§10.32 Integral Representations

10.32.1		$$I_0\left(z\right)=\frac{1}{\pi }{\int }_0^{\pi }{\mathrm{e}}^{\pm z\cos \theta }d\theta =\frac{1}{\pi }{\int }_0^{\pi }\cosh \left(z\cos \theta \right)d\theta .$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\int$: integral, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind and $z$: 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		$$\begin{array}{ll}{I}_{\nu }\left(z\right)& =\frac{{(\frac{1}{2}z)}^{\nu }}{{\pi }^{\frac{1}{2}}\mathrm{\Gamma }\left(\nu +\frac{1}{2}\right)}{\int }_0^{\pi }{\mathrm{e}}^{\pm z\cos \theta }{(\sin \theta )}^{2\nu }d\theta \\ & =\frac{{(\frac{1}{2}z)}^{\nu }}{{\pi }^{\frac{1}{2}}\mathrm{\Gamma }\left(\nu +\frac{1}{2}\right)}{\int }_{-1}^1{(1-t^2)}^{\nu -\frac{1}{2}}{\mathrm{e}}^{\pm zt}dt,\end{array}$$
$\mathrm{\Re }\nu >-\frac{1}{2}$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{\Re }$: real part, $\sin z$: sine function, $z$: 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		$$I_n\left(z\right)=\frac{1}{\pi }{\int }_0^{\pi }{\mathrm{e}}^{z\cos \theta }\cos \left(n\theta \right)d\theta .$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $n$: integer and $z$: 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		$$I_{\nu }\left(z\right)=\frac{1}{\pi }{\int }_0^{\pi }{\mathrm{e}}^{z\cos \theta }\cos \left(\nu \theta \right)d\theta -\frac{\sin \left(\nu \pi \right)}{\pi }{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-z\cosh t-\nu t}dt,$$
.
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\int$: integral, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{ph}$: phase, $\sin z$: 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 See also: Annotations for §10.32(i), §10.32 and Ch.10

10.32.5		$$K_0\left(z\right)=-\frac{1}{\pi }{\int }_0^{\pi }{\mathrm{e}}^{\pm z\cos \theta }\left(\gamma +\ln \left(2z{(\sin \theta )}^2\right)\right)d\theta .$$
ⓘ Symbols: $\gamma$: Euler’s constant, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\ln z$: principal branch of logarithm function, $\sin z$: sine function and $z$: complex variable A&S Ref: 9.6.17 Permalink: http://dlmf.nist.gov/10.32.E5 Encodings: TeX, pMML, png See also: Annotations for §10.32(i), §10.32 and Ch.10

10.32.6		$$K_0\left(x\right)={\int }_0^{\mathrm{\infty }}\cos \left(x\sinh t\right)dt={\int }_0^{\mathrm{\infty }}\frac{\cos \left(xt\right)}{\sqrt{t^2+1}}dt,$$
$x>0$.
ⓘ Symbols: $\cos z$: cosine function, $dx$: differential of $x$, $\sinh z$: hyperbolic sine function, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind and $x$: 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		$$\begin{array}{ll}{K}_{\nu }\left(x\right)& =\sec \left(\frac{1}{2}\nu \pi \right){\int }_0^{\mathrm{\infty }}\cos \left(x\sinh t\right)\cosh \left(\nu t\right)dt\\ & =\csc \left(\frac{1}{2}\nu \pi \right){\int }_0^{\mathrm{\infty }}\sin \left(x\sinh t\right)\sinh \left(\nu t\right)dt,\end{array}$$
, $x>0$.
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\cos z$: cosine function, $dx$: differential of $x$, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{\Re }$: real part, $\sec z$: secant function, $\sin z$: 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 Encodings: TeX, pMML, png See also: Annotations for §10.32(i), §10.32 and Ch.10

10.32.8		$$\begin{array}{ll}{K}_{\nu }\left(z\right)& =\frac{{\pi }^{\frac{1}{2}}{(\frac{1}{2}z)}^{\nu }}{\mathrm{\Gamma }\left(\nu +\frac{1}{2}\right)}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-z\cosh t}{(\sinh t)}^{2\nu }dt\\ & =\frac{{\pi }^{\frac{1}{2}}{(\frac{1}{2}z)}^{\nu }}{\mathrm{\Gamma }\left(\nu +\frac{1}{2}\right)}{\int }_1^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{(t^2-1)}^{\nu -\frac{1}{2}}dt,\end{array}$$
$\mathrm{\Re }\nu >-\frac{1}{2}$, .
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: phase, $\mathrm{\Re }$: 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 Encodings: TeX, pMML, png See also: Annotations for §10.32(i), §10.32 and Ch.10

10.32.9		$$K_{\nu }\left(z\right)={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-z\cosh t}\cosh \left(\nu t\right)dt,$$
.
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: phase, $z$: 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		$$K_{\nu }\left(z\right)=\frac{1}{2}{(\frac{1}{2}z)}^{\nu }{\int }_0^{\mathrm{\infty }}\exp \left(-t-\frac{z^2}{4t}\right)\frac{dt}{t^{\nu +1}},$$
.
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\exp z$: exponential function, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: phase, $z$: 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

10.32.12		$$I_{\nu }\left(z\right)=\frac{1}{2\pi \mathrm{i}}{\int }_{\mathrm{\infty }-\mathrm{i}\pi }^{\mathrm{\infty }+\mathrm{i}\pi }{\mathrm{e}}^{z\cosh t-\nu t}dt,$$
.
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{ph}$: phase, $z$: 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

10.32.15		$$I_{\mu }\left(z\right)I_{\nu }\left(z\right)=\frac{2}{\pi }{\int }_0^{\frac{1}{2}\pi }{I}_{\mu +\nu }\left(2z\cos \theta \right)\cos \left((\mu -\nu )\theta \right)d\theta ,$$
$\mathrm{\Re }\left(\mu +\nu \right)>-1$.
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\int$: integral, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{\Re }$: real part, $z$: 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		$$I_{\mu }\left(x\right)K_{\nu }\left(x\right)={\int }_0^{\mathrm{\infty }}{J}_{\mu \pm \nu }\left(2x\sinh t\right){\mathrm{e}}^{(-\mu \pm \nu )t}dt,$$
$\mathrm{\Re }\left(\mu \mp \nu \right)>-\frac{1}{2}$, $\mathrm{\Re }\left(\mu \pm \nu \right)>-1$, $x>0$.
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\sinh z$: hyperbolic sine function, $\int$: integral, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{\Re }$: real part, $x$: 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		$$K_{\mu }\left(z\right)K_{\nu }\left(z\right)=2{\int }_0^{\mathrm{\infty }}{K}_{\mu \pm \nu }\left(2z\cosh t\right)\cosh \left((\mu \mp \nu )t\right)dt,$$
.
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\cosh z$: hyperbolic cosine function, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: phase, $z$: 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		$$K_{\nu }\left(z\right)K_{\nu }\left(\zeta \right)=\frac{1}{2}{\int }_0^{\mathrm{\infty }}\exp \left(-\frac{t}{2}-\frac{z^2+{\zeta }^2}{2t}\right)K_{\nu }\left(\frac{z\zeta }{t}\right)\frac{dt}{t},$$
, , .
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\exp z$: exponential function, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: phase, $z$: 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

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).