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31: 10.43 Integrals

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10.43.2 $$\int z^{\nu }{𝒵}_{\nu }\left(z\right)dz={\pi }^{\frac{1}{2}}{2}^{\nu -1}\mathrm{\Gamma }\left(\nu +\frac{1}{2}\right)z\left({𝒵}_{\nu }\left(z\right){𝐋}_{\nu -1}\left(z\right)-{𝒵}_{\nu -1}\left(z\right){𝐋}_{\nu }\left(z\right)\right),$$ $\nu \ne -\frac{1}{2}$.

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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 $x$ , $\int$: integral, ${𝐋}_{\nu }\left(z\right)$: modified Struve function, ${𝒵}_{\nu }\left(z\right)$: modified cylinder function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

11.1.8 (Case $\nu =0$ only)

Referenced by:

§10.43(i)

Permalink:

http://dlmf.nist.gov/10.43.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.43(i), §10.43 and Ch.10

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10.43.12 $${\mathrm{Ki}}_{\alpha }\left(x\right)={\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-x\cosh t}}{{(\cosh t)}^{\alpha }}dt,$$ $x>0$.

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Symbols:

${\mathrm{Ki}}_{\alpha }\left(x\right)$: Bickley function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\int$: integral and $x$: real variable

A&S Ref:

11.2.10 (with other conditions)

Referenced by:

§10.43(iii)

Permalink:

http://dlmf.nist.gov/10.43.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §10.43(iii), §10.43 and Ch.10

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10.43.21 $${\int }_0^{\mathrm{\infty }}\sin \left(at\right)K_0\left(t\right)dt=\frac{\mathrm{arcsinh}a}{{(1+a^2)}^{\frac{1}{2}}},$$ .

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Symbols:

$dx$: differential of $x$ , $\mathrm{arcsinh}z$: inverse hyperbolic sine function, $\mathrm{\Im }$: imaginary part, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind and $\sin z$: sine function

A&S Ref:

11.4.15

Permalink:

http://dlmf.nist.gov/10.43.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §10.43(iv), §10.43 and Ch.10

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10.43.25 $${\int }_0^{\mathrm{\infty }}{K}_{\nu }\left(bt\right)\exp \left(-p^2{t}^2\right)dt=\frac{\sqrt{\pi }}{4p}\sec \left(\frac{1}{2}\pi \nu \right)\exp \left(\frac{b^2}{8p^2}\right)K_{\frac{1}{2}\nu }\left(\frac{b^2}{8p^2}\right),$$ , $\mathrm{\Re }\left(p^2\right)>0$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\exp z$: exponential function, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{\Re }$: real part, $\sec z$: secant function and $\nu$: complex parameter

A&S Ref:

11.4.32 (Case $\nu =0$)

Referenced by:

§10.43(iv)

Permalink:

http://dlmf.nist.gov/10.43.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §10.43(iv), §10.43 and Ch.10

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10.43.30 $$f(y)=\frac{2y}{{\pi }^2}\sinh \left(\pi y\right){\int }_0^{\mathrm{\infty }}\frac{g(x)}{x}{K}_{\mathrm{i}y}\left(x\right)dx.$$

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Defines:

$f(x)$: Kontorovich–Lebedev transform of $g(x)$ (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $x$: real variable, $y$: real variable and $g(x)$: function

Referenced by:

§10.43(v)

Permalink:

http://dlmf.nist.gov/10.43.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §10.43(v), §10.43 and Ch.10

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32: 29.5 Special Cases and Limiting Forms

§29.5 Special Cases and Limiting Forms

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29.5.5 $$\underset{k\to 1-}{lim}\frac{{\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)}{{\mathrm{𝐸𝑐}}_{\nu }^m(0,k^2)}=\underset{k\to 1-}{lim}\frac{{\mathrm{𝐸𝑠}}_{\nu }^{m+1}(z,k^2)}{{\mathrm{𝐸𝑠}}_{\nu }^{m+1}(0,k^2)}=\frac{1}{{(\cosh z)}^{\mu }}F(\genfrac{}{}{0pt}{}{\frac{1}{2}\mu -\frac{1}{2}\nu ,\frac{1}{2}\mu +\frac{1}{2}\nu +\frac{1}{2}}{\frac{1}{2}};{\tanh }^{2}z),$$ $m$ even,

