10 Bessel Functions Modified Bessel Functions 10.42 Zeros 10.44 Sums

§10.43 Integrals

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Referenced by:: §10.59
Permalink:: http://dlmf.nist.gov/10.43
See also:: Annotations for Ch.10

§10.43(i) Indefinite Integrals
§10.43(ii) Integrals over the Intervals $(0,x)$ and $(x,\infty)$
§10.43(iii) Fractional Integrals
§10.43(iv) Integrals over the Interval ( $0,\infty$ )
§10.43(v) Kontorovich–Lebedev Transform
§10.43(vi) Compendia

§10.43(i) Indefinite Integrals

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Keywords:: indefinite, integrals of modified Bessel functions
Notes:: Differentiate, apply (10.29.2), and also (11.4.29) and (11.4.30) in the case of (10.43.2).
Permalink:: http://dlmf.nist.gov/10.43.i
See also:: Annotations for §10.43 and Ch.10

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

10.43.1	$\displaystyle\int z^{\nu+1}\mathscr{Z}_{\nu}\left(z\right)\,\mathrm{d}z$	$\displaystyle=z^{\nu+1}\mathscr{Z}_{\nu+1}\left(z\right),$
10.43.1	$\displaystyle\int z^{-\nu+1}\mathscr{Z}_{\nu}\left(z\right)\,\mathrm{d}z$	$\displaystyle=z^{-\nu+1}\mathscr{Z}_{\nu-1}\left(z\right).$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\mathscr{Z}_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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		$\int z^{\nu}\mathscr{Z}_{\nu}\left(z\right)\,\mathrm{d}z=\pi^{\frac{1}{2}}2^{% \nu-1}\Gamma\left(\nu+\tfrac{1}{2}\right)z\*\left(\mathscr{Z}_{\nu}\left(z% \right)\mathbf{L}_{\nu-1}\left(z\right)-\mathscr{Z}_{\nu-1}\left(z\right)% \mathbf{L}_{\nu}\left(z\right)\right),$
$\nu\neq-\tfrac{1}{2}$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\mathbf{L}_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Struve function, $\mathscr{Z}_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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		$\displaystyle\int e^{\pm z}z^{\nu}\mathscr{Z}_{\nu}\left(z\right)\,\mathrm{d}z$	$\displaystyle=\frac{e^{\pm z}z^{\nu+1}}{2\nu+1}\left(\mathscr{Z}_{\nu}\left(z% \right)\mp\mathscr{Z}_{\nu+1}\left(z\right)\right),$
	$\nu\neq-\tfrac{1}{2}$ ,
		$\displaystyle\int e^{\pm z}z^{-\nu}\mathscr{Z}_{\nu}\left(z\right)\,\mathrm{d}z$	$\displaystyle=\frac{e^{\pm z}z^{-\nu+1}}{1-2\nu}\left(\mathscr{Z}_{\nu}\left(z% \right)\mp\mathscr{Z}_{\nu-1}\left(z\right)\right),$
	$\nu\neq\tfrac{1}{2}$ .
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\mathscr{Z}_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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(ii) Integrals over the Intervals $(0,x)$ and $(x,\infty)$

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Keywords:: integrals of modified Bessel functions, over finite intervals, over infinite intervals
Notes:: For (10.43.4) replace $x$ by $i x$ in (10.22.11), (10.22.12) and use (10.27.6). For (10.43.5) combine (10.22.39) and (10.22.40) by means of (10.4.3) to obtain an expansion for $\int_{x}^{\infty}({H^{(1)}_{0}}\left(t\right)/t)\,\mathrm{d}t$ ; then replace $x$ by $i x$ and use (10.27.8). For (10.43.6)–(10.43.10) differentiate, apply §10.29(i) and also verify the limiting behavior as $x\to 0$ or $x\to\infty$ .
Permalink:: http://dlmf.nist.gov/10.43.ii
See also:: Annotations for §10.43 and Ch.10

