§10.43 Integrals

Referenced by:: §10.59
Permalink:: http://dlmf.nist.gov/10.43

§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

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

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

10.43.1

$\int z^{{\nu+1}}\mathop{\mathscr{Z}_{{\nu}}\/}\nolimits\!\left(z\right)dz=z^{{\nu+1}}\mathop{\mathscr{Z}_{{\nu+1}}\/}\nolimits\!\left(z\right),$

$\int z^{{-\nu+1}}\mathop{\mathscr{Z}_{{\nu}}\/}\nolimits\!\left(z\right)dz=z^{{-\nu+1}}\mathop{\mathscr{Z}_{{\nu-1}}\/}\nolimits\!\left(z\right).$

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\mathscr{Z}_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified cylinder function, $z$ : complex variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.43.E1
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10.43.2 $\int z^{\nu}\mathop{\mathscr{Z}_{{\nu}}\/}\nolimits\!\left(z\right)dz=\pi^{\frac{1}{2}}2^{{\nu-1}}\mathop{\Gamma\/}\nolimits\!\left(\nu+\tfrac{1}{2}\right)z\*\left(\mathop{\mathscr{Z}_{{\nu}}\/}\nolimits\!\left(z\right)\mathop{\mathbf{L}_{{\nu-1}}\/}\nolimits\!\left(z\right)-\mathop{\mathscr{Z}_{{\nu-1}}\/}\nolimits\!\left(z\right)\mathop{\mathbf{L}_{{\nu}}\/}\nolimits\!\left(z\right)\right),$ $\nu\neq-\tfrac{1}{2}$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\mathbf{L}_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Struve function, $\mathop{\mathscr{Z}_{{\nu}}\/}\nolimits\!\left(z\right)$ : 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
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For the modified Struve function $\mathop{\mathbf{L}_{{\nu}}\/}\nolimits\!\left(z\right)$ see §11.2(i).

10.43.3

$\int e^{{\pm z}}z^{\nu}\mathop{\mathscr{Z}_{{\nu}}\/}\nolimits\!\left(z\right)dz=\frac{e^{{\pm z}}z^{{\nu+1}}}{2\nu+1}\left(\mathop{\mathscr{Z}_{{\nu}}\/}\nolimits\!\left(z\right)\mp\mathop{\mathscr{Z}_{{\nu+1}}\/}\nolimits\!\left(z\right)\right),$ $\nu\neq-\tfrac{1}{2}$ ,

$\int e^{{\pm z}}z^{{-\nu}}\mathop{\mathscr{Z}_{{\nu}}\/}\nolimits\!\left(z\right)dz=\frac{e^{{\pm z}}z^{{-\nu+1}}}{1-2\nu}\left(\mathop{\mathscr{Z}_{{\nu}}\/}\nolimits\!\left(z\right)\mp\mathop{\mathscr{Z}_{{\nu-1}}\/}\nolimits\!\left(z\right)\right),$ $\nu\neq\tfrac{1}{2}$ .

Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{\mathscr{Z}_{{\nu}}\/}\nolimits\!\left(z\right)$ : 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
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§10.43(ii) Integrals over the Intervals $(0,x)$ and $(x,\infty)$

Notes:: For (10.43.4) replace $x$ by $ix$ 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 $\int _{x}^{\infty}(\mathop{{H^{{(1)}}_{{0}}}\/}\nolimits\!\left(t\right)/t)dt$ ; then replace $x$ by $ix$ 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 $x\to\infty$ .
Keywords:: integrals of modified Bessel functions
Permalink:: http://dlmf.nist.gov/10.43.ii

10.43.4 $\int _{0}^{x}\frac{\mathop{I_{{0}}\/}\nolimits\!\left(t\right)-1}{t}dt=\frac{1}{2}\sum _{{k=1}}^{\infty}(-1)^{{k-1}}\frac{\mathop{\psi\/}\nolimits\!\left(k+1\right)-\mathop{\psi\/}\nolimits\!\left(1\right)}{k!}(\tfrac{1}{2}x)^{k}\mathop{I_{{k}}\/}\nolimits\!\left(x\right)=\frac{2}{x}\sum _{{k=0}}^{\infty}(-1)^{k}(2k+3)(\mathop{\psi\/}\nolimits\!\left(k+2\right)-\mathop{\psi\/}\nolimits\!\left(1\right))\mathop{I_{{2k+3}}\/}\nolimits\!\left(x\right).$

