1 Algebraic and Analytic Methods Topics of Discussion 1.8 Fourier Series 1.10 Functions of a Complex Variable

§1.9 Calculus of a Complex Variable

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Keywords:: calculus, complex variable
Referenced by:: §1.1, §1.1
Permalink:: http://dlmf.nist.gov/1.9
See also:: Annotations for Ch.1

§1.9(i) Complex Numbers
§1.9(ii) Continuity, Point Sets, and Differentiation
§1.9(iii) Integration
§1.9(iv) Conformal Mapping
§1.9(v) Infinite Sequences and Series
§1.9(vi) Power Series
§1.9(vii) Inversion of Limits

§1.9(i) Complex Numbers

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Notes:: See Copson (1935, Chapter 1), Levinson and Redheffer (1970, Chapter 1), or Markushevich (1983, pp. 14–18).
Referenced by:: §1.11(iv), §4.2(i), §4.45(ii), Erratum (V1.0.25) for Subsection 1.9(i)
Permalink:: http://dlmf.nist.gov/1.9.i
Clarification (effective with 1.0.25):: A phrase was added just below (1.9.1) which elaborates that ${\mathrm{i}}^{2}=-1$ .
See also:: Annotations for §1.9 and Ch.1

1.9.1 $z=x+\mathrm{i}y,$ $x,y\in\mathbb{R}$ ,

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Defines:: $\mathrm{i}$ : imaginary unit
Symbols:: $\in$ : element of, $\mathbb{R}$ : real line and : variable
A&S Ref:: 3.7.1
Referenced by:: §1.9(i), Erratum (V1.0.25) for Subsection 1.9(i)
Permalink:: http://dlmf.nist.gov/1.9.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §1.9(i), §1.9 and Ch.1

such that ${\mathrm{i}}^{2}=-1$ .

Real and Imaginary Parts

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Keywords:: complex numbers, imaginary part, real part
See also:: Annotations for §1.9(i), §1.9 and Ch.1

1.9.2

$\Re z=x,$

$\Im z=y.$

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Defines:: $\Im$ : imaginary part and $\Re$ : real part
Symbols:: : 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

Polar Representation

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Keywords:: complex numbers, polar representation
See also:: Annotations for §1.9(i), §1.9 and Ch.1

1.9.3

$x=r\cos\theta,$

$y=r\sin\theta,$

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Symbols:: $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, : 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 $r=(x^{2}+y^{2})^{1/2},$

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Symbols:: : 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 $z\neq 0$ ,

1.9.5 $\theta=\omega,\;\;\pi-\omega,\;\;-\pi+\omega,\mbox{ or }-\omega,$

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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 lies in the 1st, 2nd, 3rd, or 4th quadrants. Here

1.9.6 $\omega=\operatorname{arctan}\left(\left|y/x\right|\right)\in\left[0,\tfrac{1}{% 2}\pi\right].$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $[\NVar{a},\NVar{b}]$ : closed interval, $\in$ : element of, $\operatorname{arctan}\NVar{z}$ : arctangent function and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
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

Modulus and Phase

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Keywords:: complex numbers, modulus, phase
See also:: Annotations for §1.9(i), §1.9 and Ch.1

1.9.7

$\left|z\right|=r,$

$\operatorname{ph}z=\theta+2n\pi,$ $n\in\mathbb{Z}$ .

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Defines:: $\operatorname{ph}$ : phase and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$ (locally)
Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\in$ : element of, $\mathbb{Z}$ : set of all integers, : variable, : nonnegative integer, : 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 $\operatorname{ph}z$ corresponds to , that is, $-\pi\leq\operatorname{ph}z\leq\pi$ . 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 $-\pi<\operatorname{ph}z\leq\pi$ .)

1.9.8

$\left|\Re z\right|\leq\left|z\right|,$

$\left|\Im z\right|\leq\left|z\right|,$

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Symbols:: $\Im$ : imaginary part, $\Re$ : real part, : variable and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
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 $z=r{\mathrm{e}}^{\mathrm{i}\theta},$

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Symbols:: $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, : variable, : 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 ${\mathrm{e}}^{\mathrm{i}\theta}=\cos\theta+\mathrm{i}\sin\theta;$

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Symbols:: $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\sin\NVar{z}$ : 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.