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Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$: Lamé function, ${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$: Lamé function, $\cosh z$: hyperbolic cosine function, $\tanh z$: hyperbolic tangent function, $m$: nonnegative integer, $z$: complex variable, $k$: real parameter, $\nu$: real parameter and $\mu$: parameter

Permalink:

http://dlmf.nist.gov/29.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §29.5 and Ch.29

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29.5.6 $$\underset{k\to 1-}{lim}\frac{{\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)}{{d{\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)/dz|}_{z=0}}=\underset{k\to 1-}{lim}\frac{{\mathrm{𝐸𝑠}}_{\nu }^{m+1}(z,k^2)}{{d{\mathrm{𝐸𝑠}}_{\nu }^{m+1}(z,k^2)/dz|}_{z=0}}=\frac{\tanh z}{{(\cosh z)}^{\mu }}F(\genfrac{}{}{0pt}{}{\frac{1}{2}\mu -\frac{1}{2}\nu +\frac{1}{2},\frac{1}{2}\mu +\frac{1}{2}\nu +1}{\frac{3}{2}};{\tanh }^{2}z),$$ $m$ odd,

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Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$: Lamé function, ${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$: Lamé function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\cosh z$: hyperbolic cosine function, $\tanh z$: hyperbolic tangent function, $m$: nonnegative integer, $z$: complex variable, $k$: real parameter, $\nu$: real parameter and $\mu$: parameter

Permalink:

http://dlmf.nist.gov/29.5.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §29.5 and Ch.29

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33: 10.22 Integrals

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10.22.47 $${\int }_0^{\mathrm{\infty }}\frac{t^{\nu }{Y}_{\nu }\left(at\right)}{t^2+b^2}dt=-b^{\nu -1}{K}_{\nu }\left(ab\right),$$ .

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Symbols:

$Y_{\nu }\left(z\right)$: Bessel function of the second kind, $dx$: differential of $x$ , $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{\Re }$: real part and $\nu$: complex parameter

Keywords:

Mellin transform

A&S Ref:

11.4.46 (is the special case $\nu =0$.)

Referenced by:

§10.22(iv)

Permalink:

http://dlmf.nist.gov/10.22.E47

Encodings:

pMML, png, TeX

See also:

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

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10.22.48 $${\int }_0^{\mathrm{\infty }}{J}_{\mu }\left(x\cosh \varphi \right){(\cosh \varphi )}^{1-\mu }{(\sinh \varphi )}^{2\nu +1}d\varphi =2^{\nu }\mathrm{\Gamma }\left(\nu +1\right)x^{-\nu -1}{J}_{\mu -\nu -1}\left(x\right),$$ $x>0,\mathrm{\Re }\nu >-1,\mathrm{\Re }\mu >2\mathrm{\Re }\nu +\frac{1}{2}$.

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable and $\nu$: complex parameter

Referenced by:

§10.22(iv)

Permalink:

http://dlmf.nist.gov/10.22.E48

Encodings:

pMML, png, TeX

See also:

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

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Weber–Schafheitlin Discontinuous Integrals, including Special Cases

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10.22.65

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $dx$: differential of $x$ , $\int$: integral and $\ln z$: principal branch of logarithm function

A&S Ref:

11.4.43 (Case $b=0$.)

Referenced by:

§10.22(iv)

Permalink:

http://dlmf.nist.gov/10.22.E65

Encodings:

pMML, png, TeX

See also:

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

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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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34: 22.19 Physical Applications

… ►We consider the case of a particle of mass 1, initially held at rest at displacement $a$ from the origin and then released at time $t=0$. The subsequent position as a function of time, $x(t)$, for the three cases is given with results expressed in terms of $a$ and the dimensionless parameter $\eta =\frac{1}{2}\beta a^2$. ►

Case I: $V(x)=\frac{1}{2}{x}^2+\frac{1}{4}\beta x^4$

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Case II: $V(x)=\frac{1}{2}{x}^2-\frac{1}{4}\beta x^4$

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Case III: $V(x)=-\frac{1}{2}{x}^2+\frac{1}{4}\beta x^4$

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35: 36.6 Scaling Relations

§36.6 Scaling Relations

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Table 36.6.1: Special cases of scaling exponents for cuspoids.

