Watson integrals

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1: 16.24 Physical Applications

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§16.24(i) Random Walks

►Generalized hypergeometric functions and Appell functions appear in the evaluation of the so-called Watson integrals which characterize the simplest possible lattice walks. …

2: 2.3 Integrals of a Real Variable

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§2.3(ii) Watson’s Lemma

… ►(In other words, differentiation of (2.3.8) with respect to the parameter $\lambda$ (or $\mu$) is legitimate.) …

3: 2.4 Contour Integrals

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§2.4(i) Watson’s Lemma

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4: 10.54 Integral Representations

§10.54 Integral Representations

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10.54.1 $${𝗃}_n\left(z\right)=\frac{z^n}{2^{n+1}n!}{\int }_0^{\pi }\cos \left(z\cos \theta \right){(\sin \theta )}^{2n+1}d\theta .$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $\sin z$: sine function, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $n$: integer and $z$: complex variable

A&S Ref:

10.1.13

Referenced by:

§10.54

Permalink:

http://dlmf.nist.gov/10.54.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.54 and Ch.10

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10.54.2 $${𝗃}_n\left(z\right)=\frac{{(-\mathrm{i})}^n}{2}{\int }_0^{\pi }{\mathrm{e}}^{\mathrm{i}z\cos \theta }{P}_n\left(\cos \theta \right)\sin \theta d\theta .$$

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $n$: integer and $z$: complex variable

A&S Ref:

10.1.14

Permalink:

http://dlmf.nist.gov/10.54.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.54 and Ch.10

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10.54.3 $${𝗄}_n\left(z\right)=\frac{\pi }{2}{\int }_1^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{P}_n\left(t\right)dt,$$

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{ph}$: phase, ${𝗄}_n\left(z\right)$: modified spherical Bessel function, $n$: integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/10.54.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.54 and Ch.10

… ►Additional integral representations can be obtained by combining the definitions (10.47.3)–(10.47.9) with the results given in §10.9 and §10.32.

5: 10.32 Integral Representations

… ►For collections of integral representations of modified Bessel functions, or products of modified Bessel functions, see Erdélyi et al. (1953b, §§7.3, 7.12, and 7.14.2), Erdélyi et al. (1954a, pp. 48–60, 105–115, 276–285, and 357–359), Gröbner and Hofreiter (1950, pp. 193–194), Magnus et al. (1966, §3.7), Marichev (1983, pp. 191–216), and Watson (1944, Chapters 6, 12, and 13).

6: 10.9 Integral Representations

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10.9.26 $$J_{\mu }\left(z\right)J_{\nu }\left(z\right)=\frac{2}{\pi }{\int }_0^{\pi /2}{J}_{\mu +\nu }\left(2z\cos \theta \right)\cos \left((\mu -\nu )\theta \right)d\theta ,$$ $\mathrm{\Re }\left(\mu +\nu \right)>-1$.

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $z$: complex variable and $\nu$: complex parameter

Source:

Watson (1944, (5.43.1), p. 150, originally $\cos (\mu -\nu )\theta$. Clarified to $\cos \left((\mu -\nu )\theta \right)$ in 1.0.18.)

Referenced by:

Erratum (V1.0.18) for (10.9.26)

Permalink:

http://dlmf.nist.gov/10.9.E26

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.18):

The factor on the right-hand side of the integral, originally $\cos (\mu -\nu )\theta$, was replaced with $\cos \left((\mu -\nu )\theta \right)$ to clarify the meaning. Note that this ambiguity originates with Watson (1944, p. 150).

See also:

Annotations for §10.9(iii), §10.9 and Ch.10

… ►For collections of integral representations of Bessel and Hankel functions see Erdélyi et al. (1953b, §§7.3 and 7.12), Erdélyi et al. (1954a, pp. 43–48, 51–60, 99–105, 108–115, 123–124, 272–276, and 356–357), Gröbner and Hofreiter (1950, pp. 189–192), Marichev (1983, pp. 191–192 and 196–210), Magnus et al. (1966, §3.6), and Watson (1944, Chapter 6).

7: 22.14 Integrals

§22.14 Integrals

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§22.14(i) Indefinite Integrals of Jacobian Elliptic Functions

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§22.14(iii) Other Indefinite Integrals

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§22.14(iv) Definite Integrals

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8: 11.10 Anger–Weber Functions

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§11.10(i) Definitions

… ►For the Fresnel integrals $C$ and $S$ see §7.2(iii). … ►

§11.10(x) Integrals and Sums

►For collections of integral representations and integrals see Erdélyi et al. (1954a, §§4.19 and 5.17), Marichev (1983, pp. 194–195 and 214–215), Oberhettinger (1972, p. 128), Oberhettinger (1974, §§1.12 and 2.7), Oberhettinger (1990, pp. 105 and 189–190), Prudnikov et al. (1990, §§1.5 and 2.8), Prudnikov et al. (1992a, §3.18), Prudnikov et al. (1992b, §3.18), and Zanovello (1977). …

9: 13.8 Asymptotic Approximations for Large Parameters

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10: 20.9 Relations to Other Functions

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§20.9(i) Elliptic Integrals

… ►and the notation of §19.2(ii), the complete Legendre integrals of the first kind may be expressed as theta functions: … ►

$K^{\prime }\left(k\right)=-\mathrm{i}\tau K\left(k\right),$

… ►In the case of the symmetric integrals, with the notation of §19.16(i) we have … ►The relations (20.9.1) and (20.9.2) between $k$ and $\tau$ (or $q$) are solutions of Jacobi’s inversion problem; see Baker (1995) and Whittaker and Watson (1927, pp. 480–485). …