Complex Conjugate

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Keywords:: complex conjugates, complex numbers
See also:: Annotations for §1.9(i), §1.9 and Ch.1

1.9.11 $\overline{z}=x-\mathrm{i}y,$

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Defines:: $\overline{\NVar{z}}$ : complex conjugate
Symbols:: $\mathrm{i}$ : imaginary unit and : 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 $\left|\overline{z}\right|=\left|z\right|,$

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Symbols:: $\overline{\NVar{z}}$ : complex conjugate, : variable and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
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 $\operatorname{ph}\overline{z}=-\operatorname{ph}z.$

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Symbols:: $\overline{\NVar{z}}$ : complex conjugate, $\operatorname{ph}$ : phase and : 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

Arithmetic Operations

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Keywords:: arithmetic operations, complex numbers
See also:: Annotations for §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 $z_{1}\pm z_{2}=x_{1}\pm x_{2}+\mathrm{i}(y_{1}\pm y_{2}),$

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Symbols:: $\mathrm{i}$ : imaginary unit and : 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 $z_{1}z_{2}=x_{1}x_{2}-y_{1}y_{2}+\mathrm{i}(x_{1}y_{2}+x_{2}y_{1}),$

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Symbols:: $\mathrm{i}$ : imaginary unit and : 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 $\frac{z_{1}}{z_{2}}=\frac{z_{1}\overline{z}_{2}}{{\left|z_{2}\right|}^{2}}=% \frac{x_{1}x_{2}+y_{1}y_{2}+\mathrm{i}(x_{2}y_{1}-x_{1}y_{2})}{x_{2}^{2}+y_{2}% ^{2}},$

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Symbols:: $\overline{\NVar{z}}$ : complex conjugate, $\mathrm{i}$ : imaginary unit, : variable and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
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 $z_{2}\neq 0$ . Also,

1.9.17 $\left|z_{1}z_{2}\right|=\left|z_{1}\right|\;\left|z_{2}\right|,$

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Symbols:: : variable and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
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 $\operatorname{ph}\left(z_{1}z_{2}\right)=\operatorname{ph}z_{1}+\operatorname{% ph}z_{2},$

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Symbols:: $\operatorname{ph}$ : phase and : 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 $\left|\frac{z_{1}}{z_{2}}\right|=\frac{\left|z_{1}\right|}{\left|z_{2}\right|},$

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Symbols:: : variable and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
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 $\operatorname{ph}\frac{z_{1}}{z_{2}}=\operatorname{ph}z_{1}-\operatorname{ph}z% _{2}.$

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Symbols:: $\operatorname{ph}$ : phase and : 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.

Powers

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Keywords:: complex numbers, powers
See also:: Annotations for §1.9(i), §1.9 and Ch.1

1.9.21 $z^{n}=\left(x^{n}-\genfrac{(}{)}{0.0pt}{}{n}{2}x^{n-2}y^{2}+\genfrac{(}{)}{0.0% pt}{}{n}{4}x^{n-4}y^{4}-\cdots\right)+\mathrm{i}\left(\genfrac{(}{)}{0.0pt}{}{% n}{1}x^{n-1}y-\genfrac{(}{)}{0.0pt}{}{n}{3}x^{n-3}y^{3}+\cdots\right),$ $n=1,2,\dots$ .

