Mehler–Sonine integrals

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11: 10.9 Integral Representations

§10.9 Integral Representations

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Mehler–Sonine and Related Integrals

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§10.9(ii) Contour Integrals

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Hankel’s Integrals

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12: 14.12 Integral Representations

§14.12 Integral Representations

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§14.12(i)

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Mehler–Dirichlet Formula

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Neumann’s Integral

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Heine’s Integral

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13: Bibliography O

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Tables of Lebedev, Mehler and Generalized Mehler Transforms. Mathematical Note Technical Report 246, Boeing Scientific Research Lab, Seattle.

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Referenced by:

§1.14(viii).

Encodings:

BibTeX

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Complete elliptic integrals resulting from infinite integrals of Bessel functions. J. Res. Nat. Bur. Standards Sect. B 78B (3), pp. 113–135.

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External Links:

ISSN 0160-1741, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

§10.22(vi), §10.43(vi).

Encodings:

BibTeX

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Complete elliptic integrals resulting from infinite integrals of Bessel functions. II. J. Res. Nat. Bur. Standards Sect. B 79B (3-4), pp. 137–170.

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External Links:

ISSN 0160-1741, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

§10.22(vi), §10.43(vi).

Encodings:

BibTeX

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Uniform Airy-type expansions of integrals. SIAM J. Math. Anal. 25 (2), pp. 304–321.

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External Links:

ISSN 0036-1410, Document, MathReview, ZentralBlatt

Referenced by:

§2.4(v).

Encodings:

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A class of integral identities with matrix argument. Duke Math. J. 26 (2), pp. 207–213.

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External Links:

Document, MathReview (R. Bellman), ZentralBlatt

Referenced by:

§35.3(ii).

Encodings:

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14: 14.20 Conical (or Mehler) Functions

§14.20 Conical (or Mehler) Functions

… ►Solutions are known as conical or Mehler functions. … ►

§14.20(iv) Integral Representation

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§14.20(vi) Generalized Mehler–Fock Transformation

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§14.20(x) Zeros and Integrals

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15: 18.10 Integral Representations

§18.10 Integral Representations

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§18.10(i) Dirichlet–Mehler-Type Integral Representations

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Ultraspherical

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Legendre

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Jacobi

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16: 14.1 Special Notation

… ►The main functions treated in this chapter are the Legendre functions ${𝖯}_{\nu }\left(x\right)$, ${𝖰}_{\nu }\left(x\right)$, $P_{\nu }\left(z\right)$, $Q_{\nu }\left(z\right)$; Ferrers functions ${𝖯}_{\nu }^{\mu }\left(x\right)$, ${𝖰}_{\nu }^{\mu }\left(x\right)$ (also known as the Legendre functions on the cut); associated Legendre functions $P_{\nu }^{\mu }\left(z\right)$, $Q_{\nu }^{\mu }\left(z\right)$, ${𝑸}_{\nu }^{\mu }\left(z\right)$; conical functions ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, ${𝖰}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, ${\widehat{𝖰}}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, $P_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, $Q_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$ (also known as Mehler functions). …

17: Bibliography G

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Formulas of the Dirichlet-Mehler Type. In Fractional Calculus and its Applications, B. Ross (Ed.), Lecture Notes in Math., Vol. 457, pp. 207–215.

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External Links:

MathReview (Mourad E. H. Ismail), ZentralBlatt

Referenced by:

§18.10(i).

Encodings:

BibTeX

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Algorithm 471: Exponential integrals. Comm. ACM 16 (12), pp. 761–763.

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Notes:

Includes Algol60 program, maximum accuracy 10S.

External Links:

ISSN 0001-0782, Document

Referenced by:

§6.21(ii), §8.28(vi).

Encodings:

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A table of integrals of the exponential integral. J. Res. Nat. Bur. Standards Sect. B 73B, pp. 191–210.

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External Links:

MathReview (John Todd), ZentralBlatt

Referenced by:

§6.14(iii), §6.17, §6.7(iv).

Encodings:

BibTeX

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Definite integrals of the complete elliptic integral $K$ . J. Res. Nat. Bur. Standards Sect. B 80B (2), pp. 313–323.

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External Links:

MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§19.13(i).

Encodings:

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A method for evaluating certain Bessel integrals. Z. Angew. Math. Phys. 30 (4), pp. 722–723.

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External Links:

ISSN 0044-2275, Document, MathReview

Referenced by:

§10.71(ii).

Encodings:

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18: 14.34 Software

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§14.34(iv) Conical (Mehler) and/or Toroidal Functions

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19: 18.11 Relations to Other Functions

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§18.11(ii) Formulas of Mehler–Heine Type

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20: 18.18 Sums

… ►Assume also the integrals ${\int }_{-1}^1{(f(x))}^2{(1-x)}^{\alpha }{(1+x)}^{\beta }dx$ and ${\int }_{-1}^1{(f^{\prime }(x))}^2{(1-x)}^{\alpha +1}{(1+x)}^{\beta +1}dx$ converge. … ►Assume also ${\int }_0^{\mathrm{\infty }}{(f(x))}^2{\mathrm{e}}^{-x}{x}^{\alpha }dx$ converges. … ►For integral representations for products implied by (18.18.8) and (18.18.9) see (18.17.5) and (18.17.6), respectively. … ►

Hermite

… ►Formula (18.18.28) is known as the Mehler formula. …