integral representation

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21: 11.5 Integral Representations

§11.5 Integral Representations

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§11.5(i) Integrals Along the Real Line

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§11.5(ii) Contour Integrals

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Mellin–Barnes Integrals

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§11.5(iii) Compendia

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22: 12.5 Integral Representations

§12.5 Integral Representations

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§12.5(i) Integrals Along the Real Line

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§12.5(ii) Contour Integrals

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§12.5(iii) Mellin–Barnes Integrals

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§12.5(iv) Compendia

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23: 9.5 Integral Representations

§9.5 Integral Representations

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§9.5(i) Real Variable

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§9.5(ii) Complex Variable

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24: 33.7 Integral Representations

§33.7 Integral Representations

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25: 14.12 Integral Representations

§14.12 Integral Representations

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§14.12(i)

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14.12.3 $${𝖰}_{\nu }^{\mu }\left(\cos \theta \right)=\frac{{\pi }^{1/2}\mathrm{\Gamma }\left(\nu +\mu +1\right){(\sin \theta )}^{\mu }}{2^{\mu +1}\mathrm{\Gamma }\left(\mu +\frac{1}{2}\right)\mathrm{\Gamma }\left(\nu -\mu +1\right)}\left({\int }_0^{\mathrm{\infty }}\frac{{(\sinh t)}^{2\mu }}{{(\cos \theta +\mathrm{i}\sin \theta \cosh t)}^{\nu +\mu +1}}dt+{\int }_0^{\mathrm{\infty }}\frac{{(\sinh t)}^{2\mu }}{{(\cos \theta -\mathrm{i}\sin \theta \cosh t)}^{\nu +\mu +1}}dt\right),$$ , $\mathrm{\Re }\mu >-\frac{1}{2}$, $\mathrm{\Re }\nu \pm \mu >-1$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part, $\sin z$: sine function, $\mu$: general order and $\nu$: general degree

Permalink:

http://dlmf.nist.gov/14.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §14.12(i), §14.12(i), §14.12 and Ch.14

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§14.12(ii)

… ►For further integral representations see Erdélyi et al. (1953a, pp. 158–159) and Magnus et al. (1966, pp. 184–190), and for contour integrals and other representations see §14.25.

26: 13.16 Integral Representations

§13.16 Integral Representations

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§13.16(i) Integrals Along the Real Line

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§13.16(ii) Contour Integrals

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§13.16(iii) Mellin–Barnes Integrals

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27: 10.32 Integral Representations

§10.32 Integral Representations

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§10.32(i) Integrals along the Real Line

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§10.32(ii) Contour Integrals

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§10.32(iii) Products

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§10.32(iv) Compendia

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28: 35.9 Applications

… ►For applications of the integral representation (35.5.3) see McFarland and Richards (2001, 2002) (statistical estimation of misclassification probabilities for discriminating between multivariate normal populations). …

29: 9.11 Products

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§9.11(iii) Integral Representations

… ►For an integral representation of the Dirac delta involving a product of two $\mathrm{Ai}$ functions see §1.17(ii). ►For further integral representations see Reid (1995, 1997a, 1997b). …

30: 5.9 Integral Representations

§5.9 Integral Representations

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Binet’s Formula

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§5.9(ii) Psi Function, Euler’s Constant, and Derivatives

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