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Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\mathrm{i}$ : imaginary unit, : variable and : 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

DeMoivre’s Theorem

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Keywords:: DeMoivre’s theorem, complex numbers
See also:: Annotations for §1.9(i), §1.9 and Ch.1

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

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Symbols:: $\cos\NVar{z}$ : cosine function, $\in$ : element of, $\mathrm{i}$ : imaginary unit, $\mathbb{Z}$ : set of all integers, $\sin\NVar{z}$ : sine function, : 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

Triangle Inequality

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Keywords:: complex numbers, triangle inequality
See also:: Annotations for §1.9(i), §1.9 and Ch.1

1.9.23 $\left|\left|z_{1}\right|-\left|z_{2}\right|\right|\leq\left|z_{1}+z_{2}\right|% \leq\left|z_{1}\right|+\left|z_{2}\right|.$

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Symbols:: : variable and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
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

§1.9(ii) Continuity, Point Sets, and Differentiation

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Notes:: See Levinson and Redheffer (1970, Chapters 1,2 and pp. 133–138) and Copson (1935, Chapters 2,3).
Referenced by:: §1.6(iv)
Permalink:: http://dlmf.nist.gov/1.9.ii
See also:: Annotations for §1.9 and Ch.1

Continuity

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Keywords:: at a point, continuous function, disk, limits of functions, of a complex variable, of two complex variables, of two variables, open
See also:: Annotations for §1.9(ii), §1.9 and Ch.1

A function 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 $\left|f(z)-f(z_{0})\right|<\epsilon$ for all in the open disk $\left|z-z_{0}\right|<\delta$ .

A function of two complex variables 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 $\mathbb{C}$

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Keywords:: accumulation, accumulation point, boundary, boundary points, closed, closed point set, closure, connected, connected point set, continuous function, domain, interior, interior points, limit (or limiting), limit points (or limiting points), neighborhood, of point sets in complex plane, on a region, open, open point set, point sets in complex plane, points in complex plane, region
See also:: Annotations for §1.9(ii), §1.9 and Ch.1

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 in $\mathbb{C}$ (or $\mathbb{C}\cup\infty$ ) if every neighborhood of $z_{0}$ contains a point of distinct from $z_{0}$ . ( $z_{0}$ may or may not belong to .) As a consequence, every neighborhood of a limit point of contains an infinite number of points of . Also, the union of and its limit points is the closure of .

A domain , 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 is a boundary point of . 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 is continuous on a region if for each point $z_{0}$ in and any given number $\epsilon$ () we can find a neighborhood of $z_{0}$ such that $\left|f(z)-f(z_{0})\right|<\epsilon$ for all points in the intersection of the neighborhood with .

Differentiation

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Keywords:: differentiable functions
Addition (effective with 1.2.0):: A sentence was added just below (1.9.24).
Modification (effective with 1.2.0):: In the first sentence “differentiable” was replaced with “complex differentiable”.
See also:: Annotations for §1.9(ii), §1.9 and Ch.1

A function is complex differentiable at a point if the following limit exists:

1.9.24 $f^{\prime}(z)=\frac{\mathrm{d}f}{\mathrm{d}z}=\lim_{h\to 0}\frac{f(z+h)-f(z)}{% h}.$

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Symbols:: $\mathbb{C}$ : complex plane, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to and : 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 $\mathbb{C}$ .
See also:: Annotations for §1.9(ii), §1.9(ii), §1.9 and Ch.1

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

Differentiability automatically implies continuity.

Cauchy–Riemann Equations

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Keywords:: Cauchy–Riemann equations, differentiation
See also:: Annotations for §1.9(ii), §1.9 and Ch.1

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

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

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

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Symbols:: $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of with respect to , $\,\partial\NVar{x}$ : partial differential of , : function and : 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 .

Conversely, if at a given point 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 is differentiable at $z=x+\mathrm{i}y$ .

Analyticity

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Keywords:: analytic, analytic function, entire, entire functions, functions, holomorphic, holomorphic function, in a domain
See also:: Annotations for §1.9(ii), §1.9 and Ch.1

A function is said to be analytic (holomorphic) at $z=z_{0}$ if it is complex differentiable in a neighborhood of $z_{0}$ .

A function is analytic in a domain if it is analytic at each point of . A function analytic at every point of $\mathbb{C}$ is said to be entire.

