Heine%20transformations%20%28first%2C%20second%2C%20third%29

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2: 14.28 Sums

… ►where the branches of the square roots have their principal values when

z_{1},z_{2}\in(1,\infty)

and are continuous when

z_{1},z_{2}\in\mathbb{C}\setminus(0,1]

. … ►

§14.28(ii) Heine’s Formula

… ►where

\mathcal{E}_{1}

and

\mathcal{E}_{2}

are ellipses with foci at

\pm 1

\mathcal{E}_{2}

being properly interior to

\mathcal{E}_{1}

. The series converges uniformly for

z_{1}

outside or on

\mathcal{E}_{1}

, and

z_{2}

within or on

\mathcal{E}_{2}

. ►For generalizations in terms of Gegenbauer and Jacobi polynomials, see Theorem 2. …

3: Bibliography V

… ►

G. Valent (1986) An integral transform involving Heun functions and a related eigenvalue problem. SIAM J. Math. Anal. 17 (3), pp. 688–703.

ⓘ

External Links:: ISSN 0036-1410, Document, MathReview (William L. Perry), ZentralBlatt
Referenced by:: §31.10(i).
Encodings:: BibTeX

… ►

R. Vidūnas (2005) Transformations of some Gauss hypergeometric functions. J. Comput. Appl. Math. 178 (1-2), pp. 473–487.

ⓘ

External Links:: ISSN 0377-0427, Document, Link, MathReview (Wolfgang Bühring), ZentralBlatt
Referenced by:: §15.8(v).
Encodings:: BibTeX

… ►

N. Ja. Vilenkin and A. U. Klimyk (1993) Representation of Lie Groups and Special Functions. Volume 2: Class I Representations, Special Functions, and Integral Transforms. Mathematics and its Applications (Soviet Series), Vol. 74, Kluwer Academic Publishers Group, Dordrecht.

ⓘ

Notes:: Translated from the Russian by V. A. Groza and A. A. Groza
External Links:: ISBN 0-7923-1492-1, MathReview (P. D. F. Ion), ZentralBlatt
Referenced by:: §18.38(iii).
Encodings:: BibTeX

… ►

H. Volkmer (1999) Expansions in products of Heine-Stieltjes polynomials. Constr. Approx. 15 (4), pp. 467–480.

ⓘ

External Links:: ISSN 0176-4276, Document, MathReview (Luoqing Li), ZentralBlatt
Referenced by:: §31.15(iii).
Encodings:: BibTeX

… ►

H. Volkmer (2004a) Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation. Constr. Approx. 20 (1), pp. 39–54.

ⓘ

External Links:: ISSN 0176-4276, Document, MathReview, ZentralBlatt
Referenced by:: §30.16(i), §30.16(ii), §30.16(ii).
Encodings:: BibTeX

…

4: 28 Mathieu Functions and Hill’s Equation

Chapter 28 Mathieu Functions and Hill’s Equation

…

5: 20 Theta Functions

Chapter 20 Theta Functions

…

6: Bibliography K

… ►

R. B. Kearfott, M. Dawande, K. Du, and C. Hu (1994) Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.

ⓘ

External Links:: ISSN 0098-3500, Document, GAMS (module 737; package TOMS), ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

… ►

M. K. Kerimov (1980) Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.

ⓘ

External Links:: Document, MathReview (Matti Jutila), ZentralBlatt
Referenced by:: §25.18(i).
Encodings:: BibTeX

… ►

A. V. Kitaev and A. H. Vartanian (2004) Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.

ⓘ

External Links:: ISSN 0266-5611, Document, MathReview (Shun Shimomura), ZentralBlatt
Referenced by:: §32.11(iv).
Encodings:: BibTeX

… ►

T. H. Koornwinder (2009) The Askey scheme as a four-manifold with corners. Ramanujan J. 20 (3), pp. 409–439.

ⓘ

External Links:: ISSN 1382-4090, Document, FullText, MathReview (José Luis López), ZentralBlatt
Referenced by:: §18.26(v), §18.26(v).
Encodings:: BibTeX

… ►

C. Krattenthaler (1993) HYP and HYPQ. Mathematica packages for the manipulation of binomial sums and hypergeometric series respectively

q

-binomial sums and basic hypergeometric series. Séminaire Lotharingien de Combinatoire 30, pp. 61–76.

