§10.43 Integrals

Let ${𝒵}_{\nu }\left(z\right)$ be defined as in §10.25(ii). Then

10.43.1		${\displaystyle \int z^{\nu +1}{𝒵}_{\nu }\left(z\right)dz}$	$=z^{\nu +1}{𝒵}_{\nu +1}\left(z\right),$
		${\displaystyle \int z^{-\nu +1}{𝒵}_{\nu }\left(z\right)dz}$	$=z^{-\nu +1}{𝒵}_{\nu -1}\left(z\right).$
ⓘ Symbols: $dx$: differential of $x$, $\int$: integral, ${𝒵}_{\nu }\left(z\right)$: modified cylinder function, $z$: complex variable and $\nu$: complex parameter Permalink: http://dlmf.nist.gov/10.43.E1 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.43(i), §10.43 and Ch.10

10.43.2		$$\int z^{\nu }{𝒵}_{\nu }\left(z\right)dz={\pi }^{\frac{1}{2}}{2}^{\nu -1}\mathrm{\Gamma }\left(\nu +\frac{1}{2}\right)z\left({𝒵}_{\nu }\left(z\right){\mathbf{L}}_{\nu -1}\left(z\right)-{𝒵}_{\nu -1}\left(z\right){\mathbf{L}}_{\nu }\left(z\right)\right),$$
$\nu \ne -\frac{1}{2}$.
ⓘ 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, ${\mathbf{L}}_{\nu }\left(z\right)$: 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: TeX, pMML, png See also: Annotations for §10.43(i), §10.43 and Ch.10

For the modified Struve function ${\mathbf{L}}_{\nu }\left(z\right)$ see §11.2(i).

10.43.3		${\displaystyle \int {\mathrm{e}}^{\pm z}{z}^{\nu }{𝒵}_{\nu }\left(z\right)dz}$	$={\displaystyle \frac{{\mathrm{e}}^{\pm z}{z}^{\nu +1}}{2\nu +1}}\left({𝒵}_{\nu }\left(z\right)\mp {𝒵}_{\nu +1}\left(z\right)\right),$
	$\nu \ne -\frac{1}{2}$,
		${\displaystyle \int {\mathrm{e}}^{\pm z}{z}^{-\nu }{𝒵}_{\nu }\left(z\right)dz}$	$={\displaystyle \frac{{\mathrm{e}}^{\pm z}{z}^{-\nu +1}}{1-2\nu }}\left({𝒵}_{\nu }\left(z\right)\mp {𝒵}_{\nu -1}\left(z\right)\right),$
	$\nu \ne \frac{1}{2}$.
ⓘ Symbols: $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, ${𝒵}_{\nu }\left(z\right)$: modified cylinder function, $z$: complex variable and $\nu$: complex parameter A&S Ref: 11.3.12, 11.3.14, 11.3.15, 11.3.17 (modified) Permalink: http://dlmf.nist.gov/10.43.E3 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.43(i), §10.43 and Ch.10

10.43.4		$$\begin{array}{ll}{\int }_0^x\frac{I_0\left(t\right)-1}{t}dt& =\frac{1}{2}\sum _{k=1}^{\mathrm{\infty }}{(-1)}^{k-1}\frac{\psi \left(k+1\right)-\psi \left(1\right)}{k!}{(\frac{1}{2}x)}^k{I}_k\left(x\right)\\ & =\frac{2}{x}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k(2k+3)(\psi \left(k+2\right)-\psi \left(1\right))I_{2k+3}\left(x\right).\end{array}$$
ⓘ Symbols: $dx$: differential of $x$, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\int$: integral, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $k$: nonnegative integer and $x$: real variable Referenced by: §10.43(ii) Permalink: http://dlmf.nist.gov/10.43.E4 Encodings: TeX, pMML, png See also: Annotations for §10.43(ii), §10.43 and Ch.10

10.43.5		$${\int }_x^{\mathrm{\infty }}\frac{K_0\left(t\right)}{t}dt=\frac{1}{2}{\left(\ln \left(\frac{1}{2}x\right)+\gamma \right)}^2+\frac{{\pi }^2}{24}-\sum _{k=1}^{\mathrm{\infty }}\left(\psi \left(k+1\right)+\frac{1}{2k}-\ln \left(\frac{1}{2}x\right)\right)\frac{{(\frac{1}{2}x)}^{2k}}{2k{(k!)}^2},$$
ⓘ Symbols: $\gamma$: Euler’s constant, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer and $x$: real variable A&S Ref: 11.1.22 Referenced by: §10.43(ii) Permalink: http://dlmf.nist.gov/10.43.E5 Encodings: TeX, pMML, png See also: Annotations for §10.43(ii), §10.43 and Ch.10

where $\psi ={\mathrm{\Gamma }}^{\prime }/\mathrm{\Gamma }$ and $\gamma$ is Euler’s constant (§5.2).