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

Harmonic Functions

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Keywords:: functions, harmonic, harmonic functions
See also:: Annotations for §1.9(ii), §1.9 and Ch.1

If $f(z)=u(x,y)+\mathrm{i}v(x,y)$ is analytic in an open domain , then and are harmonic in , that is,

1.9.26 $\frac{{\partial}^{2}u}{{\partial x}^{2}}+\frac{{\partial}^{2}u}{{\partial y}^{% 2}}=\frac{{\partial}^{2}v}{{\partial x}^{2}}+\frac{{\partial}^{2}v}{{\partial y% }^{2}}=0,$

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Symbols:: $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of with respect to , $\,\partial\NVar{x}$ : partial differential of , : function and : 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) and satisfy

1.9.27 $\frac{{\partial}^{2}u}{{\partial r}^{2}}+\frac{1}{r}\frac{\partial u}{\partial r% }+\frac{1}{r^{2}}\frac{{\partial}^{2}u}{{\partial\theta}^{2}}=0$

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Symbols:: $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of with respect to , $\,\partial\NVar{x}$ : partial differential of , : radius, $\theta$ : angle and : 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 .

§1.9(iii) Integration

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Keywords:: arc(s), contour, infinite, integrals, integration, simple, simple closed, simple closed contour
Notes:: See Levinson and Redheffer (1970, Chapter 3 and p. 360), Copson (1935, pp. 56–69), and Ahlfors (1966, pp. 168–169). For a proof of the Jordan curve theorem see, for example, Dienes (1931, pp. 177–197). The theorem is valid with less restrictive conditions than those assumed here.
Referenced by:: §1.10(vi), §15.16, §3.4(ii)
Permalink:: http://dlmf.nist.gov/1.9.iii
See also:: Annotations for §1.9 and Ch.1

An arc is given by $z(t)=x(t)+\mathrm{i}y(t)$ , $a\leq t\leq b$ , where and are continuously differentiable. If and are continuous and $x^{\prime}(t)$ and $y^{\prime}(t)$ are piecewise continuous, then 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 , $z(t_{1})\neq z(t_{2})$ . A simple closed contour is a simple contour, except that .

Next,

1.9.28 $\int_{C}f(z)\,\mathrm{d}z=\int_{a}^{b}f(z(t))(x^{\prime}(t)+\mathrm{i}y^{% \prime}(t))\,\mathrm{d}t,$

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{i}$ : imaginary unit, $\int$ : integral and : closed contour
Permalink:: http://dlmf.nist.gov/1.9.E28
Encodings:: TeX, pMML, png
See also:: Annotations for §1.9(iii), §1.9 and Ch.1

for a contour and 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

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Keywords:: Jordan curve theorem, domain, exterior, interior, point sets in complex plane
See also:: Annotations for §1.9(iii), §1.9 and Ch.1

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

Cauchy’s Theorem

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Keywords:: Cauchy’s theorem
See also:: Annotations for §1.9(iii), §1.9 and Ch.1

If is continuous within and on a simple closed contour and analytic within , then

1.9.29 $\int_{C}f(z)\,\mathrm{d}z=0.$

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral and : 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

Cauchy’s Integral Formula

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Keywords:: Cauchy’s integral formula, for derivatives
See also:: Annotations for §1.9(iii), §1.9 and Ch.1

If is continuous within and on a simple closed contour and analytic within , and if $z_{0}$ is a point within , then

1.9.30 $f(z_{0})=\frac{1}{2\pi\mathrm{i}}\int_{C}\frac{f(z)}{z-z_{0}}\,\mathrm{d}z,$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{i}$ : imaginary unit, $\int$ : integral and : 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 $f^{(n)}(z_{0})=\frac{n!}{2\pi\mathrm{i}}\int_{C}\frac{f(z)}{(z-z_{0})^{n+1}}\,% \mathrm{d}z,$ $n=1,2,3,\dots$ ,

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , : factorial (as in ), $\mathrm{i}$ : imaginary unit, $\int$ : integral, : nonnegative integer and : 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 is described in the positive rotational (anticlockwise) sense.

Liouville’s Theorem

ⓘ

Keywords:: Liouville’s theorem, Liouville’s theorem for entire functions, entire functions
See also:: Annotations for §1.9(iii), §1.9 and Ch.1

Any bounded entire function is a constant.

