Graf addition theorem

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31: 19.35 Other Applications

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§19.35(i) Mathematical

►Generalizations of elliptic integrals appear in analysis of modular theorems of Ramanujan (Anderson et al. (2000)); analysis of Selberg integrals (Van Diejen and Spiridonov (2001)); use of Legendre’s relation (19.7.1) to compute $\pi$ to high precision (Borwein and Borwein (1987, p. 26)). …

32: 1.9 Calculus of a Complex Variable

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DeMoivre’s Theorem

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Cauchy’s Theorem

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Liouville’s Theorem

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Operations

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Dominated Convergence Theorem

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33: Bibliography C

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The Staudt-Clausen theorem. Math. Mag. 34, pp. 131–146.

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External Links:

MathReview (K. Goldberg), ZentralBlatt

Referenced by:

§24.6.

Encodings:

BibTeX

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New proof of the addition theorem for Gegenbauer polynomials. SIAM J. Math. Anal. 2, pp. 347–351.

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External Links:

ISSN 1095-7154, Document, MathReview (Mary L. Boas), ZentralBlatt

Referenced by:

§18.18(ii).

Encodings:

BibTeX

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Short proofs of three theorems on elliptic integrals. SIAM J. Math. Anal. 9 (3), pp. 524–528.

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External Links:

Document, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

§19.26(i).

Encodings:

BibTeX

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The universal von Staudt theorems. Trans. Amer. Math. Soc. 315 (2), pp. 591–603.

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External Links:

ISSN 0002-9947, FullText, MathReview (T. Metsänkylä), ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

BibTeX

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Fourier, Gegenbauer and Jacobi expansions for a power-law fundamental solution of the polyharmonic equation and polyspherical addition theorems. SIGMA Symmetry Integrability Geom. Methods Appl. 9, pp. Paper 042, 26.

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External Links:

ISSN 1815-0659, Document, MathReview (Heinrich Begehr), ZentralBlatt

Referenced by:

§14.28(ii), §14.28(ii).

Encodings:

BibTeX

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34: 25.16 Mathematical Applications

… ►The prime number theorem (27.2.3) is equivalent to the statement … ►

25.16.12 $$H(s,z)+H(z,s)=\zeta \left(s\right)\zeta \left(z\right)+\zeta \left(s+z\right),$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $H(s,z)$: generalized Euler sums, $s$: complex variable and $z$: complex variable

Keywords:

addition formula, connection formula

Source:

Apostol and Vu (1984, (11), p. 91)

Permalink:

http://dlmf.nist.gov/25.16.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §25.16(ii), §25.16 and Ch.25

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35: 18.30 Associated OP’s

… ►For other cases there may also be, in addition to a possible integral as in (18.30.10), a finite sum of discrete weights on the negative real $x$-axis each multiplied by the polynomial product evaluated at the corresponding values of $x$, as in (18.2.3). … ►

Markov’s Theorem

►The ratio $p_n^{(0)}(z)/p_n(z)$, as defined here, thus provides the same statement of Markov’s Theorem, as in (18.2.9_5), but now in terms of differently obtained numerator and denominator polynomials. … ►

18.30.25 $$\underset{n\to \mathrm{\infty }}{lim}{F}_n(x)=\underset{n\to \mathrm{\infty }}{lim}{p}_n^{(0)}(z)/p_n(z)=\frac{1}{{\mu }_0}{\int }_a^b\frac{d\mu (x)}{z-x},$$ $z\in ℂ\backslash [a,b]$.

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Symbols:

$[a,b]$: closed interval, $ℂ$: complex plane, $dx$: differential of $x$ , $\in$: element of, $\int$: integral, $p_n(x)$: polynomial of degree $n$ , $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Proved:

Chihara (1978, Chapter III, Theorem 4.3)(proved)

Referenced by:

§18.30(vi), §18.30(vii)

Permalink:

http://dlmf.nist.gov/18.30.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §18.30(vi), §18.30(vi), §18.30 and Ch.18

… ►The ratio ${\widehat{p}}_{n-1}(x;1)/{\widehat{p}}_n(x)$ is then the $F_n(x)$ of (18.2.35), leading to Markov’s theorem as stated in (18.30.25). …

36: 25 Zeta and Related Functions

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37: Adri B. Olde Daalhuis

… ►Olde Daalhuis has published numerous papers in asymptotics and special functions and in addition to his responsibilities as Mathematics Editor of the DLMF, he is the author or coauthor of the following DLMF Chapters …

38: Bibliography D

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Note on the addition theorem of parabolic cylinder functions. J. Indian Math. Soc. (N. S.) 4, pp. 29–30.

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External Links:

MathReview (M. C. Gray), ZentralBlatt

Referenced by:

§12.13(ii).

Encodings:

BibTeX

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Ramanujan’s master theorem for symmetric cones. Pacific J. Math. 175 (2), pp. 447–490.

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External Links:

ISSN 0030-8730, MathReview (Jacques Faraut), ZentralBlatt

Referenced by:

§35.7(ii), §35.8(iv), §35.8(v).

Encodings:

BibTeX

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On Vandermonde’s theorem, and some more general expansions. Proc. Edinburgh Math. Soc. 25, pp. 114–132.

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External Links:

ZentralBlatt

Referenced by:

§15.4(ii).

Encodings:

BibTeX

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39: 2.1 Definitions and Elementary Properties

… ►For example, if $f(z)$ is analytic for all sufficiently large $|z|$ in a sector $𝐒$ and $f(z)=O\left(z^{\nu }\right)$ as $z\to \mathrm{\infty }$ in $𝐒$, $\nu$ being real, then $f^{\prime }(z)=O\left(z^{\nu -1}\right)$ as $z\to \mathrm{\infty }$ in any closed sector properly interior to $𝐒$ and with the same vertex (Ritt’s theorem). This result also holds with both $O$’s replaced by $o$’s. … ►These include addition, subtraction, multiplication, and division. …

40: 3.8 Nonlinear Equations

… ►The convergence is faster when we use instead the function $f(x)=x\cos x-\sin x$; in addition, the successful interval for the starting value $x_0$ is larger. … ►Initial approximations to the zeros can often be found from asymptotic or other approximations to $f(z)$, or by application of the phase principle or Rouché’s theorem; see §1.10(iv). … …