10.43.6		$${\int }_0^x{\mathrm{e}}^{-t}{I}_n\left(t\right)dt=x{\mathrm{e}}^{-x}(I_0\left(x\right)+I_1\left(x\right))+n({\mathrm{e}}^{-x}{I}_0\left(x\right)-1)+2{\mathrm{e}}^{-x}\sum _{k=1}^{n-1}(n-k)I_k\left(x\right),$$
$n=0,1,2,\mathrm{\dots }$.
ⓘ Symbols: $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $n$: integer, $k$: nonnegative integer and $x$: real variable Referenced by: §10.43(ii) Permalink: http://dlmf.nist.gov/10.43.E6 Encodings: TeX, pMML, png See also: Annotations for §10.43(ii), §10.43 and Ch.10

10.43.7		$${\int }_0^x{\mathrm{e}}^{\pm t}{t}^{\nu }{I}_{\nu }\left(t\right)dt=\frac{{\mathrm{e}}^{\pm x}{x}^{\nu +1}}{2\nu +1}(I_{\nu }\left(x\right)\mp I_{\nu +1}\left(x\right)),$$
$\mathrm{\Re }\nu >-\frac{1}{2}$,
ⓘ Symbols: $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{\Re }$: real part, $x$: real variable and $\nu$: complex parameter Permalink: http://dlmf.nist.gov/10.43.E7 Encodings: TeX, pMML, png See also: Annotations for §10.43(ii), §10.43 and Ch.10

10.43.8		$${\int }_0^x{\mathrm{e}}^{\pm t}{t}^{-\nu }{I}_{\nu }\left(t\right)dt=-\frac{{\mathrm{e}}^{\pm x}{x}^{-\nu +1}}{2\nu -1}(I_{\nu }\left(x\right)\mp I_{\nu -1}\left(x\right))\mp \frac{2^{-\nu +1}}{(2\nu -1)\mathrm{\Gamma }\left(\nu \right)},$$
$\nu \ne \frac{1}{2}$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $x$: real variable and $\nu$: complex parameter Permalink: http://dlmf.nist.gov/10.43.E8 Encodings: TeX, pMML, png See also: Annotations for §10.43(ii), §10.43 and Ch.10

10.43.9		$${\int }_0^x{\mathrm{e}}^{\pm t}{t}^{\nu }{K}_{\nu }\left(t\right)dt=\frac{{\mathrm{e}}^{\pm x}{x}^{\nu +1}}{2\nu +1}(K_{\nu }\left(x\right)\pm K_{\nu +1}\left(x\right))\mp \frac{2^{\nu }\mathrm{\Gamma }\left(\nu +1\right)}{2\nu +1},$$
$\mathrm{\Re }\nu >-\frac{1}{2}$,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{\Re }$: real part, $x$: real variable and $\nu$: complex parameter Permalink: http://dlmf.nist.gov/10.43.E9 Encodings: TeX, pMML, png See also: Annotations for §10.43(ii), §10.43 and Ch.10

10.43.10		$${\int }_x^{\mathrm{\infty }}{\mathrm{e}}^t{t}^{-\nu }{K}_{\nu }\left(t\right)dt=\frac{{\mathrm{e}}^x{x}^{-\nu +1}}{2\nu -1}(K_{\nu }\left(x\right)+K_{\nu -1}\left(x\right)),$$
$\mathrm{\Re }\nu >\frac{1}{2}$.
ⓘ Symbols: $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{\Re }$: real part, $x$: real variable and $\nu$: complex parameter Referenced by: §10.43(ii) Permalink: http://dlmf.nist.gov/10.43.E10 Encodings: TeX, pMML, png See also: Annotations for §10.43(ii), §10.43 and Ch.10