Winding Number

ⓘ

Keywords:: of closed contour, winding number
See also:: Annotations for §1.9(iii), §1.9 and Ch.1

If is a closed contour, and $z_{0}\not\in C$ , then

1.9.32 $\frac{1}{2\pi\mathrm{i}}\int_{C}\frac{1}{z-z_{0}}\,\mathrm{d}z=\mathcal{N}(C,z% _{0}),$

ⓘ

Defines:: $\mathcal{N}(C,z_{0})$ : winding number of (locally)
Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{i}$ : imaginary unit, $\int$ : integral and : 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 $\mathcal{N}(C,z_{0})$ is an integer called the winding number of with respect to $z_{0}$ . If 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 .

Mean Value Property

ⓘ

Keywords:: harmonic functions, mean value property, mean value property for harmonic functions
See also:: Annotations for §1.9(iii), §1.9 and Ch.1

For harmonic,

1.9.33 $u(z)=\frac{1}{2\pi}\int^{2\pi}_{0}u(z+r{\mathrm{e}}^{\mathrm{i}\phi})\,\mathrm% {d}\phi.$

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, : radius and : 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

Poisson Integral

ⓘ

Keywords:: Poisson integral, harmonic functions
See also:: Annotations for §1.9(iii), §1.9 and Ch.1

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

1.9.34 $u(r{\mathrm{e}}^{\mathrm{i}\theta})=\frac{1}{2\pi}\int^{2\pi}_{0}\frac{(R^{2}-% r^{2})h(R{\mathrm{e}}^{\mathrm{i}\phi})\,\mathrm{d}\phi}{R^{2}-2Rr\cos\left(% \phi-\theta\right)+r^{2}}$

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, : radius, : harmonic function and : 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 $\left|z\right|<R$ . Also with $\left|w\right|=R$ , $\lim\limits_{z\to w}u(z)=h(w)$ as $z\to w$ within $\left|z\right|<R$ .

§1.9(iv) Conformal Mapping

ⓘ

Keywords:: analytic function, around infinity, at infinity, conformal mapping, disk, extended complex plane, neighborhood, of infinity, open disks around infinity, points in complex plane
Notes:: See Markushevich (1983, pp. 41–46), Markushevich (1985, vol. 1, §34), and Levinson and Redheffer (1970, pp. 259–277).
Permalink:: http://dlmf.nist.gov/1.9.iv
See also:: Annotations for §1.9 and Ch.1

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 $S_{r}=\{z\mid\left|z\right|>1/r\}\cup\{\infty\},$ $0<r<\infty$ .

ⓘ

Symbols:: $\cup$ : union, : variable, : radius, $S_{r}$ : neighborhood and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
Permalink:: http://dlmf.nist.gov/1.9.E35
Encodings:: TeX, pMML, png
See also:: Annotations for §1.9(iv), §1.9 and Ch.1

Each $S_{r}$ is a neighborhood of $\infty$ . Also,

1.9.36 $\infty\pm z=z\pm\infty=\infty,$

ⓘ

Symbols:: : variable
Permalink:: http://dlmf.nist.gov/1.9.E36
Encodings:: TeX, pMML, png
See also:: Annotations for §1.9(iv), §1.9 and Ch.1

1.9.37 $\infty\cdot z=z\cdot\infty=\infty,$ $z\not=0$ ,

ⓘ

Symbols:: : variable
Permalink:: http://dlmf.nist.gov/1.9.E37
Encodings:: TeX, pMML, png
See also:: Annotations for §1.9(iv), §1.9 and Ch.1

1.9.38 $z/\infty=0,$

ⓘ

Symbols:: : variable
Permalink:: http://dlmf.nist.gov/1.9.E38
Encodings:: TeX, pMML, png
See also:: Annotations for §1.9(iv), §1.9 and Ch.1

1.9.39 $z/0=\infty,$ $z\neq 0$ .