10.43.4		$\int_{0}^{x}\frac{I_{0}\left(t\right)-1}{t}\,\mathrm{d}t=\frac{1}{2}\sum_{k=1}% ^{\infty}(-1)^{k-1}\frac{\psi\left(k+1\right)-\psi\left(1\right)}{k!}(\tfrac{1% }{2}x)^{k}I_{k}\left(x\right)=\frac{2}{x}\sum_{k=0}^{\infty}(-1)^{k}(2k+3)(% \psi\left(k+2\right)-\psi\left(1\right))I_{2k+3}\left(x\right).$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $!$ : factorial (as in $n!$ ), $\int$ : integral, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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		$\int_{x}^{\infty}\frac{K_{0}\left(t\right)}{t}\,\mathrm{d}t=\frac{1}{2}\left(% \ln\left(\tfrac{1}{2}x\right)+\gamma\right)^{2}+\frac{\pi^{2}}{24}-\sum_{k=1}^% {\infty}\left(\psi\left(k+1\right)+\frac{1}{2k}-\ln\left(\tfrac{1}{2}x\right)% \right)\frac{(\tfrac{1}{2}x)^{2k}}{2k(k!)^{2}},$
ⓘ Symbols: $\gamma$ : Euler’s constant, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $!$ : factorial (as in $n!$ ), $\int$ : integral, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\ln\NVar{z}$ : 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 $\psi=\ifrac{\Gamma'}{\Gamma}$ and $\gamma$ is Euler’s constant (§5.2).

10.43.6		$\int_{0}^{x}e^{-t}I_{n}\left(t\right)\,\mathrm{d}t=xe^{-x}(I_{0}\left(x\right)% +I_{1}\left(x\right))+n(e^{-x}I_{0}\left(x\right)-1)+2e^{-x}\sum_{k=1}^{n-1}(n% -k)I_{k}\left(x\right),$
$n=0,1,2,\dotsc$ .
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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		$\int_{0}^{x}e^{\pm t}t^{\nu}I_{\nu}\left(t\right)\,\mathrm{d}t=\frac{e^{\pm x}% x^{\nu+1}}{2\nu+1}(I_{\nu}\left(x\right)\mp I_{\nu+1}\left(x\right)),$
$\Re\nu>-\tfrac{1}{2}$ ,
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $\Re$ : 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		$\int_{0}^{x}e^{\pm t}t^{-\nu}I_{\nu}\left(t\right)\,\mathrm{d}t=-\frac{e^{\pm x% }x^{-\nu+1}}{2\nu-1}(I_{\nu}\left(x\right)\mp I_{\nu-1}\left(x\right))\mp\frac% {2^{-\nu+1}}{(2\nu-1)\Gamma\left(\nu\right)},$
$\nu\neq\tfrac{1}{2}$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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		$\int_{0}^{x}e^{\pm t}t^{\nu}K_{\nu}\left(t\right)\,\mathrm{d}t=\frac{e^{\pm x}% x^{\nu+1}}{2\nu+1}(K_{\nu}\left(x\right)\pm K_{\nu+1}\left(x\right))\mp\frac{2% ^{\nu}\Gamma\left(\nu+1\right)}{2\nu+1},$
$\Re\nu>-\tfrac{1}{2}$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\Re$ : 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		$\int_{x}^{\infty}e^{t}t^{-\nu}K_{\nu}\left(t\right)\,\mathrm{d}t=\frac{e^{x}x^% {-\nu+1}}{2\nu-1}(K_{\nu}\left(x\right)+K_{\nu-1}\left(x\right)),$
$\Re\nu>\tfrac{1}{2}$ .
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\Re$ : 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

§10.43(iii) Fractional Integrals

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Keywords:: Bickley function, fractional, integrals of modified Bessel functions
Notes:: For (10.43.12) substitute into (10.43.11) by means of (10.32.9) with $\nu=0$ , invert the order of integration and apply (5.2.1). (10.43.13)–(10.43.16) follow from (10.43.12), and in the case of (10.43.16), (5.12.1). For (10.43.17) see Bickley and Nayler (1935).
Referenced by:: 2nd item
Permalink:: http://dlmf.nist.gov/10.43.iii
See also:: Annotations for §10.43 and Ch.10

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

10.43.11		$\operatorname{Ki}_{\alpha}\left(x\right)=\frac{1}{\Gamma\left(\alpha\right)}% \int_{x}^{\infty}(t-x)^{\alpha-1}K_{0}\left(t\right)\,\mathrm{d}t,$
ⓘ Defines: $\operatorname{Ki}_{\NVar{\alpha}}\left(\NVar{x}\right)$ : Bickley function Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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 $\Re\alpha>0$ and $x>0$ , and by analytic continuation elsewhere. Equivalently,

10.43.12		$\operatorname{Ki}_{\alpha}\left(x\right)=\int_{0}^{\infty}\frac{e^{-x\cosh t}}% {(\cosh t)^{\alpha}}\,\mathrm{d}t,$
$x>0$ .
ⓘ Symbols: $\operatorname{Ki}_{\NVar{\alpha}}\left(\NVar{x}\right)$ : Bickley function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\cosh\NVar{z}$ : 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