ⓘ

Notes:: Also published in J. Symbolic Comput. (1995), v. 20, no. 5–6, pp. 737–744.
External Links:: Code, Document, MathReview (G. Gasper), ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

…

7: Bibliography M

… ►

A. J. MacLeod (1996b) Rational approximations, software and test methods for sine and cosine integrals. Numer. Algorithms 12 (3-4), pp. 259–272.

ⓘ

Notes:: Includes Fortran code, maximum accuracy 20S.
External Links:: ISSN 1017-1398, Document, MathReview, ZentralBlatt
Referenced by:: §6.13, 4th item, §6.21(ii).
Encodings:: BibTeX

… ►

W. Magnus and S. Winkler (1966) Hill’s Equation. Interscience Tracts in Pure and Applied Mathematics, No. 20, Interscience Publishers John Wiley & Sons, New York-London-Sydney.

ⓘ

Notes:: Reprinted by Dover Publications, Inc., New York, 1979.
External Links:: MathReview (H. Hochstadt), ZentralBlatt
Referenced by:: §1.3(iii), §28.29(i), §28.29(ii), §28.29(ii), §28.29(iii), §28.29(iii), §28.29(iii), §28.30(i), Ch.28, §29.3(i), §29.9.
Encodings:: BibTeX

… ►

Fr. Mechel (1966) Calculation of the modified Bessel functions of the second kind with complex argument. Math. Comp. 20 (95), pp. 407–412.

ⓘ

External Links:: ISSN 0025-5718, FullText, MathReview (H. E. Fettis), ZentralBlatt
Referenced by:: §10.74(iii).
Encodings:: BibTeX

… ►

R. Metzler, J. Klafter, and J. Jortner (1999) Hierarchies and logarithmic oscillations in the temporal relaxation patterns of proteins and other complex systems. Proc. Nat. Acad. Sci. U .S. A. 96 (20), pp. 11085–11089.

ⓘ

External Links:: FullText
Referenced by:: §8.24(i).
Encodings:: BibTeX

… ►

D. S. Moak (1981) The

q

-analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.

ⓘ

External Links:: ISSN 0022-247X, Document, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §18.27(v).
Encodings:: BibTeX

…

8: 18.11 Relations to Other Functions

… ►

18.11.1

\mathsf{P}^{m}_{n}\left(x\right)={\left(\tfrac{1}{2}\right)_{m}}(-2)^{m}(1-x^{% 2})^{\frac{1}{2}m}C^{(m+\frac{1}{2})}_{n-m}\left(x\right)={\left(n+1\right)_{m% }}(-2)^{-m}(1-x^{2})^{\frac{1}{2}m}P^{(m,m)}_{n-m}\left(x\right),

0\leq m\leq n

ⓘ

Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial, $m$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §14.3(iv), §18.11(i)
Permalink:: http://dlmf.nist.gov/18.11.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §18.11(i), §18.11(i), §18.11 and Ch.18

►For the Ferrers function

\mathsf{P}^{m}_{n}\left(x\right)

, see §14.3(i). … ►

§18.11(ii) Formulas of Mehler–Heine Type

… ►

18.11.5

\lim_{n\to\infty}\frac{1}{n^{\alpha}}P^{(\alpha,\beta)}_{n}\left(1-\frac{z^{2}% }{2n^{2}}\right)=\lim_{n\to\infty}\frac{1}{n^{\alpha}}P^{(\alpha,\beta)}_{n}% \left(\cos\frac{z}{n}\right)=\frac{2^{\alpha}}{z^{\alpha}}J_{\alpha}\left(z% \right).