Symbols:: $dx$ : differential of $x$ , $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $!$ : $n!$ : factorial, $\int$ : integral, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $k$ : nonnegative integer and $x$ : real variable
Referenced by:: §10.43(ii)
Permalink:: http://dlmf.nist.gov/10.43.E4
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10.43.5 $\int _{x}^{\infty}\frac{\mathop{K_{{0}}\/}\nolimits\!\left(t\right)}{t}dt=\frac{1}{2}\left(\mathop{\ln\/}\nolimits\!\left(\tfrac{1}{2}x\right)+\EulerConstant\right)^{2}+\frac{\pi^{2}}{24}-\sum _{{k=1}}^{\infty}\left(\mathop{\psi\/}\nolimits\!\left(k+1\right)+\frac{1}{2k}-\mathop{\ln\/}\nolimits\!\left(\tfrac{1}{2}x\right)\right)\frac{(\tfrac{1}{2}x)^{{2k}}}{2k(k!)^{2}},$

Symbols:: $\EulerConstant$ : Euler’s constant, $dx$ : differential of $x$ , $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $!$ : $n!$ : factorial, $\int$ : integral, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\ln\/}\nolimits z$ : 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
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where $\mathop{\psi\/}\nolimits=\ifrac{{\mathop{\Gamma\/}\nolimits^{{\prime}}}}{\mathop{\Gamma\/}\nolimits}$ and $\EulerConstant$ is Euler’s constant (§5.2).

10.43.6 $\int _{0}^{x}e^{{-t}}\mathop{I_{{n}}\/}\nolimits\!\left(t\right)dt=xe^{{-x}}(\mathop{I_{{0}}\/}\nolimits\!\left(x\right)+\mathop{I_{{1}}\/}\nolimits\!\left(x\right))+n(e^{{-x}}\mathop{I_{{0}}\/}\nolimits\!\left(x\right)-1)+2e^{{-x}}\sum _{{k=1}}^{{n-1}}(n-k)\mathop{I_{{k}}\/}\nolimits\!\left(x\right),$ $n=0,1,2,\ldots$ .

Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $n$ : integer, $k$ : nonnegative integer and $x$ : real variable
Referenced by:: §10.43(ii)
Permalink:: http://dlmf.nist.gov/10.43.E6
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10.43.7 $\int _{0}^{x}e^{{\pm t}}t^{\nu}\mathop{I_{{\nu}}\/}\nolimits\!\left(t\right)dt=\frac{e^{{\pm x}}x^{{\nu+1}}}{2\nu+1}(\mathop{I_{{\nu}}\/}\nolimits\!\left(x\right)\mp\mathop{I_{{\nu+1}}\/}\nolimits\!\left(x\right)),$ $\realpart{\nu}>-\tfrac{1}{2}$ ,

Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\realpart{}$ : real part, $x$ : real variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.43.E7
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10.43.8 $\int _{0}^{x}e^{{\pm t}}t^{{-\nu}}\mathop{I_{{\nu}}\/}\nolimits\!\left(t\right)dt=-\frac{e^{{\pm x}}x^{{-\nu+1}}}{2\nu-1}(\mathop{I_{{\nu}}\/}\nolimits\!\left(x\right)\mp\mathop{I_{{\nu-1}}\/}\nolimits\!\left(x\right))\mp\frac{2^{{-\nu+1}}}{(2\nu-1)\mathop{\Gamma\/}\nolimits\!\left(\nu\right)},$ $\nu\neq\tfrac{1}{2}$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $x$ : real variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.43.E8
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10.43.9 $\int _{0}^{x}e^{{\pm t}}t^{\nu}\mathop{K_{{\nu}}\/}\nolimits\!\left(t\right)dt=\frac{e^{{\pm x}}x^{{\nu+1}}}{2\nu+1}(\mathop{K_{{\nu}}\/}\nolimits\!\left(x\right)\pm\mathop{K_{{\nu+1}}\/}\nolimits\!\left(x\right))\mp\frac{2^{\nu}\mathop{\Gamma\/}\nolimits\!\left(\nu+1\right)}{2\nu+1},$ $\realpart{\nu}>-\tfrac{1}{2}$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\realpart{}$ : real part, $x$ : real variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.43.E9
Encodings:: TeX, pMML, png