Properties

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

10.43.13		$\operatorname{Ki}_{\alpha}\left(x\right)=\int_{x}^{\infty}\operatorname{Ki}_{% \alpha-1}\left(t\right)\,\mathrm{d}t,$
ⓘ Symbols: $\operatorname{Ki}_{\NVar{\alpha}}\left(\NVar{x}\right)$ : Bickley function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $x$ : real variable A&S Ref: 11.2.8 Referenced by: §10.43(iii) Permalink: http://dlmf.nist.gov/10.43.E13 Encodings: TeX, pMML, png See also: Annotations for §10.43(iii), §10.43(iii), §10.43 and Ch.10

10.43.14		$\operatorname{Ki}_{0}\left(x\right)=K_{0}\left(x\right),$
ⓘ Symbols: $\operatorname{Ki}_{\NVar{\alpha}}\left(\NVar{x}\right)$ : Bickley function, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind and $x$ : real variable A&S Ref: 11.2.8 Permalink: http://dlmf.nist.gov/10.43.E14 Encodings: TeX, pMML, png See also: Annotations for §10.43(iii), §10.43(iii), §10.43 and Ch.10

10.43.15		$\operatorname{Ki}_{-n}\left(x\right)=(-1)^{n}\frac{{\mathrm{d}}^{n}}{{\mathrm{% d}x}^{n}}K_{0}\left(x\right),$
$n=1,2,3,\dotsc$ .
ⓘ Symbols: $\operatorname{Ki}_{\NVar{\alpha}}\left(\NVar{x}\right)$ : Bickley function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $n$ : integer and $x$ : real variable A&S Ref: 11.2.9 Permalink: http://dlmf.nist.gov/10.43.E15 Encodings: TeX, pMML, png See also: Annotations for §10.43(iii), §10.43(iii), §10.43 and Ch.10

10.43.16		$\operatorname{Ki}_{\alpha}\left(0\right)=\frac{\sqrt{\pi}\Gamma\left(\frac{1}{% 2}\alpha\right)}{2\Gamma\left(\frac{1}{2}\alpha+\frac{1}{2}\right)},$
$\alpha\neq 0,-2,-4,\dots$ .
ⓘ Symbols: $\operatorname{Ki}_{\NVar{\alpha}}\left(\NVar{x}\right)$ : Bickley function, $\Gamma\left(\NVar{z}\right)$ : gamma function and $\pi$ : the ratio of the circumference of a circle to its diameter A&S Ref: 11.2.12, 11.2.13 (with less restrictive conditions) Referenced by: §10.43(iii) Permalink: http://dlmf.nist.gov/10.43.E16 Encodings: TeX, pMML, png See also: Annotations for §10.43(iii), §10.43(iii), §10.43 and Ch.10

10.43.17		$\alpha\operatorname{Ki}_{\alpha+1}\left(x\right)+x\operatorname{Ki}_{\alpha}% \left(x\right)+(1-\alpha)\operatorname{Ki}_{\alpha-1}\left(x\right)-x% \operatorname{Ki}_{\alpha-2}\left(x\right)=0.$
ⓘ Symbols: $\operatorname{Ki}_{\NVar{\alpha}}\left(\NVar{x}\right)$ : Bickley function and $x$ : real variable A&S Ref: 11.2.14 Referenced by: §10.43(iii) Permalink: http://dlmf.nist.gov/10.43.E17 Encodings: TeX, pMML, png See also: Annotations for §10.43(iii), §10.43(iii), §10.43 and Ch.10

For further properties of the Bickley function, including asymptotic expansions and generalizations, see Amos (1983c, 1989) and Luke (1962, Chapter 8).

§10.43(iv) Integrals over the Interval ( $0,\infty$ )

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Keywords:: Bickley function, fractional, integrals of modified Bessel functions, over infinite intervals, products
Notes:: See Watson (1944, pp. 388, 394–395, 410). For some results it is necessary to use the connection formulas (10.27.6); for example, to obtain (10.43.23) set $a=ib$ in Watson (1944, p. 394, Eq. (4)). Equations (10.43.22) follow from Eq. (7) of Watson (1944, §13.21). For (10.43.25) see Erdélyi et al. (1953b, p. 51). For (10.43.29) combine (10.22.68), (10.27.6), (10.27.10).
Permalink:: http://dlmf.nist.gov/10.43.iv
See also:: Annotations for §10.43 and Ch.10

