§10.9 Integral Representations

10.9.1		$$J_0\left(z\right)=\frac{1}{\pi }{\int }_0^{\pi }\cos \left(z\sin \theta \right)d\theta =\frac{1}{\pi }{\int }_0^{\pi }\cos \left(z\cos \theta \right)d\theta ,$$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\int$: integral, $\sin z$: sine function and $z$: complex variable A&S Ref: 9.1.18 Permalink: http://dlmf.nist.gov/10.9.E1 Encodings: TeX, pMML, png See also: Annotations for §10.9(i), §10.9(i), §10.9 and Ch.10

10.9.2		$$J_n\left(z\right)=\frac{1}{\pi }{\int }_0^{\pi }\cos \left(z\sin \theta -n\theta \right)d\theta =\frac{{\mathrm{i}}^{-n}}{\pi }{\int }_0^{\pi }{\mathrm{e}}^{\mathrm{i}z\cos \theta }\cos \left(n\theta \right)d\theta ,$$
$n\in \mathbb{Z}$.
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathbb{Z}$: set of all integers, $\int$: integral, $\sin z$: sine function, $n$: integer and $z$: complex variable A&S Ref: 9.1.21 Referenced by: §11.10(vii) Permalink: http://dlmf.nist.gov/10.9.E2 Encodings: TeX, pMML, png See also: Annotations for §10.9(i), §10.9(i), §10.9 and Ch.10

10.9.3		$$Y_0\left(z\right)=\frac{4}{{\pi }^2}{\int }_0^{\frac{1}{2}\pi }\cos \left(z\cos \theta \right)\left(\gamma +\ln \left(2z{\sin }^2\theta \right)\right)d\theta ,$$
ⓘ Symbols: $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\gamma$: Euler’s constant, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\int$: integral, $\ln z$: principal branch of logarithm function, $\sin z$: sine function and $z$: complex variable A&S Ref: 9.19 Referenced by: §10.9(i) Permalink: http://dlmf.nist.gov/10.9.E3 Encodings: TeX, pMML, png See also: Annotations for §10.9(i), §10.9(i), §10.9 and Ch.10

where $\gamma$ is Euler’s constant (§5.2(ii)).

10.9.4		$$\begin{array}{ll}{J}_{\nu }\left(z\right)& =\frac{{(\frac{1}{2}z)}^{\nu }}{{\pi }^{\frac{1}{2}}\mathrm{\Gamma }\left(\nu +\frac{1}{2}\right)}{\int }_0^{\pi }\cos \left(z\cos \theta \right){(\sin \theta )}^{2\nu }d\theta \\ & =\frac{2{(\frac{1}{2}z)}^{\nu }}{{\pi }^{\frac{1}{2}}\mathrm{\Gamma }\left(\nu +\frac{1}{2}\right)}{\int }_0^1{(1-t^2)}^{\nu -\frac{1}{2}}\cos \left(zt\right)dt,\end{array}$$
$\mathrm{\Re }\nu >-\frac{1}{2}$.
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\int$: integral, $\mathrm{\Re }$: real part, $\sin z$: sine function, $z$: complex variable and $\nu$: complex parameter A&S Ref: 9.1.20 Referenced by: §10.54 Permalink: http://dlmf.nist.gov/10.9.E4 Encodings: TeX, pMML, png See also: Annotations for §10.9(i), §10.9(i), §10.9 and Ch.10

10.9.5		$$Y_{\nu }\left(z\right)=\frac{2{(\frac{1}{2}z)}^{\nu }}{{\pi }^{\frac{1}{2}}\mathrm{\Gamma }\left(\nu +\frac{1}{2}\right)}\left({\int }_0^1{(1-t^2)}^{\nu -\frac{1}{2}}\sin \left(zt\right)dt-{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{(1+t^2)}^{\nu -\frac{1}{2}}dt\right),$$
.
ⓘ Symbols: $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $\sin z$: sine function, $z$: complex variable and $\nu$: complex parameter Referenced by: §10.9(i) Permalink: http://dlmf.nist.gov/10.9.E5 Encodings: TeX, pMML, png See also: Annotations for §10.9(i), §10.9(i), §10.9 and Ch.10

