Heine transformations (first, second, third)

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31: 19.22 Quadratic Transformations

§19.22 Quadratic Transformations

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Bartky’s Transformation

… ►Descending Gauss transformations include, as special cases, transformations of complete integrals into complete integrals; ascending Landen transformations do not. … ►

32: 10.22 Integrals

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§10.22(v) Hankel Transform

►The Hankel transform (or Bessel transform) of a function $f(x)$ is defined as … ►For collections of Hankel transforms see Erdélyi et al. (1954b, Chapter 8) and Oberhettinger (1972). ►The following two formulas are generalizations of the Hankel transform. …This is the Weber transform. …

33: Bibliography F

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Un théorème de Paley-Wiener pour la transformation de Fourier sur un espace riemannien symétrique de rang un. J. Funct. Anal. 49 (2), pp. 230–268.

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External Links:

ISSN 0022-1236, Document, Link, MathReview (Mogens Flensted-Jensen), ZentralBlatt

Referenced by:

§18.39(i).

Encodings:

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Tables of Elliptic Integrals of the First, Second, and Third Kind. Technical report Technical Report ARL 64-232, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio.

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Notes:

Reviewed in Math. Comp. v. 1919(1965)509. Table erratum: Math. Comp. v. 20 (1966), no. 96, pp. 639-640.

Referenced by:

§19.37(ii), §19.37(ii), §19.37(iii), §19.37(iii), §19.37(iii).

Encodings:

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Calculation of elliptic integrals of the third kind by means of Gauss’ transformation. Math. Comp. 19 (89), pp. 97–104.

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External Links:

FullText, MathReview (J. E. L. Peck), ZentralBlatt

Referenced by:

§19.36(ii).

Encodings:

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On the asymptotic expansion of Mellin transforms. SIAM J. Math. Anal. 18 (1), pp. 273–282.

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External Links:

ISSN 0036-1410, Document, Link, MathReview (Archibald Brown), ZentralBlatt

Referenced by:

§2.3(vi).

Encodings:

BibTeX

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Fast computation of incomplete elliptic integral of first kind by half argument transformation. Numer. Math. 116 (4), pp. 687–719.

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External Links:

ISSN 0029-599X, Document, FullText, MathReview (Mehdi Hassani), ZentralBlatt

Referenced by:

§19.36(iv), §19.36(iv).

Encodings:

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34: 16.16 Transformations of Variables

§16.16 Transformations of Variables

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§16.16(i) Reduction Formulas

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§16.16(ii) Other Transformations

… ►For quadratic transformations of Appell functions see Carlson (1976).

35: 10.5 Wronskians and Cross-Products

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10.5.1 $$𝒲\left\{J_{\nu }\left(z\right),J_{-\nu }\left(z\right)\right\}=J_{\nu +1}\left(z\right)J_{-\nu }\left(z\right)+J_{\nu }\left(z\right)J_{-\nu -1}\left(z\right)=-2\sin \left(\nu \pi \right)/(\pi z),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.15

Permalink:

http://dlmf.nist.gov/10.5.E1

Encodings:

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See also:

Annotations for §10.5 and Ch.10

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10.5.2 $$𝒲\left\{J_{\nu }\left(z\right),Y_{\nu }\left(z\right)\right\}=J_{\nu +1}\left(z\right)Y_{\nu }\left(z\right)-J_{\nu }\left(z\right)Y_{\nu +1}\left(z\right)=2/(\pi z),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.16

Permalink:

http://dlmf.nist.gov/10.5.E2

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See also:

Annotations for §10.5 and Ch.10

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10.5.3 $$𝒲\left\{J_{\nu }\left(z\right),H_{\nu }^{(1)}\left(z\right)\right\}=J_{\nu +1}\left(z\right)H_{\nu }^{(1)}\left(z\right)-J_{\nu }\left(z\right)H_{\nu +1}^{(1)}\left(z\right)=2\mathrm{i}/(\pi z),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.5.E3

Encodings:

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See also:

Annotations for §10.5 and Ch.10

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10.5.4 $$𝒲\left\{J_{\nu }\left(z\right),H_{\nu }^{(2)}\left(z\right)\right\}=J_{\nu +1}\left(z\right)H_{\nu }^{(2)}\left(z\right)-J_{\nu }\left(z\right)H_{\nu +1}^{(2)}\left(z\right)=-2\mathrm{i}/(\pi z),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.5.E4

Encodings:

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See also:

Annotations for §10.5 and Ch.10

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10.5.5 $$𝒲\left\{H_{\nu }^{(1)}\left(z\right),H_{\nu }^{(2)}\left(z\right)\right\}=H_{\nu +1}^{(1)}\left(z\right)H_{\nu }^{(2)}\left(z\right)-H_{\nu }^{(1)}\left(z\right)H_{\nu +1}^{(2)}\left(z\right)=-4\mathrm{i}/(\pi z).$$

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Symbols:

$H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.17

Permalink:

http://dlmf.nist.gov/10.5.E5

Encodings:

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See also:

Annotations for §10.5 and Ch.10

36: Bibliography T

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LSFBTR: A subroutine for calculating spherical Bessel transforms. Comput. Phys. Comm. 30 (1), pp. 93–99.

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Notes:

Single-precision Fortran.

External Links:

ISSN 0010-4655, Document, Code

Referenced by:

9th item.

Encodings:

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Laplace type integrals: Transformation to standard form and uniform asymptotic expansions. Quart. Appl. Math. 43 (1), pp. 103–123.

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External Links:

ISSN 0033-569X, FullText, MathReview (I. Fenyő), ZentralBlatt

Referenced by:

§2.4(i).

