integrals of modified Bessel functions

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21: 10.38 Derivatives with Respect to Order

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10.38.6 $${\frac{\partial I_{\nu }\left(x\right)}{\partial \nu }|}_{\nu =\pm \frac{1}{2}}=-\frac{1}{\sqrt{2\pi x}}\left(E_1\left(2x\right){\mathrm{e}}^x\pm \mathrm{Ei}\left(2x\right){\mathrm{e}}^{-x}\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\mathrm{Ei}\left(x\right)$: exponential integral, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $x$: real variable and $\nu$: complex parameter

A&S Ref:

10.2.32 and 10.2.33 (both corrected)

Referenced by:

§10.38, Ch.10

Permalink:

http://dlmf.nist.gov/10.38.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §10.38, §10.38 and Ch.10

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10.38.7 $${\frac{\partial K_{\nu }\left(x\right)}{\partial \nu }|}_{\nu =\pm \frac{1}{2}}=\pm \sqrt{\frac{\pi }{2x}}{E}_1\left(2x\right){\mathrm{e}}^x.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $x$: real variable and $\nu$: complex parameter

A&S Ref:

10.2.34 (corrected)

Referenced by:

§10.38, Ch.10

Permalink:

http://dlmf.nist.gov/10.38.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.38, §10.38 and Ch.10

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22: Bibliography C

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Derivatives with respect to the degree and order of associated Legendre functions for $|z|>1$ using modified Bessel functions. Integral Transforms Spec. Funct. 21 (7-8), pp. 581–588.

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External Links:

ISSN 1065-2469, Document, FullText, MathReview (Dmitriy Karp), ZentralBlatt

Referenced by:

§14.11, §14.11.

Encodings:

BibTeX

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23: 13.10 Integrals

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13.10.8 $$\frac{1}{2\pi \mathrm{i}}{\int }_{-\mathrm{\infty }}^{(0+)}{\mathrm{e}}^{tz}{t}^{-a}𝐌(a,b,y/t)dt=\frac{1}{\mathrm{\Gamma }\left(a\right)}{z}^{\frac{1}{2}(2a-b-1)}{y}^{\frac{1}{2}(1-b)}{I}_{b-1}\left(2\sqrt{zy}\right),$$ $\mathrm{\Re }z>0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{\Re }$: real part, $y$: real variable and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.10.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §13.10(ii), §13.10(ii), §13.10 and Ch.13

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13.10.9 $$\frac{1}{2\pi \mathrm{i}}{\int }_{-\mathrm{\infty }}^{(0+)}{\mathrm{e}}^{tz}{t}^{-a}U(a,b,y/t)dt=\frac{2z^{\frac{1}{2}(2a-b-1)}{y}^{\frac{1}{2}(1-b)}}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(a-b+1\right)}{K}_{b-1}\left(2\sqrt{zy}\right),$$ $\mathrm{\Re }z>0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{\Re }$: real part, $y$: real variable and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.10.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §13.10(ii), §13.10(ii), §13.10 and Ch.13

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24: 13.23 Integrals

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13.23.6 $$\frac{1}{\mathrm{\Gamma }\left(1+2\mu \right)2\pi \mathrm{i}}{\int }_{-\mathrm{\infty }}^{(0+)}{\mathrm{e}}^{zt+\frac{1}{2}{t}^{-1}}{t}^{\kappa }{M}_{\kappa ,\mu }\left(t^{-1}\right)dt=\frac{z^{-\kappa -\frac{1}{2}}}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)}{I}_{2\mu }\left(2\sqrt{z}\right),$$ $\mathrm{\Re }z>0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{\Re }$: real part and $z$: complex variable

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/13.23.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §13.23(i), §13.23 and Ch.13

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13.23.7 $$\frac{1}{2\pi \mathrm{i}}{\int }_{-\mathrm{\infty }}^{(0+)}{\mathrm{e}}^{zt+\frac{1}{2}{t}^{-1}}{t}^{\kappa }{W}_{\kappa ,\mu }\left(t^{-1}\right)dt=\frac{2z^{-\kappa -\frac{1}{2}}}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)\mathrm{\Gamma }\left(\frac{1}{2}-\mu -\kappa \right)}{K}_{2\mu }\left(2\sqrt{z}\right),$$ $\mathrm{\Re }z>0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{\Re }$: real part and $z$: complex variable

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/13.23.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §13.23(i), §13.23 and Ch.13

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25: 11.5 Integral Representations

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11.5.7 $$I_{-\nu }\left(x\right)-{𝐋}_{\nu }\left(x\right)=\frac{2{(\frac{1}{2}x)}^{\nu }}{\sqrt{\pi }\mathrm{\Gamma }\left(\nu +\frac{1}{2}\right)}{\int }_0^{\mathrm{\infty }}{(1+t^2)}^{\nu -\frac{1}{2}}\sin \left(xt\right)dt,$$ $x>0$, .

