12.13(i), §25.11(iv), §3.7(ii), §3.8(i)
Permalink:: http://dlmf.nist.gov/1.10.i

Let be analytic on the disk $|z-z_{0}|<R$ . Then

1.10.1 $f(z)=\sum^{\infty}_{{n=0}}\frac{f^{{(n)}}(z_{0})}{n!}(z-z_{0})^{n}.$

Symbols:: : : factorial, : variable and : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.10.E1
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The right-hand side is the Taylor series for at $z=z_{0}$ , and its radius of convergence is at least .

¶ Examples

1.10.2 $e^{z}=1+\frac{z}{1!}+\frac{z^{2}}{2!}+\cdots,$ $|z|<\infty$ ,

Symbols:: : base of exponential function, : : factorial and : variable
Permalink:: http://dlmf.nist.gov/1.10.E2
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1.10.3 $\mathop{\ln\/}\nolimits\!\left(1+z\right)=z-\frac{z^{2}}{2}+\frac{z^{3}}{3}-\cdots,$

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and : variable
Permalink:: http://dlmf.nist.gov/1.10.E3
Encodings:: TeX, pMML, png

1.10.4 $(1-z)^{{-\alpha}}=1+\alpha z+\frac{\alpha(\alpha+1)}{2!}z^{2}+\frac{\alpha(% \alpha+1)(\alpha+2)}{3!}z^{3}+\cdots,$

Symbols:: : : factorial and : variable
Permalink:: http://dlmf.nist.gov/1.10.E4
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Again, in these examples $\mathop{\ln\/}\nolimits\!\left(1+z\right)$ and $(1-z)^{{-\alpha}}$ have their principal values; see §§4.2(i) and 4.2(iv).

¶ Zeros

Keywords:: analytic function, simple zero, zeros of analytic functions

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

§1.10(ii) Analytic Continuation

Notes:: See Levinson and Redheffer (1970, pp. 398–402) and Copson (1935, pp. 192–193).
Keywords:: analytic continuation, functions
Referenced by:: §10.11, §4.23(iv)
Permalink:: http://dlmf.nist.gov/1.10.ii

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 , then they are equal throughout 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 , $a\leq t\leq b$ , is an arc and $a=t_{0}<t_{1}<\dots<t_{n}=b$ . Suppose the subarc , $t\in[t_{{j-1}},t_{j}]$ 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 , $a\leq t\leq b$ , 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

Keywords:: Schwarz reflection principle, analytic continuation

Let 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 be analytic within , continuous within and on , and real on $\mathit{AB}$ . Then can be continued analytically across $\mathit{AB}$ by reflection, that is,

1.10.5 $f(\conj{z})=\conj{f(z)}.$

Symbols:: $\conj{z}$ : complex conjugate
Permalink:: http://dlmf.nist.gov/1.10.E5
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§1.10(iii) Laurent Series

Notes:: See Levinson and Redheffer (1970, pp. 162–170), Copson (1935, pp. 75–81, 438–440), and Markushevich (1983, pp. 234–245).
Defines:: $\Residue$ : residue
Keywords:: Laurent series, analytic function, annulus, essential singularity, functions, isolated essential singularity, isolated singularity, meromorphic function, neighborhood, pole, punctured neighborhood, removable singularity, residue, singularities, singularity
Referenced by:: ¶ ‣ §1.10(vii), §32.2(i)
Permalink:: http://dlmf.nist.gov/1.10.iii

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

1.10.6 $f(z)=\sum^{\infty}_{{n=-\infty}}a_{n}(z-z_{0})^{n},$

Symbols:: : variable and : nonnegative integer
Referenced by:: §1.10(iii)
Permalink:: http://dlmf.nist.gov/1.10.E6
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where

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

Symbols:: : differential of , $\int$ : integral, : variable, : nonnegative integer and $r_{1}$ , $r_{2}$ : annulus radii
Permalink:: http://dlmf.nist.gov/1.10.E7
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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 : $0<|z-z_{0}|<r_{2}$ , and assume that is analytic in , but not at $z_{0}$ . Then $z=z_{0}$ is an isolated singularity of . This singularity is removable if $a_{n}=0$ for all , 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 . If is the first negative integer (counting from $-\infty$ ) with $a_{{-n}}\not=0$ , then $z_{0}$ is a pole of order (or multiplicity) . Lastly, if $a_{n}\not=0$ for infinitely many negative , then $z_{0}$ is an isolated essential singularity.

The singularities of at infinity are classified in the same way as the singularities of at .

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

The coefficient $a_{{-1}}$ of $(z-z_{0})^{{-1}}$ in the Laurent series for is called the residue of 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

Keywords:: Picard’s theorem

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

§1.10(iv) Residue Theorem

Notes:: See Copson (1935, pp. 117–120).
Keywords:: residue
Referenced by:: §10.74(vi), §16.5, §3.8(v)
Permalink:: http://dlmf.nist.gov/1.10.iv

If is analytic within a simple closed contour , and continuous within and on —except in both instances for a finite number of singularities within —then

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

Symbols:: : differential of , $\int$ : integral, : variable and : closed contour
Referenced by:: ¶ ‣ §2.10(iii)
Permalink:: http://dlmf.nist.gov/1.10.E8
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Here and elsewhere in this subsection the path is described in the positive sense.