The Bickley function ${\mathrm{Ki}}_{\alpha }\left(x\right)$ is defined by

10.43.11		$${\mathrm{Ki}}_{\alpha }\left(x\right)=\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_x^{\mathrm{\infty }}{(t-x)}^{\alpha -1}{K}_0\left(t\right)dt,$$
ⓘ Defines: ${\mathrm{Ki}}_{\alpha }\left(x\right)$: Bickley function Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind and $x$: real variable A&S Ref: 11.2.11 Referenced by: §10.43(iii) Permalink: http://dlmf.nist.gov/10.43.E11 Encodings: TeX, pMML, png See also: Annotations for §10.43(iii), §10.43 and Ch.10

when $\mathrm{\Re }\alpha >0$ and $x>0$, and by analytic continuation elsewhere. Equivalently,

10.43.12		$${\mathrm{Ki}}_{\alpha }\left(x\right)={\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-x\cosh t}}{{(\cosh t)}^{\alpha }}dt,$$
$x>0$.
ⓘ 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: TeX, pMML, png See also: Annotations for §10.43(iii), §10.43 and Ch.10

10.43.18		$${\int }_0^{\mathrm{\infty }}{K}_{\nu }\left(t\right)dt=\frac{1}{2}\pi \sec \left(\frac{1}{2}\pi \nu \right),$$
.
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\int$: 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 Permalink: http://dlmf.nist.gov/10.43.E18 Encodings: TeX, pMML, png See also: Annotations for §10.43(iv), §10.43 and Ch.10

10.43.19		$${\int }_0^{\mathrm{\infty }}{t}^{\mu -1}{K}_{\nu }\left(t\right)dt=2^{\mu -2}\mathrm{\Gamma }\left(\frac{1}{2}\mu -\frac{1}{2}\nu \right)\mathrm{\Gamma }\left(\frac{1}{2}\mu +\frac{1}{2}\nu \right),$$
.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: 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: 11.4.22, 11.4.23 Permalink: http://dlmf.nist.gov/10.43.E19 Encodings: TeX, pMML, png See also: Annotations for §10.43(iv), §10.43 and Ch.10

10.43.20		${\displaystyle {\int }_0^{\mathrm{\infty }}}\cos \left(at\right)K_0\left(t\right)dt$	$={\displaystyle \frac{\pi }{2{(1+a^2)}^{\frac{1}{2}}}},$
,
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\mathrm{\Im }$: imaginary part, $\int$: integral and $K_{\nu }\left(z\right)$: modified Bessel function of the second kind A&S Ref: 11.4.14 Permalink: http://dlmf.nist.gov/10.43.E20 Encodings: TeX, pMML, png See also: Annotations for §10.43(iv), §10.43 and Ch.10
10.43.21		${\displaystyle {\int }_0^{\mathrm{\infty }}}\sin \left(at\right)K_0\left(t\right)dt$	$={\displaystyle \frac{\mathrm{arcsinh}a}{{(1+a^2)}^{\frac{1}{2}}}},$
.
ⓘ 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: TeX, pMML, png See also: Annotations for §10.43(iv), §10.43 and Ch.10

When $\mathrm{\Re }\mu >|\mathrm{\Re }\nu |$,

10.43.22
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{\Re }$: real part and $\nu$: complex parameter Keywords: Laplace transform, Mellin transform Referenced by: §10.43(iv) Permalink: http://dlmf.nist.gov/10.43.E22 Encodings: TeX, pMML, png See also: Annotations for §10.43(iv), §10.43 and Ch.10

For the second equation there is a cut in the $a$-plane along the interval $[0,1]$, and all quantities assume their principal values (§4.2(i)). For the Ferrers function $\mathsf{P}$ and the associated Legendre function $P$, see §§14.3(i) and 14.21(i).

