10 Bessel Functions Modified Bessel Functions 10.31 Power Series 10.33 Continued Fractions

§10.32 Integral Representations

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Referenced by:: §10.54
Permalink:: http://dlmf.nist.gov/10.32
See also:: Annotations for Ch.10

§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

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Keywords:: along real line, integral representations, modified Bessel functions
Notes:: See Watson (1944, pp. 79, 80, 172, and 181–183). For (10.32.7) see also Erdélyi et al. (1953b, p. 82).
Permalink:: http://dlmf.nist.gov/10.32.i
See also:: Annotations for §10.32 and Ch.10

10.32.1 $I_{0}\left(z\right)=\frac{1}{\pi}\int_{0}^{\pi}e^{\pm z\cos\theta}\,\mathrm{d}% \theta=\frac{1}{\pi}\int_{0}^{\pi}\cosh\left(z\cos\theta\right)\,\mathrm{d}\theta.$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\cosh\NVar{z}$ : hyperbolic cosine function, $\int$ : integral, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind and : complex variable
A&S Ref:: 9.6.16
Permalink:: http://dlmf.nist.gov/10.32.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §10.32(i), §10.32 and Ch.10

10.32.2 $I_{\nu}\left(z\right)=\frac{(\frac{1}{2}z)^{\nu}}{\pi^{\frac{1}{2}}\Gamma\left% (\nu+\frac{1}{2}\right)}\int_{0}^{\pi}e^{\pm z\cos\theta}(\sin\theta)^{2\nu}\,% \mathrm{d}\theta=\frac{(\frac{1}{2}z)^{\nu}}{\pi^{\frac{1}{2}}\Gamma\left(\nu+% \frac{1}{2}\right)}\int_{-1}^{1}(1-t^{2})^{\nu-\frac{1}{2}}e^{\pm zt}\,\mathrm% {d}t,$ $\Re\nu>-\tfrac{1}{2}$ .

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $\Re$ : real part, $\sin\NVar{z}$ : sine function, : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.6.18
Permalink:: http://dlmf.nist.gov/10.32.E2
Encodings:: TeX, pMML, png
See also:: Annotations for §10.32(i), §10.32 and Ch.10

10.32.3 $I_{n}\left(z\right)=\frac{1}{\pi}\int_{0}^{\pi}e^{z\cos\theta}\cos\left(n% \theta\right)\,\mathrm{d}\theta.$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, : integer and : complex variable
A&S Ref:: 9.6.19
Permalink:: http://dlmf.nist.gov/10.32.E3
Encodings:: TeX, pMML, png
See also:: Annotations for §10.32(i), §10.32 and Ch.10

10.32.4 $I_{\nu}\left(z\right)=\frac{1}{\pi}\int_{0}^{\pi}e^{z\cos\theta}\cos\left(\nu% \theta\right)\,\mathrm{d}\theta-\frac{\sin\left(\nu\pi\right)}{\pi}\int_{0}^{% \infty}e^{-z\cosh t-\nu t}\,\mathrm{d}t,$ $|\operatorname{ph}z|<\tfrac{1}{2}\pi$ .

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\cosh\NVar{z}$ : hyperbolic cosine function, $\int$ : integral, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $\operatorname{ph}$ : phase, $\sin\NVar{z}$ : sine function, : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.6.20
Permalink:: http://dlmf.nist.gov/10.32.E4
Encodings:: TeX, pMML, png
See also:: Annotations for §10.32(i), §10.32 and Ch.10

10.32.5 $K_{0}\left(z\right)=-\frac{1}{\pi}\int_{0}^{\pi}e^{\pm z\cos\theta}\left(% \gamma+\ln\left(2z(\sin\theta)^{2}\right)\right)\,\mathrm{d}\theta.$

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Symbols:: $\gamma$ : Euler’s constant, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\ln\NVar{z}$ : principal branch of logarithm function, $\sin\NVar{z}$ : sine function and : complex variable
A&S Ref:: 9.6.17
Permalink:: http://dlmf.nist.gov/10.32.E5
Encodings:: TeX, pMML, png
See also:: Annotations for §10.32(i), §10.32 and Ch.10