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, $\cos\NVar{z}$ : cosine function, $z$ : complex variable and $n$ : nonnegative integer
A&S Ref:: 22.15.1
Referenced by:: §18.11(ii), §18.11(ii)
Permalink:: http://dlmf.nist.gov/18.11.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §18.11(ii), §18.11(ii), §18.11 and Ch.18

… ►

18.11.6

\lim_{n\to\infty}\frac{1}{n^{\alpha}}L^{(\alpha)}_{n}\left(\frac{z}{n}\right)=% \frac{1}{z^{\frac{1}{2}\alpha}}J_{\alpha}\left(2z^{\frac{1}{2}}\right).

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, $z$ : complex variable and $n$ : nonnegative integer
A&S Ref:: 22.15.2
Referenced by:: §18.11(ii)
Permalink:: http://dlmf.nist.gov/18.11.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §18.11(ii), §18.11(ii), §18.11 and Ch.18

…

9: 17.6 ${{}_{2}\phi_{1}}$ Function

… ►

Heine’s First Transformation

… ►

Heine’s Second Tranformation

… ►

Heine’s Third Transformation

… ►

Fine’s Second Transformation

… ►

Heine’s Contiguous Relations

…

10: 19.36 Methods of Computation

… ►

§19.36(ii) Quadratic Transformations

… ►Thompson (1997, pp. 499, 504) uses descending Landen transformations for both

F\left(\phi,k\right)

and

E\left(\phi,k\right)

. … ►The function

\operatorname{el2}\left(x,k_{c},a,b\right)

is computed by descending Landen transformations if

x

is real, or by descending Gauss transformations if

x

is complex (Bulirsch (1965b)). … ►Bulirsch (1969a, b) extend Bartky’s transformation to

\operatorname{el3}\left(x,k_{c},p\right)

by expressing it in terms of the first incomplete integral, a complete integral of the third kind, and a more complicated integral to which Bartky’s method can be applied. … ►For computation of Legendre’s integral of the third kind, see Abramowitz and Stegun (1964, §§17.7 and 17.8, Examples 15, 17, 19, and 20). …

Heine%20transformations%20%28first%2C%20second%2C%20third%29

1—10 of 858 matching pages

1: 1.14 Integral Transforms

§1.14 Integral Transforms

§1.14(i) Fourier Transform

§1.14(iii) Laplace Transform

Fourier Transform

Laplace Transform

2: 14.28 Sums

§14.28(ii) Heine’s Formula

3: Bibliography V

4: 28 Mathieu Functions and Hill’s Equation

Chapter 28 Mathieu Functions and Hill’s Equation

5: 20 Theta Functions

Chapter 20 Theta Functions

6: Bibliography K

7: Bibliography M

8: 18.11 Relations to Other Functions

§18.11(ii) Formulas of Mehler–Heine Type

9: 17.6 ${{}_{2}\phi_{1}}$ Function

Heine’s First Transformation

Heine’s Second Tranformation

Heine’s Third Transformation

Fine’s Second Transformation

Heine’s Contiguous Relations

10: 19.36 Methods of Computation

§19.36(ii) Quadratic Transformations

Heine%20transformations%20%28first%2C%20second%2C%20third%29

1—10 of 858 matching pages

1: 1.14 Integral Transforms

§1.14 Integral Transforms

§1.14(i) Fourier Transform

§1.14(iii) Laplace Transform

Fourier Transform

Laplace Transform

2: 14.28 Sums

§14.28(ii) Heine’s Formula

3: Bibliography V

4: 28 Mathieu Functions and Hill’s Equation

Chapter 28 Mathieu Functions and Hill’s Equation

5: 20 Theta Functions

Chapter 20 Theta Functions

6: Bibliography K

7: Bibliography M

8: 18.11 Relations to Other Functions

§18.11(ii) Formulas of Mehler–Heine Type

9: 17.6 ϕ 1 2 Function

Heine’s First Transformation

Heine’s Second Tranformation

Heine’s Third Transformation

Fine’s Second Transformation

Heine’s Contiguous Relations

10: 19.36 Methods of Computation

§19.36(ii) Quadratic Transformations

9: 17.6 ${{}_{2}\phi_{1}}$ Function