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41: 18.18 Sums

… ►Let $f(z)$ be analytic within an ellipse $E$ with foci $z=\pm 1$, and … ►

18.18.2 $$f(z)=\sum _{n=0}^{\mathrm{\infty }}{a}_n{P}_n^{(\alpha ,\beta )}\left(z\right),$$

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Symbols:

$P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $z$: complex variable, $n$: nonnegative integer, $f(z)$: analytic function and $a_n$

Referenced by:

§18.18(i), §18.18(i), §18.18(i), §18.18(i)

Permalink:

http://dlmf.nist.gov/18.18.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §18.18(i), §18.18(i), §18.18 and Ch.18

… ►

18.18.3 $$a_n=\left(n+\frac{1}{2}\right){\int }_{-1}^{1}f(x)P_n\left(x\right)dx.$$

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, $dx$: differential of $x$ , $\int$: integral, $n$: nonnegative integer, $f(z)$: analytic function, $a_n$ and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.18.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §18.18(i), §18.18(i), §18.18 and Ch.18

… ►

18.18.4 $$f(x)=\sum _{n=0}^{\mathrm{\infty }}{b}_n{L}_n^{(\alpha )}\left(x\right),$$ ,

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $n$: nonnegative integer, $f(z)$: analytic function and $x$: real variable

Referenced by:

§18.18(i), §18.18(i)

Permalink:

http://dlmf.nist.gov/18.18.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §18.18(i), §18.18(i), §18.18 and Ch.18

… ►

18.18.6 $$f(x)=\sum _{n=0}^{\mathrm{\infty }}{d}_n{H}_n\left(x\right),$$ ,

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Symbols:

$H_n\left(x\right)$: Hermite polynomial, $n$: nonnegative integer, $f(z)$: analytic function, $d_n$ and $x$: real variable

Referenced by:

§18.18(i), §18.18(i)

Permalink:

http://dlmf.nist.gov/18.18.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §18.18(i), §18.18(i), §18.18 and Ch.18

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42: 4.37 Inverse Hyperbolic Functions

… ►

4.37.6 $$\mathrm{Arccoth}z=\mathrm{Arctanh}\left(1/z\right).$$

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Defines:

$\mathrm{Arccoth}z$: general inverse hyperbolic cotangent function

Symbols:

$\mathrm{Arctanh}z$: general inverse hyperbolic tangent function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.37.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §4.37(i), §4.37 and Ch.4

… ►These functions are analytic in the cut plane depicted in Figure 4.37.1(iv), (v), (vi), respectively. … ►It should be noted that the imaginary axis is not a cut; the function defined by (4.37.19) and (4.37.20) is analytic everywhere except on $(-\mathrm{\infty },1]$. …

43: 23.2 Definitions and Periodic Properties

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§23.2(ii) Weierstrass Elliptic Functions

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44: 13.2 Definitions and Basic Properties

… ► … ►

§13.2(ii) Analytic Continuation

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45: 13.14 Definitions and Basic Properties

… ► … ►

§13.14(ii) Analytic Continuation

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46: 11.4 Basic Properties

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§11.4(iii) Analytic Continuation

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47: Bibliography

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Complex Analysis: An Introduction of the Theory of Analytic Functions of One Complex Variable. 2nd edition, McGraw-Hill Book Co., New York.

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Notes:

A third edition was published in 1978 by McGraw-Hill, New York.

External Links:

MathReview (P. Lelong), ZentralBlatt

Referenced by:

§1.9(iii), §23.18.

Encodings:

BibTeX

… ►

Hypergeometric Functions and Elliptic Integrals. In Current Topics in Analytic Function Theory, H. M. Srivastava and S. Owa (Eds.), pp. 48–85.

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External Links:

MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§19.9(i).

Encodings:

BibTeX

…

48: 14 Legendre and Related Functions

Chapter 14 Legendre and Related Functions

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49: 10.47 Definitions and Basic Properties

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50: 4.13 Lambert $W$-Function

… ► $W_0\left(z\right)$ is a single-valued analytic function on $ℂ\setminus (-\mathrm{\infty },-{\mathrm{e}}^{-1}]$, real-valued when $z>-{\mathrm{e}}^{-1}$, and has a square root branch point at $z=-{\mathrm{e}}^{-1}$. …The other branches $W_k\left(z\right)$ are single-valued analytic functions on $ℂ\setminus (-\mathrm{\infty },0]$, have a logarithmic branch point at $z=0$, and, in the case $k=\pm 1$, have a square root branch point at $z=-{\mathrm{e}}^{-1}\mp 0\mathrm{i}$ respectively. …