Watson integrals

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11: 5.9 Integral Representations

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5.9.10_2 $$\mathrm{Ln}\mathrm{\Gamma }\left(z\right)=\left(z-\frac{1}{2}\right)\ln z-z+\frac{1}{2}\ln \left(2\pi \right)+{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}\left(\frac{1}{{\mathrm{e}}^t-1}-\frac{1}{t}+\frac{1}{2}\right)\frac{dt}{t},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{Ln}z$: general logarithm function, $\ln z$: principal branch of logarithm function and $z$: complex variable

Source:

Whittaker and Watson (1927, §12.31)

Referenced by:

§5.9(i), Erratum (V1.1.7) for Additions

Permalink:

http://dlmf.nist.gov/5.9.E10_2

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

This equation was added.

See also:

Annotations for §5.9(i), §5.9(i), §5.9 and Ch.5

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12: 19 Elliptic Integrals

Chapter 19 Elliptic Integrals

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13: 11.7 Integrals and Sums

§11.7 Integrals and Sums

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§11.7(i) Indefinite Integrals

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§11.7(ii) Definite Integrals

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§11.7(iv) Integrals with Respect to Order

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14: 6.12 Asymptotic Expansions

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§6.12(i) Exponential and Logarithmic Integrals

… ►For the function $\chi$ see §9.7(i). … ►

§6.12(ii) Sine and Cosine Integrals

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15: 10.43 Integrals

… ►For collections of integrals of the functions $I_{\nu }\left(z\right)$ and $K_{\nu }\left(z\right)$, including integrals with respect to the order, see Apelblat (1983, §12), Erdélyi et al. (1953b, §§7.7.1–7.7.7 and 7.14–7.14.2), Erdélyi et al. (1954a, b), Gradshteyn and Ryzhik (2000, §§5.5, 6.5–6.7), Gröbner and Hofreiter (1950, pp. 197–203), Luke (1962), Magnus et al. (1966, §3.8), Marichev (1983, pp. 191–216), Oberhettinger (1972), Oberhettinger (1974, §§1.11 and 2.7), Oberhettinger (1990, §§1.17–1.20 and 2.17–2.20), Oberhettinger and Badii (1973, §§1.15 and 2.13), Okui (1974, 1975), Prudnikov et al. (1986b, §§1.11–1.12, 2.15–2.16, 3.2.8–3.2.10, and 3.4.1), Prudnikov et al. (1992a, §§3.15, 3.16), Prudnikov et al. (1992b, §§3.15, 3.16), Watson (1944, Chapter 13), and Wheelon (1968).

16: 22.8 Addition Theorems

… ►A geometric interpretation of (22.8.20) analogous to that of (23.10.5) is given in Whittaker and Watson (1927, p. 530). … ►

22.8.22 $$z_1+z_2+z_3+z_4=2K\left(k\right).$$

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Symbols:

$K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $z$: complex and $k$: modulus

Permalink:

http://dlmf.nist.gov/22.8.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §22.8(iii), §22.8 and Ch.22

… ►If sums/differences of the $z_j$’s are rational multiples of $K\left(k\right)$, then further relations follow. … ►

22.8.24 $$z_1-z_2=z_2-z_3=\frac{2}{3}K\left(k\right),$$

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Symbols:

$K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $z$: complex and $k$: modulus

Permalink:

http://dlmf.nist.gov/22.8.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §22.8(iii), §22.8 and Ch.22

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22.8.26 $$z_1-z_2=z_2-z_3=z_3-z_4=\frac{1}{2}K\left(k\right),$$

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Symbols:

$K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $z$: complex and $k$: modulus

Permalink:

http://dlmf.nist.gov/22.8.E26

Encodings:

pMML, png, TeX

See also:

Annotations for §22.8(iii), §22.8 and Ch.22

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17: 7.12 Asymptotic Expansions

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§7.12(ii) Fresnel Integrals

►The asymptotic expansions of $C\left(z\right)$ and $S\left(z\right)$ are given by (7.5.3), (7.5.4), and … ►They are bounded by $|\csc \left(4\mathrm{ph}z\right)|$ times the first neglected terms when . … ►

§7.12(iii) Goodwin–Staton Integral

►See Olver (1997b, p. 115) for an expansion of $G\left(z\right)$ with bounds for the remainder for real and complex values of $z$.

18: 8.21 Generalized Sine and Cosine Integrals

§8.21 Generalized Sine and Cosine Integrals

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§8.21(iii) Integral Representations

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§8.21(iv) Interrelations

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§8.21(v) Special Values

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19: 11.5 Integral Representations

… ►For further integral representations see Babister (1967, §§3.3, 3.14), Erdélyi et al. (1954a, §§5.17, 15.3), Magnus et al. (1966, p. 114), Oberhettinger (1972), Oberhettinger (1974, §2.7), Oberhettinger and Badii (1973, §2.14), and Watson (1944, pp. 330, 374, and 426).