10.43.10 $\int _{x}^{\infty}e^{t}t^{{-\nu}}\mathop{K_{{\nu}}\/}\nolimits\!\left(t\right)dt=\frac{e^{x}x^{{-\nu+1}}}{2\nu-1}(\mathop{K_{{\nu}}\/}\nolimits\!\left(x\right)+\mathop{K_{{\nu-1}}\/}\nolimits\!\left(x\right)),$ $\realpart{\nu}>\tfrac{1}{2}$ .

Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\realpart{}$ : real part, $x$ : real variable and $\nu$ : complex parameter
Referenced by:: §10.43(ii)
Permalink:: http://dlmf.nist.gov/10.43.E10
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§10.43(iii) Fractional Integrals

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).
Keywords:: Bickley function, integrals of modified Bessel functions
Referenced by:: §10.75(vii)
Permalink:: http://dlmf.nist.gov/10.43.iii

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

10.43.11 $\mathop{\mathrm{Ki}_{{\alpha}}\/}\nolimits\!\left(x\right)=\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(\alpha\right)}\int _{x}^{\infty}(t-x)^{{\alpha-1}}\mathop{K_{{0}}\/}\nolimits\!\left(t\right)dt,$

Defines:: $\mathop{\mathrm{Ki}_{{\alpha}}\/}\nolimits\!\left(x\right)$ : Bickley function
Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function and $x$ : real variable
A&S Ref:: 11.2.11
Referenced by:: §10.43(iii)
Permalink:: http://dlmf.nist.gov/10.43.E11
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when $\realpart{\alpha}>0$ and $x>0$ , and by analytic continuation elsewhere. Equivalently,

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

Symbols:: $\mathop{\mathrm{Ki}_{{\alpha}}\/}\nolimits\!\left(x\right)$ : Bickley function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\mathop{\cosh\/}\nolimits z$ : 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
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¶ Properties

10.43.13 $\mathop{\mathrm{Ki}_{{\alpha}}\/}\nolimits\!\left(x\right)=\int _{x}^{\infty}\mathop{\mathrm{Ki}_{{\alpha-1}}\/}\nolimits\!\left(t\right)dt,$

Symbols:: $\mathop{\mathrm{Ki}_{{\alpha}}\/}\nolimits\!\left(x\right)$ : Bickley function, $dx$ : 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
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10.43.14 $\mathop{\mathrm{Ki}_{{0}}\/}\nolimits\!\left(x\right)=\mathop{K_{{0}}\/}\nolimits\!\left(x\right),$

Symbols:: $\mathop{\mathrm{Ki}_{{\alpha}}\/}\nolimits\!\left(x\right)$ : Bickley function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function and $x$ : real variable
A&S Ref:: 11.2.8
Permalink:: http://dlmf.nist.gov/10.43.E14
Encodings:: TeX, pMML, png

10.43.15 $\mathop{\mathrm{Ki}_{{-n}}\/}\nolimits\!\left(x\right)=(-1)^{n}\frac{{d}^{n}}{{dx}^{n}}\mathop{K_{{0}}\/}\nolimits\!\left(x\right),$ $n=1,2,3,\ldots$ .

Symbols:: $\mathop{\mathrm{Ki}_{{\alpha}}\/}\nolimits\!\left(x\right)$ : Bickley function, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $n$ : integer and $x$ : real variable
A&S Ref:: 11.2.9
Permalink:: http://dlmf.nist.gov/10.43.E15
Encodings:: TeX, pMML, png

10.43.16 $\mathop{\mathrm{Ki}_{{\alpha}}\/}\nolimits\!\left(0\right)=\frac{\sqrt{\pi}\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}\alpha\right)}{2\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}\alpha+\frac{1}{2}\right)},$ $\alpha\neq 0,-2,-4,\dots$ .