10.43.23		${\displaystyle {\int }_0^{\mathrm{\infty }}}{t}^{\nu +1}{I}_{\nu }\left(bt\right)\exp \left(-p^2{t}^2\right)dt$	$={\displaystyle \frac{b^{\nu }}{{(2p^2)}^{\nu +1}}}\exp \left({\displaystyle \frac{b^2}{4p^2}}\right),$
$\mathrm{\Re }\nu >-1,\mathrm{\Re }\left(p^2\right)>0$,
ⓘ Symbols: $dx$: differential of $x$, $\exp z$: exponential function, $\int$: integral, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{\Re }$: real part and $\nu$: complex parameter Keywords: Mellin transform Referenced by: §10.43(iv) Permalink: http://dlmf.nist.gov/10.43.E23 Encodings: TeX, pMML, png See also: Annotations for §10.43(iv), §10.43 and Ch.10
10.43.24		${\displaystyle {\int }_0^{\mathrm{\infty }}}{I}_{\nu }\left(bt\right)\exp \left(-p^2{t}^2\right)dt$	$={\displaystyle \frac{\sqrt{\pi }}{2p}}\exp \left({\displaystyle \frac{b^2}{8p^2}}\right)I_{\frac{1}{2}\nu }\left({\displaystyle \frac{b^2}{8p^2}}\right),$
$\mathrm{\Re }\nu >-1$, $\mathrm{\Re }\left(p^2\right)>0$,
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\exp z$: exponential function, $\int$: integral, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{\Re }$: real part and $\nu$: complex parameter A&S Ref: 11.4.31 Permalink: http://dlmf.nist.gov/10.43.E24 Encodings: TeX, pMML, png See also: Annotations for §10.43(iv), §10.43 and Ch.10
10.43.25		${\displaystyle {\int }_0^{\mathrm{\infty }}}{K}_{\nu }\left(bt\right)\exp \left(-p^2{t}^2\right)dt$	$={\displaystyle \frac{\sqrt{\pi }}{4p}}\sec \left(\frac{1}{2}\pi \nu \right)\exp \left({\displaystyle \frac{b^2}{8p^2}}\right)K_{\frac{1}{2}\nu }\left({\displaystyle \frac{b^2}{8p^2}}\right),$
, $\mathrm{\Re }\left(p^2\right)>0$.
ⓘ 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: TeX, pMML, png See also: Annotations for §10.43(iv), §10.43 and Ch.10
10.43.26		${\displaystyle {\int }_0^{\mathrm{\infty }}}{\displaystyle \frac{K_{\mu }\left(at\right)J_{\nu }\left(bt\right)}{t^{\lambda }}}dt$	$={\displaystyle \frac{b^{\nu }\mathrm{\Gamma }\left(\frac{1}{2}\nu -\frac{1}{2}\lambda +\frac{1}{2}\mu +\frac{1}{2}\right)\mathrm{\Gamma }\left(\frac{1}{2}\nu -\frac{1}{2}\lambda -\frac{1}{2}\mu +\frac{1}{2}\right)}{2^{\lambda +1}{a}^{\nu -\lambda +1}}}\mathbf{F}({\displaystyle \frac{\nu -\lambda +\mu +1}{2}},{\displaystyle \frac{\nu -\lambda -\mu +1}{2}};\nu +1;-{\displaystyle \frac{b^2}{a^2}}),$
$\mathrm{\Re }\left(\nu +1-\lambda \right)>|\mathrm{\Re }\mu |,\mathrm{\Re }a>|\mathrm{\Im }b|$.
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$, $\mathrm{\Im }$: imaginary part, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{\Re }$: real part, $\mathbf{F}(a,b;c;z)$ or $\mathbf{F}(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2\mathbf{F}_1(a,b;c;z)$ Olver’s hypergeometric function and $\nu$: complex parameter Keywords: Mellin transform Permalink: http://dlmf.nist.gov/10.43.E26 Encodings: TeX, pMML, png See also: Annotations for §10.43(iv), §10.43 and Ch.10

For the hypergeometric function $\mathbf{F}$ see §15.2(i).