20: 10.22 Integrals

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10.22.72 $${\int }_0^{\mathrm{\infty }}{J}_{\mu }\left(at\right)J_{\nu }\left(bt\right)J_{\nu }\left(ct\right)t^{1-\mu }dt=\frac{{(bc)}^{\mu -1}\sin \left((\mu -\nu )\pi \right){(\sinh \chi )}^{\mu -\frac{1}{2}}}{{(\frac{1}{2}{\pi }^3)}^{\frac{1}{2}}{a}^{\mu }}{\mathrm{e}}^{(\mu -\frac{1}{2})\mathrm{i}\pi }{Q}_{\nu -\frac{1}{2}}^{\frac{1}{2}-\mu }\left(\cosh \chi \right),$$ $\mathrm{\Re }\mu >-\frac{1}{2},\mathrm{\Re }\nu >-1,a>b+c,\cosh \chi =(a^2-b^2-c^2)/(2bc)$.

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Symbols:

$J_{\nu }\left(z\right)$: Bessel 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, $\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part, $\sin z$: sine function and $\nu$: complex parameter

Keywords:

Mellin transform

Referenced by:

§10.22(iv), §10.22(iv), Erratum (V1.0.21) for Equation (10.22.72)

Permalink:

http://dlmf.nist.gov/10.22.E72

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.21):

Originally, the factor on the right-hand side was written as $\frac{{(bc)}^{\mu -1}\cos \left(\nu \pi \right){(\sinh \chi )}^{\mu -\frac{1}{2}}}{{(\frac{1}{2}{\pi }^3)}^{\frac{1}{2}}{a}^{\mu }}$, which was taken directly from Watson (1944, p. 412, (13.46.5)), who uses a different normalization for the associated Legendre functions of the second kind $Q_{\nu }^{\mu }$. Watson’s $Q_{\nu }^{\mu }$ equals $\frac{\sin \left((\nu +\mu )\pi \right)}{\sin \left(\nu \pi \right)}{\mathrm{e}}^{-\mu \pi \mathrm{i}}{Q}_{\nu }^{\mu }$ in the DLMF.

See also:

Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10

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10.22.78 $$f(x)={\int }_0^{\mathrm{\infty }}{(xt)}^{\frac{1}{2}}\frac{J_{\nu }\left(xt\right)Y_{\nu }\left(at\right)-Y_{\nu }\left(xt\right)J_{\nu }\left(at\right)}{J_{\nu }^2\left(at\right)+Y_{\nu }^2\left(at\right)}{\int }_a^{\mathrm{\infty }}{(yt)}^{\frac{1}{2}}\left(J_{\nu }\left(yt\right)Y_{\nu }\left(at\right)-Y_{\nu }\left(yt\right)J_{\nu }\left(at\right)\right)f(y)dydt,$$ $a>0$.

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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, $dx$: differential of $x$ , $\int$: integral, $x$: real variable, $y$: real variable, $\nu$: complex parameter and $f(x)$: function

Sources:

Titchmarsh (1962a, p. 87); Watson (1944, §14.52)

Referenced by:

§1.18(ix), §1.18(vi), §10.22(vi), Erratum (V1.2.0) for Chapter 10 Additions

Permalink:

http://dlmf.nist.gov/10.22.E78

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §10.22(v), §10.22 and Ch.10

… ►For collections of integrals of the functions $J_{\nu }\left(z\right)$, $Y_{\nu }\left(z\right)$, $H_{\nu }^{(1)}\left(z\right)$, and $H_{\nu }^{(2)}\left(z\right)$, including integrals with respect to the order, see Andrews et al. (1999, pp. 216–225), Apelblat (1983, §12), Erdélyi et al. (1953b, §§7.7.1–7.7.7 and 7.14–7.14.2), Erdélyi et al. (1954a, b), Gradshteyn and Ryzhik (2000, §§5.5 and 6.5–6.7), Gröbner and Hofreiter (1950, pp. 196–204), Luke (1962), Magnus et al. (1966, §3.8), Marichev (1983, pp. 191–216), Oberhettinger (1974, §§1.10 and 2.7), Oberhettinger (1990, §§1.13–1.16 and 2.13–2.16), Oberhettinger and Badii (1973, §§1.14 and 2.12), Okui (1974, 1975), Prudnikov et al. (1986b, §§1.8–1.10, 2.12–2.14, 3.2.4–3.2.7, 3.3.2, and 3.4.1), Prudnikov et al. (1992a, §§3.12–3.14), Prudnikov et al. (1992b, §§3.12–3.14), Watson (1944, Chapters 5, 12, 13, and 14), and Wheelon (1968).