Symbols:: $\mathop{\mathrm{Ki}_{{\alpha}}\/}\nolimits\!\left(x\right)$ : Bickley function and $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function
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
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10.43.17 $\alpha\mathop{\mathrm{Ki}_{{\alpha+1}}\/}\nolimits\!\left(x\right)+x\mathop{\mathrm{Ki}_{{\alpha}}\/}\nolimits\!\left(x\right)+(1-\alpha)\mathop{\mathrm{Ki}_{{\alpha-1}}\/}\nolimits\!\left(x\right)-x\mathop{\mathrm{Ki}_{{\alpha-2}}\/}\nolimits\!\left(x\right)=0.$

Symbols:: $\mathop{\mathrm{Ki}_{{\alpha}}\/}\nolimits\!\left(x\right)$ : 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

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 ( $0,\infty$ )

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).
Keywords:: Bickley function, integrals of modified Bessel functions
Permalink:: http://dlmf.nist.gov/10.43.iv

10.43.18 $\int _{0}^{\infty}\mathop{K_{{\nu}}\/}\nolimits\!\left(t\right)dt=\tfrac{1}{2}\pi\mathop{\sec\/}\nolimits(\tfrac{1}{2}\pi\nu),$ $|\realpart{\nu}|<1$ .

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\realpart{}$ : real part, $\mathop{\sec\/}\nolimits z$ : secant function and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.43.E18
Encodings:: TeX, pMML, png

10.43.19 $\int _{0}^{\infty}t^{{\mu-1}}\mathop{K_{{\nu}}\/}\nolimits\!\left(t\right)dt=2^{{\mu-2}}\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}\mu-\tfrac{1}{2}\nu\right)\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}\mu+\tfrac{1}{2}\nu\right),$ $|\realpart{\nu}|<\realpart{\mu}$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\realpart{}$ : real part and $\nu$ : complex parameter
A&S Ref:: 11.4.22, 11.4.23
Permalink:: http://dlmf.nist.gov/10.43.E19
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10.43.20 $\int _{0}^{\infty}\mathop{\cos\/}\nolimits\!\left(at\right)\mathop{K_{{0}}\/}\nolimits\!\left(t\right)dt=\frac{\pi}{2(1+a^{2})^{{\frac{1}{2}}}},$ $|\imagpart{a}|<1$ ,

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\imagpart{}$ : imaginary part, $\int$ : integral and $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function
A&S Ref:: 11.4.14
Permalink:: http://dlmf.nist.gov/10.43.E20
Encodings:: TeX, pMML, png

10.43.21 $\int _{0}^{\infty}\mathop{\sin\/}\nolimits\!\left(at\right)\mathop{K_{{0}}\/}\nolimits\!\left(t\right)dt=\frac{\mathop{\mathrm{arcsinh}\/}\nolimits a}{(1+a^{2})^{{\frac{1}{2}}}},$ $|\imagpart{a}|<1$ .

Symbols:: $dx$ : differential of $x$ , $\mathop{\mathrm{arcsinh}\/}\nolimits z$ : inverse hyperbolic sine function, $\imagpart{}$ : imaginary part, $\int$ : integral, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function and $\mathop{\sin\/}\nolimits z$ : sine function
A&S Ref:: 11.4.15
Permalink:: http://dlmf.nist.gov/10.43.E21
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When $\realpart{\mu}>|\realpart{\nu}|$ ,

10.43.22 $\int _{0}^{\infty}t^{{\mu-1}}e^{{-at}}\mathop{K_{{\nu}}\/}\nolimits\!\left(t\right)dt=\begin{cases}\left(\frac{1}{2}\pi\right)^{{\frac{1}{2}}}\mathop{\Gamma\/}\nolimits\!\left(\mu-\nu\right)\mathop{\Gamma\/}\nolimits\!\left(\mu+\nu\right)(1-a^{2})^{{-\frac{1}{2}\mu+\frac{1}{4}}}\mathop{\mathsf{P}^{{-\mu+\frac{1}{2}}}_{{\nu-\frac{1}{2}}}\/}\nolimits\!\left(a\right),&-1<a<1,\\ \left(\frac{1}{2}\pi\right)^{{\frac{1}{2}}}\mathop{\Gamma\/}\nolimits\!\left(\mu-\nu\right)\mathop{\Gamma\/}\nolimits\!\left(\mu+\nu\right)(a^{2}-1)^{{-\frac{1}{2}\mu+\frac{1}{4}}}\mathop{P^{{-\mu+\frac{1}{2}}}_{{\nu-\frac{1}{2}}}\/}\nolimits\!\left(a\right),&\realpart{a}\geq 0,a\neq 1.\end{cases}$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\realpart{}$ : real part and $\nu$ : complex parameter
Referenced by:: §10.43(iv)
Permalink:: http://dlmf.nist.gov/10.43.E22
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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 $\mathop{\mathsf{P}\/}\nolimits$ and the associated Legendre function $\mathop{P\/}\nolimits$ , see §§14.3(i) and 14.21(i).

10.43.23 $\int _{0}^{\infty}t^{{\nu+1}}\mathop{I_{{\nu}}\/}\nolimits\!\left(bt\right)\mathop{\exp\/}\nolimits(-p^{2}t^{2})dt=\frac{b^{\nu}}{(2p^{2})^{{\nu+1}}}\mathop{\exp\/}\nolimits\left(\frac{b^{2}}{4p^{2}}\right),$ $\realpart{\nu}>-1,\realpart{(p^{2})}>0$ ,

Symbols:: $dx$ : differential of $x$ , $\mathop{\exp\/}\nolimits z$ : exponential function, $\int$ : integral, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\realpart{}$ : real part and $\nu$ : complex parameter
Referenced by:: §10.43(iv)
Permalink:: http://dlmf.nist.gov/10.43.E23
Encodings:: TeX, pMML, png

10.43.24 $\int _{0}^{\infty}\mathop{I_{{\nu}}\/}\nolimits\!\left(bt\right)\mathop{\exp\/}\nolimits\!\left(-p^{2}t^{2}\right)dt=\frac{\sqrt{\pi}}{2p}\mathop{\exp\/}\nolimits\!\left(\frac{b^{2}}{8p^{2}}\right)\mathop{I_{{\frac{1}{2}\nu}}\/}\nolimits\!\left(\frac{b^{2}}{8p^{2}}\right),$ $\realpart{\nu}>-1$ , $\realpart{(p^{2})}>0$ ,

Symbols:: $dx$ : differential of $x$ , $\mathop{\exp\/}\nolimits z$ : exponential function, $\int$ : integral, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\realpart{}$ : real part and $\nu$ : complex parameter
A&S Ref:: 11.4.31
Permalink:: http://dlmf.nist.gov/10.43.E24
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10.43.25 $\int _{0}^{\infty}\mathop{K_{{\nu}}\/}\nolimits\!\left(bt\right)\mathop{\exp\/}\nolimits\!\left(-p^{2}t^{2}\right)dt=\frac{\sqrt{\pi}}{4p}\mathop{\sec\/}\nolimits\!\left(\tfrac{1}{2}\pi\nu\right)\mathop{\exp\/}\nolimits\!\left(\frac{b^{2}}{8p^{2}}\right)\mathop{K_{{\frac{1}{2}\nu}}\/}\nolimits\!\left(\frac{b^{2}}{8p^{2}}\right),$ $|\realpart{\nu}|<1$ , $\realpart{(p^{2})}>0$ .

Symbols:: $dx$ : differential of $x$ , $\mathop{\exp\/}\nolimits z$ : exponential function, $\int$ : integral, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\realpart{}$ : real part, $\mathop{\sec\/}\nolimits z$ : 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