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singularity	$K$	${\beta }_K$	${\gamma }_{1K}$	${\gamma }_{2K}$	${\gamma }_{3K}$	${\gamma }_K$
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36: 11.5 Integral Representations

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11.5.3 $${𝐊}_0\left(z\right)=\frac{2}{\pi }{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-z\sinh t}dt,$$ $\mathrm{\Re }z>0$,

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Symbols:

${𝐊}_{\nu }\left(z\right)$: Struve function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\sinh z$: hyperbolic sine function, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

Referenced by:

§11.5(i)

Permalink:

http://dlmf.nist.gov/11.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §11.5(i), §11.5 and Ch.11

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11.5.6 $${𝐋}_{\nu }\left(z\right)=\frac{2{(\frac{1}{2}z)}^{\nu }}{\sqrt{\pi }\mathrm{\Gamma }\left(\nu +\frac{1}{2}\right)}{\int }_0^{\pi /2}\sinh \left(z\cos \theta \right){(\sin \theta )}^{2\nu }d\theta ,$$ $\mathrm{\Re }\nu >-\frac{1}{2}$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\sinh z$: hyperbolic sine function, $\int$: integral, ${𝐋}_{\nu }\left(z\right)$: modified Struve function, $\mathrm{\Re }$: real part, $\sin z$: sine function, $z$: complex variable and $\nu$: real or complex order

Referenced by:

§11.5(i)

Permalink:

http://dlmf.nist.gov/11.5.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §11.5(i), §11.5 and Ch.11

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37: 4.15 Graphics

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►

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	A	B	C	$\overline{\mathrm{C}}$	D	$\overline{\mathrm{D}}$	E	$\overline{\mathrm{E}}$	F
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$w$	$0$	$1$	$\cosh r+\mathrm{i}0$	$\cosh r-\mathrm{i}0$	$\mathrm{i}\sinh r$	$-\mathrm{i}\sinh r$	$-\cosh r+\mathrm{i}0$	$-\cosh r-\mathrm{i}0$	$-1$

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Figure 4.15.7: Conformal mapping of sine and inverse sine. … Magnify

… ►they can be obtained by translating the surfaces shown in Figures 4.15.8, 4.15.10, 4.15.12 by $-\frac{1}{2}\pi$ parallel to the $x$-axis, and adjusting the phase coloring in the case of Figure 4.15.10. …

38: 13.18 Relations to Other Functions

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13.18.1 $$M_{0,\frac{1}{2}}\left(2z\right)=2\sinh z,$$

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Symbols:

$M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\sinh z$: hyperbolic sine function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.18.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.18(i), §13.18 and Ch.13

… ►Special cases are the error functions … ►Special cases of §13.18(iv) are as follows. …

39: 13.6 Relations to Other Functions

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13.6.2 $$M(1,2,2z)=\frac{{\mathrm{e}}^z}{z}\sinh z,$$

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Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\sinh z$: hyperbolic sine function and $z$: complex variable

A&S Ref:

13.6.14

Permalink:

http://dlmf.nist.gov/13.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §13.6(i), §13.6 and Ch.13

… ►Special cases are the error functions … ►and in the case that $b-2a$ is an integer we have …Note that (13.6.11_1) and (13.6.11_2) are special cases of (13.11.1) and (13.11.2), respectively … ►Special cases of §13.6(iv) are as follows. …

40: 36.12 Uniform Approximation of Integrals

… ►In the cuspoid case (one integration variable) … ►

§36.12(ii) Special Case

… ►For further information concerning integrals with several coalescing saddle points see Arnol’d et al. (1988), Berry and Howls (1993, 1994), Bleistein (1967), Duistermaat (1974), Ludwig (1966), Olde Daalhuis (2000), and Ursell (1972, 1980).