¶ Phase (or Argument) Principle

Keywords:: argument principle, phase principle, principle of the argument

If the singularities within are poles and is analytic and nonvanishing on , then

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

Symbols:: : differential of , $\int$ : integral, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, : variable, : closed contour, : number of zeros and : number of zeros
Permalink:: http://dlmf.nist.gov/1.10.E9
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where and are respectively the numbers of zeros and poles, counting multiplicity, of within , and $\Delta_{C}(\mathop{\mathrm{ph}\/}\nolimits f(z))$ is the change in any continuous branch of $\mathop{\mathrm{ph}\/}\nolimits\!\left(f(z)\right)$ as passes once around in the positive sense. For examples of applications see Olver (1997b, pp. 252–254).

In addition,

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

Symbols:: : differential of , $\int$ : integral, : variable and : closed contour
Permalink:: http://dlmf.nist.gov/1.10.E10
Encodings:: TeX, pMML, png

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

¶ Rouché’s Theorem

Keywords:: Rouché’s theorem

If and are analytic on and inside a simple closed contour , and on , then and have the same number of zeros inside .

§1.10(v) Maximum-Modulus Principle

Notes:: See Titchmarsh (1962, pp. 165–169).
Permalink:: http://dlmf.nist.gov/1.10.v

¶ Analytic Functions

Keywords:: maximum-modulus principle

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

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

¶ Harmonic Functions

Keywords:: harmonic functions, maximum-modulus principle

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

¶ Schwarz’s Lemma

Keywords:: Schwarz’s lemma, maximum-modulus principle

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

1.10.11 $|f(z)|\leq\frac{M|z|}{R}\;\mbox{ and }\;|f^{{\prime}}(0)|\leq\frac{M}{R}.$

Symbols:: : variable and : radius
Permalink:: http://dlmf.nist.gov/1.10.E11
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Equalities hold iff , where is a constant such that $\left|A\right|=M/R$ .

§1.10(vi) Multivalued Functions

Notes:: See Levinson and Redheffer (1970, pp. 64–77), or Markushevich (1983, pp. 106–121).
Keywords:: branch, functions, many-valued function, multivalued function
Referenced by:: ¶ ‣ §10.72(i), Methodology, Common Notations and Definitions
Permalink:: http://dlmf.nist.gov/1.10.vi

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

¶ Example

Keywords:: branch, branch point, cut, domain, neighborhood, singularity

$F(z)=\sqrt{z}$ is two-valued for $z\not=0$ . If $D=\Complex\setminus(-\infty,0]$ 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 $D=\Complex\setminus[0,\infty)$ , 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 is multivalued and is a point such that there exists a branch of in a cut neighborhood of , but there does not exist a branch of in any punctured neighborhood of . Then is a branch point of . For example, 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 to be single-valued in the cut plane (or domain).

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

If the path circles a branch point at , 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

Keywords:: branch

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 can be defined, for example, in the cut plane obtained from $\Complex$ 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)).

Figure 1.10.1: Domain

Symbols:: : domain
Referenced by:: ¶ ‣ §1.10(vi)
Permalink:: http://dlmf.nist.gov/1.10.F1
Encodings:: pdf, png

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

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

§1.10(vii) Inverse Functions

Notes:: For (1.10.13) and (1.10.14) see Copson (1935, §6.23). See also Andrews et al. (1999, pp. 629–631) and Henrici (1974, pp. 57–59). The Extended Inversion Theorem is proved in a similar way.
Keywords:: functions, inverse function
Permalink:: http://dlmf.nist.gov/1.10.vii

¶ Lagrange Inversion Theorem

Keywords:: Lagrange inversion theorem, inverse function

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

1.10.12

Symbols:: : variable and : variable
Referenced by:: ¶ ‣ §1.10(vii), ¶ ‣ §1.10(vii)
Permalink:: http://dlmf.nist.gov/1.10.E12
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has a unique solution analytic at $w=w_{0}$ , and

1.10.13 $F(w)=z_{0}+\sum^{\infty}_{{n=1}}F_{n}(w-w_{0})^{n}$

Symbols:: : variable, : variable, : nonnegative integer, : inverse function of and $F_{n}$ : residue
Referenced by:: §1.10(vii)
Permalink:: http://dlmf.nist.gov/1.10.E13
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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 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:: : variable, : variable, : nonnegative integer, : inverse function of and $G_{n}$ : residue
Referenced by:: §1.10(vii)
Permalink:: http://dlmf.nist.gov/1.10.E14
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where $nG_{n}$ is the residue of $g^{{\prime}}(z)/(f(z)-f(z_{0}))^{n}$ at $z=z_{0}$ .

¶ Extended Inversion Theorem

Keywords:: Lagrange inversion theorem, inverse function

Suppose that

1.10.15 $f(z)=f(z_{0})+\sum^{\infty}_{{n=0}}f_{n}(z-z_{0})^{{\mu+n}},$

Symbols:: : variable, : nonnegative integer, $f_{n}$ : residue and $\mu$ : parameter
Referenced by:: §2.3(iii)
Permalink:: http://dlmf.nist.gov/1.10.E15
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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 , where

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

Symbols:: : variable, : variable, : nonnegative integer, : inverse function of , $F_{n}$ : residue and $\mu$ : parameter
Referenced by:: ¶ ‣ §1.10(vii)
Permalink:: http://dlmf.nist.gov/1.10.E16
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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 is analytic at $z_{0}$ , then

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

Symbols:: : variable, : variable, : nonnegative integer, : inverse function of , $\mu$ : parameter and $G_{n}$ : residue
Referenced by:: §2.3(iii)
Permalink:: http://dlmf.nist.gov/1.10.E17
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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

Notes:: See Copson (1935, pp. 106–113).
Keywords:: convergence, differentiation, functions, integrals
Permalink:: http://dlmf.nist.gov/1.10.viii

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

1.10.18 $F(z)=\int^{b}_{a}f(z,t)dt$

Symbols:: : differential of , $\int$ : integral and : variable
Referenced by:: ¶ ‣ §1.10(viii), §1.10(viii)
Permalink:: http://dlmf.nist.gov/1.10.E18
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is analytic in 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 has a singularity at , with the following conditions. For each $t\in[a,b)$ , is analytic in ; is a continuous function of both variables when $z\in D$ and $t\in[a,b)$ ; the integral (1.10.18) converges at , and this convergence is uniform with respect to in every compact subset of .

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

1.10.19 $\left|\int_{{a_{1}}}^{b}f(z,t)dt\right|<\epsilon,$

Symbols:: : differential of , $\int$ : integral, : variable and $\epsilon$ : positive number
Permalink:: http://dlmf.nist.gov/1.10.E19
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for all $a_{1}\in[a_{0},b)$ and all $z\in S$ ; compare §1.5(iv).

¶ -test

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

§1.10(ix) Infinite Products

Notes:: See Titchmarsh (1962, pp. 13–19, 246–250).
Keywords:: convergence, infinite products
Permalink:: http://dlmf.nist.gov/1.10.ix

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 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 , converges iff $\sum^{\infty}_{{n=1}}\mathop{\ln\/}\nolimits\!\left(1+a_{n}\right)$ 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.

¶ -test

Keywords:: -test for uniform convergence, Weierstrass -test, infinite products

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

1.10.20 $|\mathop{\ln\/}\nolimits\!\left(1+a_{n}(z)\right)|\leq M_{n},$ $n\geq N$ ,

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, : variable and : nonnegative integer
Referenced by:: ¶ ‣ §1.10(ix)
Permalink:: http://dlmf.nist.gov/1.10.E20
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and

1.10.21 $\sum^{\infty}_{{n=1}}M_{n}<\infty,$

Symbols:: : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.10.E21
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then the product $\prod^{\infty}_{{n=1}}(1+a_{n}(z))$ converges uniformly to an analytic function in , and only when at least one of the factors $1+a_{n}(z)$ is zero in . This conclusion remains true if, in place of (1.10.20), $|a_{n}(z)|\leq M_{n}$ for all , and again $\sum^{\infty}_{{n=1}}M_{n}<\infty$ .

¶ Weierstrass Product

Keywords:: Weierstrass product, infinite products

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:: : base of exponential function, : variable and : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.10.E22
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is an entire function with zeros at $z_{n}$ .

§1.10(x) Infinite Partial Fractions

Notes:: See Levinson and Redheffer (1970, pp. 392–395).
Keywords:: infinite partial fractions, infinite products
Permalink:: http://dlmf.nist.gov/1.10.x

Suppose is a domain, and

1.10.23 $F(z)=\prod^{\infty}_{{n=1}}a_{n}(z),$ $z\in D$ ,

Symbols:: $\in$ : element of, : variable, : nonnegative integer, : domain and : function
Permalink:: http://dlmf.nist.gov/1.10.E23
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where $a_{n}(z)$ is analytic for all $n\geq 1$ , and the convergence of the product is uniform in any compact subset of . Then is analytic in .

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

1.10.24 $\frac{F^{{\prime}}(z)}{F(z)}=\sum^{\infty}_{{n=1}}\frac{a_{n}^{{\prime}}(z)}{a% _{n}(z)}.$

Symbols:: : variable, : nonnegative integer and : function
Permalink:: http://dlmf.nist.gov/1.10.E24
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¶ Mittag-Leffler’s Expansion

Keywords:: Mittag-Leffler’s expansion, functions, infinite partial fractions

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