ⓘ

Symbols:: : variable
Permalink:: http://dlmf.nist.gov/1.9.E39
Encodings:: TeX, pMML, png
See also:: Annotations for §1.9(iv), §1.9 and Ch.1

A function is analytic at $\infty$ if is analytic at , and we set $f^{\prime}(\infty)=g^{\prime}(0)$ .

Conformal Transformation

ⓘ

Keywords:: angle between, angle between arcs, arc(s), conformal mapping, linear transformation
See also:: Annotations for §1.9(iv), §1.9 and Ch.1

Suppose is analytic in a domain and $C_{1},C_{2}$ are two arcs in passing through $z_{0}$ . Let $C^{\prime}_{1},C^{\prime}_{2}$ be the images of $C_{1}$ and $C_{2}$ under the mapping . 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 is conformal (angle-preserving) at $z_{0}$ .

The linear transformation , $a\not=0$ , has $f^{\prime}(z)=a$ and maps $\mathbb{C}$ conformally onto $\mathbb{C}$ .

Bilinear Transformation

ⓘ

Keywords:: Möbius transformation, bilinear transformation, cross ratio, fractional linear transformation, homographic transformation
See also:: Annotations for §1.9(iv), §1.9 and Ch.1

1.9.40 $w=f(z)=\frac{az+b}{cz+d},$ $ad-bc\not=0$ , $c\not=0$ .

ⓘ

Symbols:: : variable and : 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

$f(-d/c)=\infty,$

$f(\infty)=a/c.$

ⓘ

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 $f^{\prime}(z)=\frac{ad-bc}{(cz+d)^{2}},$ $z\not=-d/c$ .

ⓘ

Symbols:: : 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 $f^{\prime}(\infty)=\frac{bc-ad}{c^{2}}.$

ⓘ

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:: : variable and : 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 $\mathbb{C}\,\cup\,\{\infty\}$ 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:: : 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.

§1.9(v) Infinite Sequences and Series

ⓘ

Keywords:: absolute, convergence, divergent, infinite sequences, infinite series, pointwise, uniform
Notes:: See Copson (1935, pp. 19–24, 92–98).
Permalink:: http://dlmf.nist.gov/1.9.v
See also:: Annotations for §1.9 and Ch.1

A sequence $\{z_{n}\}$ converges to if $\lim\limits_{n\to\infty}z_{n}=z$ . For $z_{n}=x_{n}+\mathrm{i}y_{n}$ , 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 $\sum^{\infty}_{n=0}\left|z_{n}\right|$ converges. A series $\sum^{\infty}_{n=0}z_{n}$ converges (diverges) absolutely when $\lim\limits_{n\to\infty}{\left|z_{n}\right|}^{1/n}<1$ (), or when $\lim\limits_{n\to\infty}\left|\ifrac{z_{n+1}}{z_{n}}\right|<1$ (). Absolutely convergent series are also convergent.

Let $\{f_{n}(z)\}$ be a sequence of functions defined on a set . This sequence converges pointwise to a function if

1.9.46 $f(z)=\lim_{n\to\infty}f_{n}(z)$

ⓘ

Symbols:: : variable and : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.9.E46
Encodings:: TeX, pMML, png
See also:: Annotations for §1.9(v), §1.9 and Ch.1

for each $z\in S$ . The sequence converges uniformly on , if for every $\epsilon>0$ there exists an integer , independent of , such that

1.9.47 $\left|f_{n}(z)-f(z)\right|<\epsilon$

ⓘ

Symbols:: : variable, : nonnegative integer, $\epsilon$ : positive number and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
Permalink:: http://dlmf.nist.gov/1.9.E47
Encodings:: TeX, pMML, png
See also:: Annotations for §1.9(v), §1.9 and Ch.1

for all $z\in S$ and $n\geq N$ .

A series $\sum^{\infty}_{n=0}f_{n}(z)$ converges uniformly on , if the sequence $s_{n}(z)=\sum^{n}_{k=0}f_{k}(z)$ converges uniformly on .