10.32.6 $K_{0}\left(x\right)=\int_{0}^{\infty}\cos\left(x\sinh t\right)\,\mathrm{d}t=% \int_{0}^{\infty}\frac{\cos\left(xt\right)}{\sqrt{t^{2}+1}}\,\mathrm{d}t,$

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Symbols:: $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of , $\sinh\NVar{z}$ : hyperbolic sine function, $\int$ : integral, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind and : real variable
A&S Ref:: 9.6.21
Permalink:: http://dlmf.nist.gov/10.32.E6
Encodings:: TeX, pMML, png
See also:: Annotations for §10.32(i), §10.32 and Ch.10

10.32.7 $K_{\nu}\left(x\right)=\sec\left(\tfrac{1}{2}\nu\pi\right)\int_{0}^{\infty}\cos% \left(x\sinh t\right)\cosh\left(\nu t\right)\,\mathrm{d}t=\csc\left(\tfrac{1}{% 2}\nu\pi\right)\int_{0}^{\infty}\sin\left(x\sinh t\right)\sinh\left(\nu t% \right)\,\mathrm{d}t,$ $|\Re\nu|<1$ ,

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\csc\NVar{z}$ : cosecant function, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of , $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine 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, $\sin\NVar{z}$ : sine function, : 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
Encodings:: TeX, pMML, png
See also:: Annotations for §10.32(i), §10.32 and Ch.10

10.32.8 $K_{\nu}\left(z\right)=\frac{\pi^{\frac{1}{2}}(\frac{1}{2}z)^{\nu}}{\Gamma\left% (\nu+\frac{1}{2}\right)}\int_{0}^{\infty}e^{-z\cosh t}(\sinh t)^{2\nu}\,% \mathrm{d}t=\frac{\pi^{\frac{1}{2}}(\frac{1}{2}z)^{\nu}}{\Gamma\left(\nu+\frac% {1}{2}\right)}\int_{1}^{\infty}e^{-zt}(t^{2}-1)^{\nu-\frac{1}{2}}\,\mathrm{d}t,$ $\Re\nu>-\tfrac{1}{2}$ , $|\operatorname{ph}z|<\tfrac{1}{2}\pi$ .

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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 , $\mathrm{e}$ : base of natural logarithm, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\int$ : integral, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\operatorname{ph}$ : phase, $\Re$ : real part, : 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
Encodings:: TeX, pMML, png
See also:: Annotations for §10.32(i), §10.32 and Ch.10

10.32.9 $K_{\nu}\left(z\right)=\int_{0}^{\infty}e^{-z\cosh t}\cosh\left(\nu t\right)\,% \mathrm{d}t,$ $|\operatorname{ph}z|<\tfrac{1}{2}\pi$ .

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\cosh\NVar{z}$ : hyperbolic cosine function, $\int$ : integral, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\operatorname{ph}$ : phase, : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.6.24
Referenced by:: §10.43(iii)
Permalink:: http://dlmf.nist.gov/10.32.E9
Encodings:: TeX, pMML, png
See also:: Annotations for §10.32(i), §10.32 and Ch.10

10.32.10 $K_{\nu}\left(z\right)=\tfrac{1}{2}(\tfrac{1}{2}z)^{\nu}\int_{0}^{\infty}\exp% \left(-t-\frac{z^{2}}{4t}\right)\frac{\,\mathrm{d}t}{t^{\nu+1}},$ $|\operatorname{ph}z|<\tfrac{1}{4}\pi$ .

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\exp\NVar{z}$ : exponential function, $\int$ : integral, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\operatorname{ph}$ : phase, : complex variable and $\nu$ : complex parameter
Referenced by:: §6.7(iii), §7.14(i)
Permalink:: http://dlmf.nist.gov/10.32.E10
Encodings:: TeX, pMML, png
See also:: Annotations for §10.32(i), §10.32 and Ch.10

Basset’s Integral

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Keywords:: Basset’s integral, along real line, integral representations, modified Bessel functions
See also:: Annotations for §10.32(i), §10.32 and Ch.10

10.32.11 $K_{\nu}\left(xz\right)=\frac{\Gamma\left(\nu+\frac{1}{2}\right)(2z)^{\nu}}{\pi% ^{\frac{1}{2}}x^{\nu}}\int_{0}^{\infty}\frac{\cos\left(xt\right)\,\mathrm{d}t}% {(t^{2}+z^{2})^{\nu+\frac{1}{2}}},$ $\Re\nu>-\tfrac{1}{2}$ ,

, $|\operatorname{ph}z|<\tfrac{1}{2}\pi$ .

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\operatorname{ph}$ : phase, $\Re$ : real part, : real variable, : 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
See also:: Annotations for §10.32(i), §10.32(i), §10.32 and Ch.10

§10.32(ii) Contour Integrals

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Keywords:: contour integrals, integral representations, modified Bessel functions
Notes:: See Watson (1944, pp. 181, 191, and 193) and apply (10.27.8). See also Paris and Kaminski (2001, p. 114).
Permalink:: http://dlmf.nist.gov/10.32.ii
See also:: Annotations for §10.32 and Ch.10

10.32.12 $I_{\nu}\left(z\right)=\frac{1}{2\pi i}\int_{\infty-i\pi}^{\infty+i\pi}e^{z% \cosh t-\nu t}\,\mathrm{d}t,$ $|\operatorname{ph}z|<\tfrac{1}{2}\pi$ .

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\cosh\NVar{z}$ : hyperbolic cosine function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $\operatorname{ph}$ : phase, : complex variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.32.E12
Encodings:: TeX, pMML, png
See also:: Annotations for §10.32(ii), §10.32 and Ch.10

Mellin–Barnes Type

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Keywords:: Mellin–Barnes type, integral representations, modified Bessel functions
See also:: Annotations for §10.32(ii), §10.32 and Ch.10

10.32.13 $K_{\nu}\left(z\right)=\frac{(\frac{1}{2}z)^{\nu}}{4\pi i}\int_{c-i\infty}^{c+i% \infty}\Gamma\left(t\right)\Gamma\left(t-\nu\right)(\tfrac{1}{2}z)^{-2t}\,% \mathrm{d}t,$ $c>\max(\Re\nu,0),|\operatorname{ph}z|<\frac{1}{2}\pi$ .

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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 , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\operatorname{ph}$ : phase, $\Re$ : real part, : complex variable and $\nu$ : complex parameter
Keywords:: Mellin transform
Referenced by:: Erratum (V1.0.11) for Equation (10.32.13)
Permalink:: http://dlmf.nist.gov/10.32.E13
Encodings:: TeX, pMML, png
Errata (effective with 1.0.11):: Originally the constraint $|\operatorname{ph}z|<\frac{1}{2}\pi$ was written incorrectly as $|\operatorname{ph}z|<\pi$ .
Reported 2015-05-20 by Richard Paris
See also:: Annotations for §10.32(ii), §10.32(ii), §10.32 and Ch.10

10.32.14 $K_{\nu}\left(z\right)=\frac{1}{2\pi^{2}i}\left(\frac{\pi}{2z}\right)^{\frac{1}% {2}}e^{-z}\cos\left(\nu\pi\right)\*\int_{-i\infty}^{i\infty}\Gamma\left(t% \right)\Gamma\left(\tfrac{1}{2}-t-\nu\right)\Gamma\left(\tfrac{1}{2}-t+\nu% \right)(2z)^{t}\,\mathrm{d}t,$ $\nu-\tfrac{1}{2}\notin\mathbb{Z},|\operatorname{ph}z|<\tfrac{3}{2}\pi$ .

In (10.32.14) the integration contour separates the poles of $\Gamma\left(t\right)$ from the poles of $\Gamma\left(\frac{1}{2}-t-\nu\right)\Gamma\left(\frac{1}{2}-t+\nu\right)$ .

§10.32(iii) Products

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Keywords:: integral representations, modified Bessel functions, 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)).
Permalink:: http://dlmf.nist.gov/10.32.iii
See also:: Annotations for §10.32 and Ch.10

10.32.15 $I_{\mu}\left(z\right)I_{\nu}\left(z\right)=\frac{2}{\pi}\int_{0}^{\frac{1}{2}% \pi}I_{\mu+\nu}\left(2z\cos\theta\right)\cos\left((\mu-\nu)\theta\right)\,% \mathrm{d}\theta,$ $\Re\left(\mu+\nu\right)>-1$ .