Encodings:

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Asymptotic expansions of Kummer hypergeometric functions for large values of the parameters. Integral Transforms Spec. Funct. 33 (1), pp. 16–31.

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External Links:

ISSN 1065-2469, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§13.8(iv).

Encodings:

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Introduction to the Theory of Fourier Integrals. Third edition, Chelsea Publishing Co., New York.

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Notes:

Original edition was published in 1937 by Oxford University Press. Expanded bibliography and some improvements were made for the second and third editions.

External Links:

ISBN 0-8284-0324-4, MathReview, ZentralBlatt

Referenced by:

§1.14(i), §1.14(ii), §1.14(iv), §1.14(v), §1.14(vii), §1.8(iv), §10.22(v), §10.32(iii), §5.13.

Encodings:

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Algebraic transformations of hypergeometric functions and automorphic forms on Shimura curves. Trans. Amer. Math. Soc. 365 (12), pp. 6697–6729.

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External Links:

ISSN 0002-9947, Document, FullText, MathReview (John M. Voight), ZentralBlatt

Referenced by:

§15.8(v), §15.8(v).

Encodings:

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37: 19.1 Special Notation

… ►All derivatives are denoted by differentials, not by primes. … ►of the first, second, and third kinds, respectively, and Legendre’s incomplete integrals …of the first, second, and third kinds, respectively. … ►However, it should be noted that in Chapter 8 of Abramowitz and Stegun (1964) the notation used for elliptic integrals differs from Chapter 17 and is consistent with that used in the present chapter and the rest of the NIST Handbook and DLMF. … ►The first three functions are incomplete integrals of the first, second, and third kinds, and the $\mathrm{cel}$ function includes complete integrals of all three kinds.

38: Bibliography K

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A Fortran computer program for calculating the prolate spheroidal radial functions of the first and second kind and their first derivatives. NRL Report No. 7012 Naval Res. Lab.  Washingtion, D.C..

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Referenced by:

§30.18(iii).

Encodings:

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A Fortran computer program for calculating the prolate and oblate angle functions of the first kind and their first and second derivatives. NRL Report No. 7161 Naval Res. Lab.  Washingtion, D.C..

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Referenced by:

§30.18(iii).

Encodings:

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Elliptic asymptotics of the first and second Painlevé transcendents. Uspekhi Mat. Nauk 49 (1(295)), pp. 77–140 (Russian).

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Notes:

In Russian. English translation: Russian Math. Surveys 49(1994), no. 1, pp. 81–150.

External Links:

ISSN 0042-1316, Document, MathReview (V. A. Golubeva), ZentralBlatt

Referenced by:

§32.11(i), §32.12(ii).

Encodings:

BibTeX

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A new proof of a Paley-Wiener type theorem for the Jacobi transform. Ark. Mat. 13, pp. 145–159.

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External Links:

Document, MathReview (G. Gasper), ZentralBlatt

Referenced by:

§18.10(i).

Encodings:

BibTeX

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A Handbook of Methods of Approximate Fourier Transformation and Inversion of the Laplace Transformation. Mir, Moscow.

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Notes:

Translated from the Russian by George Yankovsky. Reprint of the 1977 translation

External Links:

MathReview, ZentralBlatt

Referenced by:

§3.5(viii).

Encodings:

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39: 10.2 Definitions

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§10.2(ii) Standard Solutions

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Bessel Function of the First Kind

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Bessel Function of the Second Kind (Weber’s Function)

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Bessel Functions of the Third Kind (Hankel Functions)

►These solutions of (10.2.1) are denoted by $H_{\nu }^{(1)}\left(z\right)$ and $H_{\nu }^{(2)}\left(z\right)$, and their defining properties are given by …

40: 10.11 Analytic Continuation

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10.11.2 $$Y_{\nu }\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)={\mathrm{e}}^{-m\nu \pi \mathrm{i}}{Y}_{\nu }\left(z\right)+2\mathrm{i}\sin \left(m\nu \pi \right)\cot \left(\nu \pi \right)J_{\nu }\left(z\right).$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $m$: integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.36

Referenced by:

§10.11, §10.47(v)

Permalink:

http://dlmf.nist.gov/10.11.E2

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Annotations for §10.11 and Ch.10

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10.11.6 $$Y_n\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)={(-1)}^{mn}(Y_n\left(z\right)+2\mathrm{i}m{J}_n\left(z\right)),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $m$: integer, $n$: integer and $z$: complex variable

Referenced by:

§10.11

Permalink:

http://dlmf.nist.gov/10.11.E6

Encodings:

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See also:

Annotations for §10.11 and Ch.10

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10.11.7 $$H_n^{(1)}\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)={(-1)}^{mn-1}((m-1)H_n^{(1)}\left(z\right)+m{H}_n^{(2)}\left(z\right)),$$

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Symbols:

$H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $m$: integer, $n$: integer and $z$: complex variable

Referenced by:

§10.17(i)

Permalink:

http://dlmf.nist.gov/10.11.E7

Encodings:

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See also:

Annotations for §10.11 and Ch.10

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10.11.8 $$H_n^{(2)}\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)={(-1)}^{mn}(m{H}_n^{(1)}\left(z\right)+(m+1)H_n^{(2)}\left(z\right)).$$

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Symbols:

$H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $m$: integer, $n$: integer and $z$: complex variable

Referenced by:

§10.11, §10.17(i)

Permalink:

http://dlmf.nist.gov/10.11.E8

Encodings:

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See also:

Annotations for §10.11 and Ch.10

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$H_{\nu }^{(1)}\left(\overline{z}\right)=\overline{H_{\nu }^{(2)}\left(z\right)},$ $H_{\nu }^{(2)}\left(\overline{z}\right)=\overline{H_{\nu }^{(1)}\left(z\right)}.$

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