Weierstrass -test

ⓘ

Keywords:: -test for uniform convergence, Weierstrass -test, convergence, doubly-infinite, infinite series
See also:: Annotations for §1.9(v), §1.9 and Ch.1

Suppose $\{M_{n}\}$ is a sequence of real numbers such that $\sum^{\infty}_{n=0}M_{n}$ converges and $\left|f_{n}(z)\right|\leq M_{n}$ for all $z\in S$ and all $n\geq 0$ . Then the series $\sum^{\infty}_{n=0}f_{n}(z)$ converges uniformly on .

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

§1.9(vi) Power Series

ⓘ

Keywords:: circle of, convergence, power series, radius of
Notes:: See Copson (1935, pp. 37–40), Levinson and Redheffer (1970, pp. 349–351), or Markushevich (1983, pp. 131–135). For the operations on series, see Henrici (1974, Chapter 1) or Olver (1997b, pp. 19–22).
Referenced by:: §22.10(ii)
Permalink:: http://dlmf.nist.gov/1.9.vi
See also:: Annotations for §1.9 and Ch.1

For a series $\sum^{\infty}_{n=0}a_{n}(z-z_{0})^{n}$ there is a number , $0\leq R\leq\infty$ , such that the series converges for all in $\left|z-z_{0}\right|<R$ and diverges for in $\left|z-z_{0}\right|>R$ . The circle $\left|z-z_{0}\right|=R$ is called the circle of convergence of the series, and is the radius of convergence. Inside the circle the sum of the series is an analytic function . For in $\left|z-z_{0}\right|\leq\rho$ (), the convergence is absolute and uniform. Moreover,

1.9.48 $a_{n}=\frac{f^{(n)}(z_{0})}{n!},$

ⓘ

Symbols:: : factorial (as in ), : variable and : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.9.E48
Encodings:: TeX, pMML, png
See also:: Annotations for §1.9(vi), §1.9 and Ch.1

and

1.9.49 $R=\liminf_{n\to\infty}{\left|a_{n}\right|}^{-1/n}.$

ⓘ

Defines:: : radius of convergence (locally)
Symbols:: $\liminf$ : least limit point, : nonnegative integer and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
Permalink:: http://dlmf.nist.gov/1.9.E49
Encodings:: TeX, pMML, png
See also:: Annotations for §1.9(vi), §1.9 and Ch.1

For the converse of this result see §1.10(i).

Operations

ⓘ

Keywords:: addition, differentiation, multiplication, of logarithms, of powers, of reciprocals, power series, subtraction
See also:: Annotations for §1.9(vi), §1.9 and Ch.1

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:: : variable and : 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 $\left(\sum^{\infty}_{n=0}a_{n}z^{n}\right)\left(\sum^{\infty}_{n=0}b_{n}z^{n}% \right)=\sum^{\infty}_{n=0}c_{n}z^{n},$

ⓘ

Symbols:: : variable and : 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 $c_{n}=\sum^{n}_{k=0}a_{k}b_{n-k}.$

ⓘ

Symbols:: : integer and : 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 $f(z)=a_{0}+a_{1}z+a_{2}z^{2}+\cdots,$ $a_{0}\not=0$ .

ⓘ

Symbols:: : 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 $\frac{1}{f(z)}=b_{0}+b_{1}z+b_{2}z^{2}+\cdots,$

ⓘ

Symbols:: : 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 $b_{n}=-(a_{1}b_{n-1}+a_{2}b_{n-2}+\dots+a_{n}b_{0})/a_{0},$ $n\geq 1$ .

ⓘ

Symbols:: : 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

With $a_{0}=1$ ,

1.9.57 $\ln f(z)=q_{1}z+q_{2}z^{2}+q_{3}z^{3}+\cdots,$

ⓘ

Symbols:: $\ln\NVar{z}$ : principal branch of logarithm function, : 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), §1.9(vi), §1.9 and Ch.1

(principal value), where

1.9.58

$q_{1}=a_{1},$

$q_{2}=(2a_{2}-a_{1}^{2})/2,$

$q_{3}=(3a_{3}-3a_{1}a_{2}+a_{1}^{3})/3,$

ⓘ

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), §1.9(vi), §1.9 and Ch.1

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:: : 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), §1.9(vi), §1.9 and Ch.1