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $\Re$ : real part, : complex variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.32.E15
Encodings:: TeX, pMML, png
See also:: Annotations for §10.32(iii), §10.32 and Ch.10

10.32.16 $I_{\mu}\left(x\right)K_{\nu}\left(x\right)=\int_{0}^{\infty}J_{\mu\pm\nu}\left% (2x\sinh t\right)e^{(-\mu\pm\nu)t}\,\mathrm{d}t,$ $\Re\left(\mu\mp\nu\right)>-\tfrac{1}{2}$ , $\Re\left(\mu\pm\nu\right)>-1$ ,

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Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\sinh\NVar{z}$ : hyperbolic sine 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, $\Re$ : real part, : real variable and $\nu$ : complex parameter
Keywords:: Laplace transform
Referenced by:: §10.32(iii)
Permalink:: http://dlmf.nist.gov/10.32.E16
Encodings:: TeX, pMML, png
See also:: Annotations for §10.32(iii), §10.32 and Ch.10

10.32.17 $K_{\mu}\left(z\right)K_{\nu}\left(z\right)=2\int_{0}^{\infty}K_{\mu\pm\nu}% \left(2z\cosh t\right)\cosh\left((\mu\mp\nu)t\right)\,\mathrm{d}t,$ $|\operatorname{ph}z|<\tfrac{1}{2}\pi$ .

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\cosh\NVar{z}$ : hyperbolic cosine function, $\int$ : integral, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\operatorname{ph}$ : phase, : complex variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.32.E17
Encodings:: TeX, pMML, png
See also:: Annotations for §10.32(iii), §10.32 and Ch.10

10.32.18 $K_{\nu}\left(z\right)K_{\nu}\left(\zeta\right)=\frac{1}{2}\int_{0}^{\infty}% \exp\left(-\frac{t}{2}-\frac{z^{2}+\zeta^{2}}{2t}\right)K_{\nu}\left(\frac{z% \zeta}{t}\right)\frac{\,\mathrm{d}t}{t},$ $|\operatorname{ph}z|<\pi$ , $|\operatorname{ph}\zeta|<\pi$ , $|\operatorname{ph}\left(z+\zeta\right)|<\tfrac{1}{4}\pi$ .

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\exp\NVar{z}$ : exponential function, $\int$ : integral, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\operatorname{ph}$ : phase, : complex variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.32.E18
Encodings:: TeX, pMML, png
See also:: Annotations for §10.32(iii), §10.32 and Ch.10

Mellin–Barnes Type

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See also:: Annotations for §10.32(iii), §10.32 and Ch.10

10.32.19 $K_{\mu}\left(z\right)K_{\nu}\left(z\right)=\frac{1}{8\pi i}\int_{c-i\infty}^{c% +i\infty}\frac{\Gamma\left(t+\frac{1}{2}\mu+\frac{1}{2}\nu\right)\Gamma\left(t% +\frac{1}{2}\mu-\frac{1}{2}\nu\right)\Gamma\left(t-\frac{1}{2}\mu+\frac{1}{2}% \nu\right)\Gamma\left(t-\frac{1}{2}\mu-\frac{1}{2}\nu\right)}{\Gamma\left(2t% \right)}(\tfrac{1}{2}z)^{-2t}\,\mathrm{d}t,$ $c>\tfrac{1}{2}(|\Re\mu|+|\Re\nu|),|\operatorname{ph}z|<\tfrac{1}{2}\pi$ .

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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 , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\operatorname{ph}$ : phase, $\Re$ : real part, : complex variable and $\nu$ : complex parameter
Keywords:: Mellin transform
Referenced by:: §10.32(iii)
Permalink:: http://dlmf.nist.gov/10.32.E19
Encodings:: TeX, pMML, png
See also:: Annotations for §10.32(iii), §10.32(iii), §10.32 and Ch.10

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

§10.32(iv) Compendia

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Keywords:: compendia, integral representations, modified Bessel functions
Permalink:: http://dlmf.nist.gov/10.32.iv
See also:: Annotations for §10.32 and Ch.10

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).

10.31 Power Series 10.33 Continued Fractions

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§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