10.43.18		$\int_{0}^{\infty}K_{\nu}\left(t\right)\,\mathrm{d}t=\tfrac{1}{2}\pi\sec\left(% \tfrac{1}{2}\pi\nu\right),$
$\|\Re\nu\|<1$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\Re$ : real part, $\sec\NVar{z}$ : 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		$\int_{0}^{\infty}t^{\mu-1}K_{\nu}\left(t\right)\,\mathrm{d}t=2^{\mu-2}\Gamma% \left(\tfrac{1}{2}\mu-\tfrac{1}{2}\nu\right)\Gamma\left(\tfrac{1}{2}\mu+\tfrac% {1}{2}\nu\right),$
$\|\Re\nu\|<\Re\mu$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\Re$ : 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	$\displaystyle\int_{0}^{\infty}\cos\left(at\right)K_{0}\left(t\right)\,\mathrm{% d}t$	$\displaystyle=\frac{\pi}{2(1+a^{2})^{\frac{1}{2}}},$
$\|\Im a\|<1$ ,
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\Im$ : imaginary part, $\int$ : integral and $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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	$\displaystyle\int_{0}^{\infty}\sin\left(at\right)K_{0}\left(t\right)\,\mathrm{% d}t$	$\displaystyle=\frac{\operatorname{arcsinh}a}{(1+a^{2})^{\frac{1}{2}}},$
$\|\Im a\|<1$ .
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\operatorname{arcsinh}\NVar{z}$ : inverse hyperbolic sine function, $\Im$ : imaginary part, $\int$ : integral, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind and $\sin\NVar{z}$ : 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 $\Re\mu>|\Re\nu|$ ,

10.43.22		$\int_{0}^{\infty}t^{\mu-1}e^{-at}K_{\nu}\left(t\right)\,\mathrm{d}t=\begin{% cases}\left(\frac{1}{2}\pi\right)^{\frac{1}{2}}\Gamma\left(\mu-\nu\right)% \Gamma\left(\mu+\nu\right)(1-a^{2})^{-\frac{1}{2}\mu+\frac{1}{4}}\mathsf{P}^{-% \mu+\frac{1}{2}}_{\nu-\frac{1}{2}}\left(a\right),&-1<a<1,\\ \left(\frac{1}{2}\pi\right)^{\frac{1}{2}}\Gamma\left(\mu-\nu\right)\Gamma\left% (\mu+\nu\right)(a^{2}-1)^{-\frac{1}{2}\mu+\frac{1}{4}}P^{-\mu+\frac{1}{2}}_{% \nu-\frac{1}{2}}\left(a\right),&\Re a\geq 0,a\neq 1.\end{cases}$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\Re$ : 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	$\displaystyle\int_{0}^{\infty}t^{\nu+1}I_{\nu}\left(bt\right)\exp\left(-p^{2}t% ^{2}\right)\,\mathrm{d}t$	$\displaystyle=\frac{b^{\nu}}{(2p^{2})^{\nu+1}}\exp\left(\frac{b^{2}}{4p^{2}}% \right),$
$\Re\nu>-1,\Re\left(p^{2}\right)>0$ ,
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\int$ : integral, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $\Re$ : 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	$\displaystyle\int_{0}^{\infty}I_{\nu}\left(bt\right)\exp\left(-p^{2}t^{2}% \right)\,\mathrm{d}t$	$\displaystyle=\frac{\sqrt{\pi}}{2p}\exp\left(\frac{b^{2}}{8p^{2}}\right)I_{% \frac{1}{2}\nu}\left(\frac{b^{2}}{8p^{2}}\right),$
$\Re\nu>-1$ , $\Re\left(p^{2}\right)>0$ ,
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\int$ : integral, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $\Re$ : 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	$\displaystyle\int_{0}^{\infty}K_{\nu}\left(bt\right)\exp\left(-p^{2}t^{2}% \right)\,\mathrm{d}t$	$\displaystyle=\frac{\sqrt{\pi}}{4p}\sec\left(\tfrac{1}{2}\pi\nu\right)\exp% \left(\frac{b^{2}}{8p^{2}}\right)K_{\frac{1}{2}\nu}\left(\frac{b^{2}}{8p^{2}}% \right),$
$\|\Re\nu\|<1$ , $\Re\left(p^{2}\right)>0$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\int$ : integral, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\Re$ : real part, $\sec\NVar{z}$ : 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	$\displaystyle\int_{0}^{\infty}\frac{K_{\mu}\left(at\right)J_{\nu}\left(bt% \right)}{t^{\lambda}}\,\mathrm{d}t$	$\displaystyle=\frac{b^{\nu}\Gamma\left(\frac{1}{2}\nu-\frac{1}{2}\lambda+\frac% {1}{2}\mu+\frac{1}{2}\right)\Gamma\left(\frac{1}{2}\nu-\frac{1}{2}\lambda-% \frac{1}{2}\mu+\frac{1}{2}\right)}{2^{\lambda+1}a^{\nu-\lambda+1}}\*\mathbf{F}% \left(\frac{\nu-\lambda+\mu+1}{2},\frac{\nu-\lambda-\mu+1}{2};\nu+1;-\frac{b^{% 2}}{a^{2}}\right),$
$\Re\left(\nu+1-\lambda\right)>\|\Re\mu\|,\Re a>\|\Im b\|$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\Im$ : imaginary part, $\int$ : integral, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\Re$ : real part, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ 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	$\displaystyle\int_{0}^{\infty}t^{\mu+\nu+1}K_{\mu}\left(at\right)J_{\nu}\left(% bt\right)\,\mathrm{d}t$	$\displaystyle=\frac{(2a)^{\mu}(2b)^{\nu}\Gamma\left(\mu+\nu+1\right)}{(a^{2}+b% ^{2})^{\mu+\nu+1}},$
$\Re\left(\nu+1\right)>\|\Re\mu\|,\Re a>\|\Im b\|$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\Im$ : imaginary part, $\int$ : integral, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\Re$ : 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	$\displaystyle\int_{0}^{\infty}t\exp\left(-p^{2}t^{2}\right)I_{\nu}\left(at% \right)I_{\nu}\left(bt\right)\,\mathrm{d}t$	$\displaystyle=\frac{1}{2p^{2}}\exp\left(\frac{a^{2}+b^{2}}{4p^{2}}\right)I_{% \nu}\left(\frac{ab}{2p^{2}}\right),$
$\Re\nu>-1,\Re\left(p^{2}\right)>0$ ,
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\int$ : integral, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $\Re$ : 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	$\displaystyle\int_{0}^{\infty}t\exp\left(-p^{2}t^{2}\right)I_{0}\left(at\right% )K_{0}\left(at\right)\,\mathrm{d}t$	$\displaystyle=\frac{1}{4p^{2}}\exp\left(\frac{a^{2}}{2p^{2}}\right)K_{0}\left(% \frac{a^{2}}{2p^{2}}\right),$
$\Re\left(p^{2}\right)>0$ .
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\int$ : integral, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind and $\Re$ : 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).