Also,

1.9.60 $(f(z))^{\nu}=p_{0}+p_{1}z+p_{2}z^{2}+\cdots,$

ⓘ

Symbols:: : 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), §1.9(vi), §1.9 and Ch.1

(principal value), where $\nu\in\mathbb{C}$ ,

1.9.61

$p_{0}=1,$

$p_{1}=\nu a_{1},$

$p_{2}=\nu((\nu-1)a_{1}^{2}+2a_{2})/2,$

ⓘ

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), §1.9(vi), §1.9 and Ch.1

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:: : 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), §1.9(vi), §1.9 and Ch.1

For the definitions of the principal values of $\ln f(z)$ 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 $f^{(m)}(z)=\sum_{n=0}^{\infty}{\left(n+1\right)_{m}}a_{n+m}(z-z_{0})^{n},$ $\left|z-z_{0}\right|<R$ , $m=0,1,2,\dots$ .

ⓘ

Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), : variable, : nonnegative integer, : nonnegative integer, : radius of convergence and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
Permalink:: http://dlmf.nist.gov/1.9.E63
Encodings:: TeX, pMML, png
See also:: Annotations for §1.9(vi), §1.9(vi), §1.9 and Ch.1

§1.9(vii) Inversion of Limits

ⓘ

Keywords:: convergence, double, infinite sequences, infinite series
Notes:: See Copson (1935, pp. 27–30, 95–97). For (1.9.69)–(1.9.71), see Titchmarsh (1962b, §1.77).
Referenced by:: §1.16(i), §1.3(iii)
Permalink:: http://dlmf.nist.gov/1.9.vii
See also:: Annotations for §1.9 and Ch.1

Double Sequences and Series

ⓘ

Keywords:: convergence, double, double sequence, double series, infinite sequences, relation to infinite double series
See also:: Annotations for §1.9(vii), §1.9 and Ch.1

A set of complex numbers $\{z_{m,n}\}$ where and take all positive integer values is called a double sequence. It converges to if for every $\epsilon>0$ , there is an integer such that

1.9.64 $\left|z_{m,n}-z\right|<\epsilon$

ⓘ

Symbols:: : variable, : nonnegative integer, : nonnegative integer, $\epsilon$ : positive number and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
Permalink:: http://dlmf.nist.gov/1.9.E64
Encodings:: TeX, pMML, png
See also:: Annotations for §1.9(vii), §1.9(vii), §1.9 and Ch.1

for all $m,n\geq N$ . Suppose $\{z_{m,n}\}$ converges to 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:: : variable, : nonnegative integer and : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.9.E65
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §1.9(vii), §1.9(vii), §1.9 and Ch.1

exist. Then both repeated limits equal .

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:: : variable, : nonnegative integer, : 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), §1.9(vii), §1.9 and Ch.1

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 $\left|\zeta_{m,n}\right|$ .

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:: : nonnegative integer, : 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), §1.9(vii), §1.9 and Ch.1

Term-by-Term Integration

ⓘ

Keywords:: compact, compact set, infinite series, integration, point sets in complex plane, term by term, term-by-term integration
See also:: Annotations for §1.9(vii), §1.9 and Ch.1

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 , that is, every closed and bounded set in . Then

1.9.68 $\int_{C}\sum^{\infty}_{n=0}f_{n}(z)\,\mathrm{d}z=\sum^{\infty}_{n=0}\int_{C}f_% {n}(z)\,\mathrm{d}z$

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, : variable, : nonnegative integer and : finite contour in
Permalink:: http://dlmf.nist.gov/1.9.E68
Encodings:: TeX, pMML, png
See also:: Annotations for §1.9(vii), §1.9(vii), §1.9 and Ch.1

for any finite contour in .

Dominated Convergence Theorem

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Keywords:: calculus, complex variable, dominated convergence theorem, infinite series
See also:: Annotations for §1.9(vii), §1.9 and Ch.1

Let 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 , and at least one of the following two conditions is satisfied: