§10.32 Integral Representations

Referenced by:: §10.54
Permalink:: http://dlmf.nist.gov/10.32

§10.32(i) Integrals along the Real Line
§10.32(ii) Contour Integrals
§10.32(iii) Products
§10.32(iv) Compendia

§10.32(i) Integrals along the Real Line

Notes:: See Watson (1944, pp. 79, 80, 172, and 181–183). For (10.32.7) see also Erdélyi et al. (1953b, p. 82).
Keywords:: modified Bessel functions
Permalink:: http://dlmf.nist.gov/10.32.i

10.32.1 $\mathop{I_{{0}}\/}\nolimits\!\left(z\right)=\frac{1}{\pi}\int _{0}^{\pi}e^{{\pm z\mathop{\cos\/}\nolimits\theta}}d\theta=\frac{1}{\pi}\int _{0}^{\pi}\mathop{\cosh\/}\nolimits\!\left(z\mathop{\cos\/}\nolimits\theta\right)d\theta.$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\int$ : integral, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function and $z$ : complex variable
A&S Ref:: 9.6.16
Permalink:: http://dlmf.nist.gov/10.32.E1
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10.32.2 $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)=\frac{(\frac{1}{2}z)^{\nu}}{\pi^{{\frac{1}{2}}}\mathop{\Gamma\/}\nolimits\!\left(\nu+\frac{1}{2}\right)}\int _{0}^{\pi}e^{{\pm z\mathop{\cos\/}\nolimits\theta}}(\mathop{\sin\/}\nolimits\theta)^{{2\nu}}d\theta=\frac{(\frac{1}{2}z)^{\nu}}{\pi^{{\frac{1}{2}}}\mathop{\Gamma\/}\nolimits\!\left(\nu+\frac{1}{2}\right)}\int _{{-1}}^{1}(1-t^{2})^{{\nu-\frac{1}{2}}}e^{{\pm zt}}dt,$ $\realpart{\nu}>-\tfrac{1}{2}$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\realpart{}$ : real part, $\mathop{\sin\/}\nolimits z$ : sine function, $z$ : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.6.18
Permalink:: http://dlmf.nist.gov/10.32.E2
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10.32.3 $\mathop{I_{{n}}\/}\nolimits\!\left(z\right)=\frac{1}{\pi}\int _{0}^{\pi}e^{{z\mathop{\cos\/}\nolimits\theta}}\mathop{\cos\/}\nolimits\!\left(n\theta\right)d\theta.$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $n$ : integer and $z$ : complex variable
A&S Ref:: 9.6.19
Permalink:: http://dlmf.nist.gov/10.32.E3
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10.32.4 $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)=\frac{1}{\pi}\int _{0}^{\pi}e^{{z\mathop{\cos\/}\nolimits\theta}}\mathop{\cos\/}\nolimits\!\left(\nu\theta\right)d\theta-\frac{\mathop{\sin\/}\nolimits\!\left(\nu\pi\right)}{\pi}\int _{0}^{\infty}e^{{-z\mathop{\cosh\/}\nolimits t-\nu t}}dt,$ $|\mathop{\mathrm{ph}\/}\nolimits z|<\tfrac{1}{2}\pi$ .

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\int$ : integral, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\mathop{\sin\/}\nolimits z$ : sine function, $z$ : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.6.20
Permalink:: http://dlmf.nist.gov/10.32.E4
Encodings:: TeX, pMML, png

10.32.5 $\mathop{K_{{0}}\/}\nolimits\!\left(z\right)=-\frac{1}{\pi}\int _{0}^{\pi}e^{{\pm z\mathop{\cos\/}\nolimits\theta}}\left(\EulerConstant+\mathop{\ln\/}\nolimits\!\left(2z(\mathop{\sin\/}\nolimits\theta)^{2}\right)\right)d\theta.$

Symbols:: $\EulerConstant$ : Euler’s constant, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\mathop{\sin\/}\nolimits z$ : sine function and $z$ : complex variable
A&S Ref:: 9.6.17
Permalink:: http://dlmf.nist.gov/10.32.E5
Encodings:: TeX, pMML, png

10.32.6 $\mathop{K_{{0}}\/}\nolimits\!\left(x\right)=\int _{0}^{\infty}\mathop{\cos\/}\nolimits\!\left(x\mathop{\sinh\/}\nolimits t\right)dt=\int _{0}^{\infty}\frac{\mathop{\cos\/}\nolimits\!\left(xt\right)}{\sqrt{t^{2}+1}}dt,$ $x>0$ .

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\int$ : integral, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function and $x$ : real variable
A&S Ref:: 9.6.21
Permalink:: http://dlmf.nist.gov/10.32.E6
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10.32.7 $\mathop{K_{{\nu}}\/}\nolimits\!\left(x\right)=\mathop{\sec\/}\nolimits\!\left(\tfrac{1}{2}\nu\pi\right)\int _{0}^{\infty}\mathop{\cos\/}\nolimits\!\left(x\mathop{\sinh\/}\nolimits t\right)\mathop{\cosh\/}\nolimits\!\left(\nu t\right)dt=\mathop{\csc\/}\nolimits\!\left(\tfrac{1}{2}\nu\pi\right)\int _{0}^{\infty}\mathop{\sin\/}\nolimits\!\left(x\mathop{\sinh\/}\nolimits t\right)\mathop{\sinh\/}\nolimits\!\left(\nu t\right)dt,$ $|\realpart{\nu}|<1$ , $x>0$ .

Symbols:: $\mathop{\csc\/}\nolimits z$ : cosecant function, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\int$ : integral, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\realpart{}$ : real part, $\mathop{\sec\/}\nolimits z$ : secant function, $\mathop{\sin\/}\nolimits z$ : sine function, $x$ : real variable and $\nu$ : complex parameter
A&S Ref:: 9.6.22
Referenced by:: §10.32(i)
Permalink:: http://dlmf.nist.gov/10.32.E7
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10.32.8 $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)=\frac{\pi^{{\frac{1}{2}}}(\frac{1}{2}z)^{\nu}}{\mathop{\Gamma\/}\nolimits\!\left(\nu+\frac{1}{2}\right)}\int _{0}^{\infty}e^{{-z\mathop{\cosh\/}\nolimits t}}(\mathop{\sinh\/}\nolimits t)^{{2\nu}}dt=\frac{\pi^{{\frac{1}{2}}}(\frac{1}{2}z)^{\nu}}{\mathop{\Gamma\/}\nolimits\!\left(\nu+\frac{1}{2}\right)}\int _{1}^{\infty}e^{{-zt}}(t^{2}-1)^{{\nu-\frac{1}{2}}}dt,$ $\realpart{\nu}>-\tfrac{1}{2}$ , $|\mathop{\mathrm{ph}\/}\nolimits z|<\tfrac{1}{2}\pi$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\int$ : integral, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\realpart{}$ : real part, $z$ : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.6.23
Referenced by:: §10.74(iii)
Permalink:: http://dlmf.nist.gov/10.32.E8
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10.32.9 $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)=\int _{0}^{\infty}e^{{-z\mathop{\cosh\/}\nolimits t}}\mathop{\cosh\/}\nolimits\!\left(\nu t\right)dt,$ $|\mathop{\mathrm{ph}\/}\nolimits z|<\tfrac{1}{2}\pi$ .

Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\int$ : integral, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $z$ : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.6.24
Referenced by:: §10.43(iii), §9.5(ii)
Permalink:: http://dlmf.nist.gov/10.32.E9
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10.32.10 $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)=\tfrac{1}{2}(\tfrac{1}{2}z)^{\nu}\int _{0}^{\infty}\mathop{\exp\/}\nolimits\left(-t-\frac{z^{2}}{4t}\right)\frac{dt}{t^{{\nu+1}}},$ $|\mathop{\mathrm{ph}\/}\nolimits z|<\tfrac{1}{4}\pi$ .

Symbols:: $dx$ : differential of $x$ , $\mathop{\exp\/}\nolimits z$ : exponential function, $\int$ : integral, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $z$ : complex variable and $\nu$ : complex parameter
Referenced by:: §6.7(iii), §7.14(i)
Permalink:: http://dlmf.nist.gov/10.32.E10
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¶ Basset’s Integral

Keywords:: Basset’s integral, modified Bessel functions

10.32.11 $\mathop{K_{{\nu}}\/}\nolimits\!\left(xz\right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(\nu+\frac{1}{2}\right)(2z)^{\nu}}{\pi^{{\frac{1}{2}}}x^{\nu}}\int _{0}^{\infty}\frac{\mathop{\cos\/}\nolimits\!\left(xt\right)dt}{(t^{2}+z^{2})^{{\nu+\frac{1}{2}}}},$ $\realpart{\nu}>-\tfrac{1}{2}$ , $x>0$ , $|\mathop{\mathrm{ph}\/}\nolimits z|<\tfrac{1}{2}\pi$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\realpart{}$ : real part, $x$ : real variable, $z$ : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.6.25 (corrected)
Permalink:: http://dlmf.nist.gov/10.32.E11
Encodings:: TeX, pMML, png

§10.32(ii) Contour Integrals

Notes:: See Watson (1944, pp. 181, 191, and 193) and apply (10.27.8). See also Paris and Kaminski (2001, p. 114).
Keywords:: modified Bessel functions
Permalink:: http://dlmf.nist.gov/10.32.ii

10.32.12 $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)=\frac{1}{2\pi i}\int _{{\infty-i\pi}}^{{\infty+i\pi}}e^{{z\mathop{\cosh\/}\nolimits t-\nu t}}dt,$ $|\mathop{\mathrm{ph}\/}\nolimits z|<\tfrac{1}{2}\pi$ .

Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\int$ : integral, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $z$ : complex variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.32.E12
Encodings:: TeX, pMML, png

¶ Mellin–Barnes Type

Keywords:: modified Bessel functions

10.32.13 $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)=\frac{(\frac{1}{2}z)^{\nu}}{4\pi i}\int _{{c-i\infty}}^{{c+i\infty}}\mathop{\Gamma\/}\nolimits\!\left(t\right)\mathop{\Gamma\/}\nolimits\!\left(t-\nu\right)(\tfrac{1}{2}z)^{{-2t}}dt,$ $c>\max(\realpart{\nu},0),|\mathop{\mathrm{ph}\/}\nolimits z|<\pi$ .

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, $(a,b)$ : open interval, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\realpart{}$ : real part, $z$ : complex variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.32.E13
Encodings:: TeX, pMML, png

10.32.14 $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)=\frac{1}{2\pi^{2}i}\left(\frac{\pi}{2z}\right)^{{\frac{1}{2}}}e^{{-z}}\mathop{\cos\/}\nolimits(\nu\pi)\*\int _{{-i\infty}}^{{i\infty}}\mathop{\Gamma\/}\nolimits\!\left(t\right)\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}-t-\nu\right)\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}-t+\nu\right)(2z)^{t}dt,$ $\nu-\tfrac{1}{2}\notin\Integer,|\mathop{\mathrm{ph}\/}\nolimits z|<\tfrac{3}{2}\pi$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\Integer$ : set of all integers, $\int$ : integral, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\notin$ : not an element of, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $z$ : complex variable and $\nu$ : complex parameter
Referenced by:: ¶ ‣ §10.32(ii), §10.74(iii)
Permalink:: http://dlmf.nist.gov/10.32.E14
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In (10.32.14) the integration contour separates the poles of $\mathop{\Gamma\/}\nolimits\!\left(t\right)$ from the poles of $\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}-t-\nu\right)\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}-t+\nu\right)$ .

§10.32(iii) Products

Notes:: See Watson (1944, pp. 439–441) and Erdélyi et al. (1953b, pp. 97–98). For (10.32.16) see Dixon and Ferrar (1930). (An error in the conditions has been corrected.) For (10.32.19) see Titchmarsh (1986a, Eq. (7.10.2)).
Keywords:: modified Bessel functions
Permalink:: http://dlmf.nist.gov/10.32.iii

10.32.15 $\mathop{I_{{\mu}}\/}\nolimits\!\left(z\right)\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)=\frac{2}{\pi}\int _{0}^{{\frac{1}{2}\pi}}\mathop{I_{{\mu+\nu}}\/}\nolimits\!\left(2z\mathop{\cos\/}\nolimits\theta\right)\mathop{\cos\/}\nolimits((\mu-\nu)\theta)d\theta,$ $\realpart{(\mu+\nu)}>-1$ .

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\realpart{}$ : real part, $z$ : complex variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.32.E15
Encodings:: TeX, pMML, png

10.32.16 $\mathop{I_{{\mu}}\/}\nolimits\!\left(x\right)\mathop{K_{{\nu}}\/}\nolimits\!\left(x\right)=\int _{0}^{\infty}\mathop{J_{{\mu\pm\nu}}\/}\nolimits\!\left(2x\mathop{\sinh\/}\nolimits t\right)e^{{(-\mu\pm\nu)t}}dt,$ $\realpart{(\mu\mp\nu)}>-\tfrac{1}{2}$ , $\realpart{(\mu\pm\nu)}>-1$ , $x>0$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $dx$ : differential of $x$ , $e$ : base of exponential function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\int$ : integral, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\realpart{}$ : real part, $x$ : real variable and $\nu$ : complex parameter
Referenced by:: §10.32(iii)
Permalink:: http://dlmf.nist.gov/10.32.E16
Encodings:: TeX, pMML, png

10.32.17 $\mathop{K_{{\mu}}\/}\nolimits\!\left(z\right)\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)=2\int _{0}^{\infty}\mathop{K_{{\mu\pm\nu}}\/}\nolimits\!\left(2z\mathop{\cosh\/}\nolimits t\right)\mathop{\cosh\/}\nolimits((\mu\mp\nu)t)dt,$ $|\mathop{\mathrm{ph}\/}\nolimits z|<\tfrac{1}{2}\pi$ .

Symbols:: $dx$ : differential of $x$ , $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\int$ : integral, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $z$ : complex variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.32.E17
Encodings:: TeX, pMML, png

10.32.18 $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)\mathop{K_{{\nu}}\/}\nolimits\!\left(\zeta\right)=\frac{1}{2}\int _{0}^{\infty}\mathop{\exp\/}\nolimits\left(-\frac{t}{2}-\frac{z^{2}+\zeta^{2}}{2t}\right)\mathop{K_{{\nu}}\/}\nolimits\left(\frac{z\zeta}{t}\right)\frac{dt}{t},$ $|\mathop{\mathrm{ph}\/}\nolimits z|<\pi$ , $|\mathop{\mathrm{ph}\/}\nolimits\zeta|<\pi$ , $|\mathop{\mathrm{ph}\/}\nolimits(z+\zeta)|<\tfrac{1}{4}\pi$ .

Symbols:: $dx$ : differential of $x$ , $\mathop{\exp\/}\nolimits z$ : exponential function, $\int$ : integral, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $z$ : complex variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.32.E18
Encodings:: TeX, pMML, png

¶ Mellin–Barnes Type

10.32.19 $\mathop{K_{{\mu}}\/}\nolimits\!\left(z\right)\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)=\frac{1}{8\pi i}\int _{{c-i\infty}}^{{c+i\infty}}\frac{\mathop{\Gamma\/}\nolimits\!\left(t+\frac{1}{2}\mu+\frac{1}{2}\nu\right)\mathop{\Gamma\/}\nolimits\!\left(t+\frac{1}{2}\mu-\frac{1}{2}\nu\right)\mathop{\Gamma\/}\nolimits\!\left(t-\frac{1}{2}\mu+\frac{1}{2}\nu\right)\mathop{\Gamma\/}\nolimits\!\left(t-\frac{1}{2}\mu-\frac{1}{2}\nu\right)}{\mathop{\Gamma\/}\nolimits\!\left(2t\right)}(\tfrac{1}{2}z)^{{-2t}}dt,$ $c>\tfrac{1}{2}(|\realpart{\mu}|+|\realpart{\nu}|),|\mathop{\mathrm{ph}\/}\nolimits z|<\tfrac{1}{2}\pi$ .

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, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\realpart{}$ : real part, $z$ : complex variable and $\nu$ : complex parameter
Referenced by:: §10.32(iii)
Permalink:: http://dlmf.nist.gov/10.32.E19
Encodings:: TeX, pMML, png

For similar integrals for $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ and $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ see Paris and Kaminski (2001, p. 116).

§10.32(iv) Compendia

Keywords:: modified Bessel functions
Permalink:: http://dlmf.nist.gov/10.32.iv

For collections of integral representations of modified Bessel functions, or products of modified Bessel functions, see Erdélyi et al. (1953b, §§7.3, 7.12, and 7.14.2), Erdélyi et al. (1954a, pp. 48–60, 105–115, 276–285, and 357–359), Gröbner and Hofreiter (1950, pp. 193–194), Magnus et al. (1966, §3.7), Marichev (1983, pp. 191–216), and Watson (1944, Chapters 6, 12, and 13).

§10.32 Integral Representations

Contents

§10.32(i) Integrals along the Real Line

¶ Basset’s Integral

§10.32(ii) Contour Integrals

¶ Mellin–Barnes Type

§10.32(iii) Products

¶ Mellin–Barnes Type

§10.32(iv) Compendia