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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, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, ${𝐋}_{\nu }\left(z\right)$: modified Struve function, $\mathrm{\Re }$: real part, $\sin z$: sine function, $x$: real variable and $\nu$: real or complex order

Referenced by:

§11.5(i), §11.5(ii)

Permalink:

http://dlmf.nist.gov/11.5.E7

Encodings:

pMML, png, TeX

See also:

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

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26: 10.54 Integral Representations

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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

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27: 11.14 Tables

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§11.14(ii) Struve Functions

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Abramowitz and Stegun (1964, Chapter 12) tabulates ${𝐇}_n\left(x\right)$, ${𝐇}_n\left(x\right)-Y_n\left(x\right)$, and $I_n\left(x\right)-{𝐋}_n\left(x\right)$ for $n=0,1$ and $x=0(.1)5$, $x^{-1}=0(.01)0.2$ to 6D or 7D.

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§11.14(iii) Integrals

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Abramowitz and Stegun (1964, Chapter 12) tabulates ${\int }_0^x(I_0\left(t\right)-{𝐋}_0\left(t\right))dt$ and $(2/\pi ){\int }_x^{\mathrm{\infty }}{t}^{-1}{𝐇}_0\left(t\right)dt$ for $x=0(.1)5$ to 5D or 7D; ${\int }_0^x({𝐇}_0\left(t\right)-Y_0\left(t\right))dt-(2/\pi )\ln x$, ${\int }_0^x(I_0\left(t\right)-{𝐋}_0\left(t\right))dt-(2/\pi )\ln x$, and ${\int }_x^{\mathrm{\infty }}{t}^{-1}({𝐇}_0\left(t\right)-Y_0\left(t\right))dt$ for $x^{-1}=0(.01)0.2$ to 6D.

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§11.14(v) Incomplete Functions

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28: 12.7 Relations to Other Functions

§12.7 Relations to Other Functions

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§12.7(i) Hermite Polynomials

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§12.7(ii) Error Functions, Dawson’s Integral, and Probability Function

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§12.7(iii) Modified Bessel Functions

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§12.7(iv) Confluent Hypergeometric Functions

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29: 11.9 Lommel Functions

§11.9 Lommel Functions

… ►The inhomogeneous Bessel differential equation … ►

§11.9(ii) Expansions in Series of Bessel Functions

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30: 11.15 Approximations

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§11.15(i) Expansions in Chebyshev Series

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Luke (1975, pp. 416–421) gives Chebyshev-series expansions for ${𝐇}_n\left(x\right)$, ${𝐋}_n\left(x\right)$, $0\le \left|x\right|\le 8$, and ${𝐇}_n\left(x\right)-Y_n\left(x\right)$, $x\ge 8$, for $n=0,1$; ${\int }_0^x{t}^{-m}{𝐇}_0\left(t\right)dt$, ${\int }_0^x{t}^{-m}{𝐋}_0\left(t\right)dt$, $0\le \left|x\right|\le 8$, $m=0,1$ and ${\int }_0^x({𝐇}_0\left(t\right)-Y_0\left(t\right))dt$, ${\int }_x^{\mathrm{\infty }}{t}^{-1}({𝐇}_0\left(t\right)-Y_0\left(t\right))dt$, $x\ge 8$; the coefficients are to 20D.

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MacLeod (1993) gives Chebyshev-series expansions for ${𝐋}_0\left(x\right)$, ${𝐋}_1\left(x\right)$, $0\le x\le 16$, and $I_0\left(x\right)-{𝐋}_0\left(x\right)$, $I_1\left(x\right)-{𝐋}_1\left(x\right)$, $x\ge 16$; the coefficients are to 20D.

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Newman (1984) gives polynomial approximations for ${𝐇}_n\left(x\right)$ for $n=0,1$, $0\le x\le 3$, and rational-fraction approximations for ${𝐇}_n\left(x\right)-Y_n\left(x\right)$ for $n=0,1$, $x\ge 3$. The maximum errors do not exceed 1.2×10⁻⁸ for the former and 2.5×10⁻⁸ for the latter.