10.9.6		$$J_{\nu }\left(z\right)=\frac{1}{\pi }{\int }_0^{\pi }\cos \left(z\sin \theta -\nu \theta \right)d\theta -\frac{\sin \left(\nu \pi \right)}{\pi }{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-z\sinh t-\nu t}dt,$$
,
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: 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, $\sinh z$: hyperbolic sine function, $\int$: integral, $\mathrm{ph}$: phase, $\sin z$: sine function, $z$: complex variable and $\nu$: complex parameter A&S Ref: 9.1.22 Permalink: http://dlmf.nist.gov/10.9.E6 Encodings: TeX, pMML, png See also: Annotations for §10.9(i), §10.9(i), §10.9 and Ch.10

10.9.7		$$Y_{\nu }\left(z\right)=\frac{1}{\pi }{\int }_0^{\pi }\sin \left(z\sin \theta -\nu \theta \right)d\theta -\frac{1}{\pi }{\int }_0^{\mathrm{\infty }}\left({\mathrm{e}}^{\nu t}+{\mathrm{e}}^{-\nu t}\cos \left(\nu \pi \right)\right){\mathrm{e}}^{-z\sinh t}dt,$$
.
ⓘ Symbols: $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: 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, $\sinh z$: hyperbolic sine function, $\int$: integral, $\mathrm{ph}$: phase, $\sin z$: sine function, $z$: complex variable and $\nu$: complex parameter A&S Ref: 9.1.22 Permalink: http://dlmf.nist.gov/10.9.E7 Encodings: TeX, pMML, png See also: Annotations for §10.9(i), §10.9(i), §10.9 and Ch.10

10.9.8		$J_{\nu }\left(x\right)$	$={\displaystyle \frac{2}{\pi }}{\displaystyle {\int }_0^{\mathrm{\infty }}}\sin \left(x\cosh t-\frac{1}{2}\nu \pi \right)\cosh \left(\nu t\right)dt,$
		$Y_{\nu }\left(x\right)$	$=-{\displaystyle \frac{2}{\pi }}{\displaystyle {\int }_0^{\mathrm{\infty }}}\cos \left(x\cosh t-\frac{1}{2}\nu \pi \right)\cosh \left(\nu t\right)dt,$
	.
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\cosh z$: hyperbolic cosine function, $\int$: integral, $\mathrm{\Re }$: real part, $\sin z$: sine function, $x$: real variable and $\nu$: complex parameter Permalink: http://dlmf.nist.gov/10.9.E8 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.9(i), §10.9(i), §10.9 and Ch.10

In particular,

10.9.9		$J_0\left(x\right)$	$={\displaystyle \frac{2}{\pi }}{\displaystyle {\int }_0^{\mathrm{\infty }}}\sin \left(x\cosh t\right)dt,$
	$x>0$,
		$Y_0\left(x\right)$	$=-{\displaystyle \frac{2}{\pi }}{\displaystyle {\int }_0^{\mathrm{\infty }}}\cos \left(x\cosh t\right)dt,$
	$x>0$.
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\cosh z$: hyperbolic cosine function, $\int$: integral, $\sin z$: sine function and $x$: real variable A&S Ref: 9.1.23 Permalink: http://dlmf.nist.gov/10.9.E9 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.9(i), §10.9(i), §10.9 and Ch.10

10.9.10		$$H_{\nu }^{(1)}\left(z\right)=\frac{{\mathrm{e}}^{-\frac{1}{2}\nu \pi \mathrm{i}}}{\pi \mathrm{i}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{\mathrm{i}z\cosh t-\nu t}dt,$$
,
ⓘ Symbols: $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel 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, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{ph}$: phase, $z$: complex variable and $\nu$: complex parameter Referenced by: §10.9(i) Permalink: http://dlmf.nist.gov/10.9.E10 Encodings: TeX, pMML, png See also: Annotations for §10.9(i), §10.9(i), §10.9 and Ch.10

10.9.11		$$H_{\nu }^{(2)}\left(z\right)=-\frac{{\mathrm{e}}^{\frac{1}{2}\nu \pi \mathrm{i}}}{\pi \mathrm{i}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-\mathrm{i}z\cosh t-\nu t}dt,$$
.
ⓘ Symbols: $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel 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, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{ph}$: phase, $z$: complex variable and $\nu$: complex parameter Referenced by: §10.9(i) Permalink: http://dlmf.nist.gov/10.9.E11 Encodings: TeX, pMML, png See also: Annotations for §10.9(i), §10.9(i), §10.9 and Ch.10