10.43.27		${\displaystyle {\int }_0^{\mathrm{\infty }}}{t}^{\mu +\nu +1}{K}_{\mu }\left(at\right)J_{\nu }\left(bt\right)dt$	$={\displaystyle \frac{{(2a)}^{\mu }{(2b)}^{\nu }\mathrm{\Gamma }\left(\mu +\nu +1\right)}{{(a^2+b^2)}^{\mu +\nu +1}}},$
$\mathrm{\Re }\left(\nu +1\right)>|\mathrm{\Re }\mu |,\mathrm{\Re }a>|\mathrm{\Im }b|$.
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$, $\mathrm{\Im }$: imaginary part, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{\Re }$: real part and $\nu$: complex parameter Keywords: Mellin transform Permalink: http://dlmf.nist.gov/10.43.E27 Encodings: TeX, pMML, png See also: Annotations for §10.43(iv), §10.43 and Ch.10
10.43.28		${\displaystyle {\int }_0^{\mathrm{\infty }}}t\exp \left(-p^2{t}^2\right)I_{\nu }\left(at\right)I_{\nu }\left(bt\right)dt$	$={\displaystyle \frac{1}{2p^2}}\exp \left({\displaystyle \frac{a^2+b^2}{4p^2}}\right)I_{\nu }\left({\displaystyle \frac{ab}{2p^2}}\right),$
$\mathrm{\Re }\nu >-1,\mathrm{\Re }\left(p^2\right)>0$,
ⓘ Symbols: $dx$: differential of $x$, $\exp z$: exponential function, $\int$: integral, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{\Re }$: real part and $\nu$: complex parameter Permalink: http://dlmf.nist.gov/10.43.E28 Encodings: TeX, pMML, png See also: Annotations for §10.43(iv), §10.43 and Ch.10
10.43.29		${\displaystyle {\int }_0^{\mathrm{\infty }}}t\exp \left(-p^2{t}^2\right)I_0\left(at\right)K_0\left(at\right)dt$	$={\displaystyle \frac{1}{4p^2}}\exp \left({\displaystyle \frac{a^2}{2p^2}}\right)K_0\left({\displaystyle \frac{a^2}{2p^2}}\right),$
$\mathrm{\Re }\left(p^2\right)>0$.
ⓘ Symbols: $dx$: differential of $x$, $\exp z$: exponential function, $\int$: integral, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind and $\mathrm{\Re }$: real part Referenced by: §10.43(iv) Permalink: http://dlmf.nist.gov/10.43.E29 Encodings: TeX, pMML, png See also: Annotations for §10.43(iv), §10.43 and Ch.10

For infinite integrals of triple products of modified and unmodified Bessel functions, see Gervois and Navelet (1984, 1985a, 1985b, 1986a, 1986b).

The Kontorovich–Lebedev transform of a function $g(x)$ is defined as

10.43.30		$$f(y)=\frac{2y}{{\pi }^2}\sinh \left(\pi y\right){\int }_0^{\mathrm{\infty }}\frac{g(x)}{x}{K}_{\mathrm{i}y}\left(x\right)dx.$$
ⓘ 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: TeX, pMML, png See also: Annotations for §10.43(v), §10.43 and Ch.10

Then

10.43.31		$$g(x)={\int }_0^{\mathrm{\infty }}f(y)K_{\mathrm{i}y}\left(x\right)dy,$$
ⓘ Symbols: $dx$: differential of $x$, $\mathrm{i}$: imaginary unit, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $x$: real variable, $y$: real variable, $g(x)$: function and $f(x)$: Kontorovich–Lebedev transform of $g(x)$ Referenced by: §10.43(v) Permalink: http://dlmf.nist.gov/10.43.E31 Encodings: TeX, pMML, png See also: Annotations for §10.43(v), §10.43 and Ch.10

provided that either of the following sets of conditions is satisfied:

• (a)

  On the interval , $x^{-1}g(x)$ is continuously differentiable and each of $xg(x)$ and $xd(x^{-1}g(x))/dx$ is absolutely integrable.

• (b)

  $g(x)$ is piecewise continuous and of bounded variation on every compact interval in $(0,\mathrm{\infty })$, and each of the following integrals

10.43.32		${\displaystyle {\int }_0^{\frac{1}{2}}}{\displaystyle \frac{g(x)}{x}}\ln \left({\displaystyle \frac{1}{x}}\right)dx,$
		${\displaystyle {\int }_{\frac{1}{2}}^{\mathrm{\infty }}}{\displaystyle \frac{|g(x)|}{x^{\frac{1}{2}}}}dx,$
ⓘ Symbols: $dx$: differential of $x$, $\int$: integral, $\ln z$: principal branch of logarithm function, $x$: real variable and $g(x)$: function Permalink: http://dlmf.nist.gov/10.43.E32 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.43(v), §10.43 and Ch.10

• converges.

For asymptotic expansions of the direct transform (10.43.30) see Wong (1981), and for asymptotic expansions of the inverse transform (10.43.31) see Naylor (1990, 1996).

For collections of the Kontorovich–Lebedev transform, see Erdélyi et al. (1954b, Chapter 12), Prudnikov et al. (1986b, pp. 404–412), and Oberhettinger (1972, Chapter 5).

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).