10.43.26 $\int _{0}^{\infty}\frac{\mathop{K_{{\mu}}\/}\nolimits\!\left(at\right)\mathop{J_{{\nu}}\/}\nolimits\!\left(bt\right)}{t^{\lambda}}dt=\frac{b^{\nu}\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}\nu-\frac{1}{2}\lambda+\frac{1}{2}\mu+\frac{1}{2}\right)\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}\nu-\frac{1}{2}\lambda-\frac{1}{2}\mu+\frac{1}{2}\right)}{2^{{\lambda+1}}a^{{\nu-\lambda+1}}}\*\mathop{\mathbf{F}\/}\nolimits\!\left(\frac{\nu-\lambda+\mu+1}{2},\frac{\nu-\lambda-\mu+1}{2};\nu+1;-\frac{b^{2}}{a^{2}}\right),$ $\realpart{(\nu+1-\lambda)}>|\realpart{\mu}|,\realpart{a}>|\imagpart{b}|$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $\imagpart{}$ : imaginary part, $\int$ : integral, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\realpart{}$ : real part, $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ : Olver’s hypergeometric function and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.43.E26
Encodings:: TeX, pMML, png

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

10.43.27 $\int _{0}^{\infty}t^{{\mu+\nu+1}}\mathop{K_{{\mu}}\/}\nolimits\!\left(at\right)\mathop{J_{{\nu}}\/}\nolimits\!\left(bt\right)dt=\frac{(2a)^{\mu}(2b)^{\nu}\mathop{\Gamma\/}\nolimits\!\left(\mu+\nu+1\right)}{(a^{2}+b^{2})^{{\mu+\nu+1}}},$ $\realpart{(\nu+1)}>|\realpart{\mu}|,\realpart{a}>|\imagpart{b}|$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $\imagpart{}$ : imaginary part, $\int$ : integral, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\realpart{}$ : real part and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.43.E27
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10.43.28 $\int _{0}^{\infty}t\mathop{\exp\/}\nolimits(-p^{2}t^{2})\mathop{I_{{\nu}}\/}\nolimits\!\left(at\right)\mathop{I_{{\nu}}\/}\nolimits\!\left(bt\right)dt=\frac{1}{2p^{2}}\mathop{\exp\/}\nolimits\left(\frac{a^{2}+b^{2}}{4p^{2}}\right)\mathop{I_{{\nu}}\/}\nolimits\left(\frac{ab}{2p^{2}}\right),$ $\realpart{\nu}>-1,\realpart{(p^{2})}>0$ ,

Symbols:: $dx$ : differential of $x$ , $\mathop{\exp\/}\nolimits z$ : exponential function, $\int$ : integral, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\realpart{}$ : real part and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.43.E28
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10.43.29 $\int _{0}^{\infty}t\mathop{\exp\/}\nolimits(-p^{2}t^{2})\mathop{I_{{0}}\/}\nolimits\!\left(at\right)\mathop{K_{{0}}\/}\nolimits\!\left(at\right)dt=\frac{1}{4p^{2}}\mathop{\exp\/}\nolimits\left(\frac{a^{2}}{2p^{2}}\right)\mathop{K_{{0}}\/}\nolimits\left(\frac{a^{2}}{2p^{2}}\right),$ $\realpart{(p^{2})}>0$ .

Symbols:: $dx$ : differential of $x$ , $\mathop{\exp\/}\nolimits z$ : exponential function, $\int$ : integral, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function and $\realpart{}$ : real part
Referenced by:: §10.43(iv)
Permalink:: http://dlmf.nist.gov/10.43.E29
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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

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

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

10.43.30 $f(y)=\frac{2y}{\pi^{2}}\mathop{\sinh\/}\nolimits(\pi y)\int _{0}^{\infty}\frac{g(x)}{x}\mathop{K_{{iy}}\/}\nolimits\!\left(x\right)dx.$

Symbols:: $dx$ : differential of $x$ , $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\int$ : integral, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $x$ : 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.E30
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Then

10.43.31 $g(x)=\int _{0}^{\infty}f(y)\mathop{K_{{iy}}\/}\nolimits\!\left(x\right)dy,$

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $x$ : 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
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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 $x\ifrac{d(x^{{-1}}g(x))}{dx}$ 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

$\int _{0}^{{\frac{1}{2}}}\frac{g(x)}{x}\mathop{\ln\/}\nolimits\left(\frac{1}{x}\right)dx,$

$\int _{{\frac{1}{2}}}^{\infty}\frac{|g(x)|}{x^{{\frac{1}{2}}}}dx,$

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $x$ : real variable and $g(x)$ : function
Permalink:: http://dlmf.nist.gov/10.43.E32
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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

For collections of integrals of the functions $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ and $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ , 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).