§10.43(v) Kontorovich–Lebedev Transform

ⓘ

Keywords:: Kontorovich–Lebedev transform, integrals of modified Bessel functions, modified Bessel functions
Notes:: For Conditions (a) see Sneddon (1972, pp. 359–361). For Conditions (b) see Lebedev et al. (1965, pp. 194–196).
Referenced by:: Ch.10, §18.39(i), §3.5(ix)
Permalink:: http://dlmf.nist.gov/10.43.v
See also:: Annotations for §10.43 and Ch.10

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

10.43.30		$f(y)=\frac{2y}{\pi^{2}}\sinh\left(\pi y\right)\int_{0}^{\infty}\frac{g(x)}{x}K% _{iy}\left(x\right)\,\mathrm{d}x.$
ⓘ Defines: $f(x)$ : Kontorovich–Lebedev transform of $g(x)$ (locally) Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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		$g(x)=\int_{0}^{\infty}f(y)K_{iy}\left(x\right)\,\mathrm{d}y,$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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 $0<x<\infty$ , $x^{-1}g(x)$ is continuously differentiable and each of $xg(x)$ and $x\ifrac{\mathrm{d}(x^{-1}g(x))}{\mathrm{d}x}$ is absolutely integrable.
(b)

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

10.43.32		$\int_{0}^{\frac{1}{2}}\frac{g(x)}{x}\ln\left(\frac{1}{x}\right)\,\mathrm{d}x,$
10.43.32		$\int_{\frac{1}{2}}^{\infty}\frac{\|g(x)\|}{x^{\frac{1}{2}}}\,\mathrm{d}x,$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\ln\NVar{z}$ : 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).

§10.43(vi) Compendia

ⓘ

Permalink:: http://dlmf.nist.gov/10.43.vi
See also:: Annotations for §10.43 and Ch.10

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 (2015, §§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).

10.42 Zeros 10.44 Sums

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