Mehler%E2%80%93Dirichlet%20formula

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1: 14.31 Other Applications

… ►Applications of toroidal functions include expansion of vacuum magnetic fields in stellarators and tokamaks (van Milligen and López Fraguas (1994)), analytic solutions of Poisson’s equation in channel-like geometries (Hoyles et al. (1998)), and Dirichlet problems with toroidal symmetry (Gil et al. (2000)). ►

§14.31(ii) Conical Functions

… ►These functions are also used in the Mehler–Fock integral transform (§14.20(vi)) for problems in potential and heat theory, and in elementary particle physics (Sneddon (1972, Chapter 7) and Braaksma and Meulenbeld (1967)). The conical functions and Mehler–Fock transform generalize to Jacobi functions and the Jacobi transform; see Koornwinder (1984a) and references therein. …

2: 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:

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Fourier Expansions. A Collection of Formulas. Academic Press, New York-London.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§1.8(v), §22.11, §4.27.

Encodings:

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An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series. J. Inst. Math. Appl. 20 (3), pp. 379–391.

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External Links:

Document, MathReview (David L. Elliott), ZentralBlatt

Referenced by:

§3.11(ii).

Encodings:

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Connection formulas for second-order differential equations with multiple turning points. SIAM J. Math. Anal. 8 (1), pp. 127–154.

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External Links:

Document, MathReview (Anthony Leung), ZentralBlatt

Referenced by:

§2.8(v), (9.13.13), (9.13.17), §9.13(i), §9.13(i).

Encodings:

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Numerical evaluation of the dilogarithm of complex argument. Celestial Mech. Dynam. Astronom. 62 (1), pp. 93–98.

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External Links:

ISSN 0923-2958, Document, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§25.18(i).

Encodings:

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3: 18.11 Relations to Other Functions

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§18.11(i) Explicit Formulas

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§18.11(ii) Formulas of Mehler–Heine Type

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4: 14.12 Integral Representations

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§14.12(i)

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Mehler–Dirichlet Formula

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5: 14.20 Conical (or Mehler) Functions

§14.20 Conical (or Mehler) Functions

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

14.20.2 $${\widehat{𝖰}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)=\mathrm{\Re }\left({\mathrm{e}}^{\mu \pi \mathrm{i}}{𝖰}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)\right)-\frac{1}{2}\pi \sin \left(\mu \pi \right){𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right).$$

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Defines:

${\widehat{𝖰}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)$: conical function

Symbols:

: Ferrers function of the first kind, : Ferrers 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, $\mathrm{\Re }$: real part, $\sin z$: sine function, $x$: real variable, $\tau$: real variable and $\mu$: general order

Referenced by:

§14.20(i), §14.23

Permalink:

http://dlmf.nist.gov/14.20.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §14.20(i), §14.20 and Ch.14

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14.20.6 $$P_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)=\frac{\mathrm{i}{\mathrm{e}}^{-\mu \pi \mathrm{i}}}{\sinh \left(\tau \pi \right){\left|\mathrm{\Gamma }\left(\mu +\frac{1}{2}+\mathrm{i}\tau \right)\right|}^2}\left(Q_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)-Q_{-\frac{1}{2}-\mathrm{i}\tau }^{\mu }\left(x\right)\right),$$ $\tau \ne 0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $Q_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $x$: real variable, $\tau$: real variable and $\mu$: general order

Referenced by:

§14.20(i)

Permalink:

http://dlmf.nist.gov/14.20.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §14.20(i), §14.20 and Ch.14

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§14.20(vi) Generalized Mehler–Fock Transformation

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6: 18.10 Integral Representations

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

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7: 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). …

8: 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:

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Construction of Gauss-Christoffel quadrature formulas. Math. Comp. 22, pp. 251–270.

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External Links:

ISSN 0025-5718, Document, Link, MathReview (R. E. Barnhill)

Referenced by:

§18.39(iii).

Encodings:

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Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.

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Notes:

Includes Fortran program claiming 12S accuracy

External Links:

ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt

Referenced by:

3rd item, §10.77(ix).

Encodings:

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Dirichlet convolution of cotangent numbers and relative class number formulas. Monatsh. Math. 110 (3-4), pp. 231–256.

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External Links:

ISSN 0026-9255, MathReview (Peter Stevenhagen), ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

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Explicit formulas for Bernoulli numbers. Amer. Math. Monthly 79, pp. 44–51.

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External Links:

FullText, MathReview (B. M. Stewart), ZentralBlatt

Referenced by:

§24.15(iii), §24.6.

Encodings:

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9: 14.34 Software

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

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10: 18.18 Sums

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§18.18(v) Linearization Formulas

… ►Formula (18.18.27) is known as the Hille–Hardy formula. ►

Hermite

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