10.9.12		$J_{\nu }\left(x\right)$	$={\displaystyle \frac{2{(\frac{1}{2}x)}^{-\nu }}{{\pi }^{\frac{1}{2}}\mathrm{\Gamma }\left(\frac{1}{2}-\nu \right)}}{\displaystyle {\int }_1^{\mathrm{\infty }}}{\displaystyle \frac{\sin \left(xt\right)dt}{{(t^2-1)}^{\nu +\frac{1}{2}}}},$
		$Y_{\nu }\left(x\right)$	$=-{\displaystyle \frac{2{(\frac{1}{2}x)}^{-\nu }}{{\pi }^{\frac{1}{2}}\mathrm{\Gamma }\left(\frac{1}{2}-\nu \right)}}{\displaystyle {\int }_1^{\mathrm{\infty }}}{\displaystyle \frac{\cos \left(xt\right)dt}{{(t^2-1)}^{\nu +\frac{1}{2}}}},$
	, $x>0$.
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\int$: integral, $\mathrm{\Re }$: real part, $\sin z$: sine function, $x$: real variable and $\nu$: complex parameter A&S Ref: 9.1.24 Permalink: http://dlmf.nist.gov/10.9.E12 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.9(i), §10.9(i), §10.9 and Ch.10

10.9.13		$${\left(\frac{z+\zeta }{z-\zeta }\right)}^{\frac{1}{2}\nu }{J}_{\nu }\left({(z^2-{\zeta }^2)}^{\frac{1}{2}}\right)=\begin{array}{l}\frac{1}{\pi }{\int }_0^{\pi }{\mathrm{e}}^{\zeta \cos \theta }\cos \left(z\sin \theta -\nu \theta \right)d\theta \\ -\frac{\sin \left(\nu \pi \right)}{\pi }{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-\zeta \cosh t-z\sinh t-\nu t}dt,\end{array}$$
$\mathrm{\Re }\left(z+\zeta \right)>0$,
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: 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, $\sinh z$: hyperbolic sine function, $\int$: integral, $\mathrm{\Re }$: real part, $\sin z$: sine function, $z$: complex variable and $\nu$: complex parameter Referenced by: §10.9(i) Permalink: http://dlmf.nist.gov/10.9.E13 Encodings: TeX, pMML, png See also: Annotations for §10.9(i), §10.9(i), §10.9 and Ch.10

10.9.14		$$\begin{array}{l}{\left(\frac{z+\zeta }{z-\zeta }\right)}^{\frac{1}{2}\nu }{Y}_{\nu }\left({(z^2-{\zeta }^2)}^{\frac{1}{2}}\right)\\ =\begin{array}{l}\frac{1}{\pi }{\int }_0^{\pi }{\mathrm{e}}^{\zeta \cos \theta }\sin \left(z\sin \theta -\nu \theta \right)d\theta \\ -\frac{1}{\pi }{\int }_0^{\mathrm{\infty }}\left({\mathrm{e}}^{\nu t+\zeta \cosh t}+{\mathrm{e}}^{-\nu t-\zeta \cosh t}\cos \left(\nu \pi \right)\right){\mathrm{e}}^{-z\sinh t}dt,\end{array}\end{array}$$
$\mathrm{\Re }\left(z\pm \zeta \right)>0$.
ⓘ Symbols: $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: 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, $\sinh z$: hyperbolic sine function, $\int$: integral, $\mathrm{\Re }$: real part, $\sin z$: sine function, $z$: complex variable and $\nu$: complex parameter Referenced by: §10.9(i) Permalink: http://dlmf.nist.gov/10.9.E14 Encodings: TeX, pMML, png See also: Annotations for §10.9(i), §10.9(i), §10.9 and Ch.10

10.9.15		$${\left(\frac{z+\zeta }{z-\zeta }\right)}^{\frac{1}{2}\nu }{H}_{\nu }^{(1)}\left({(z^2-{\zeta }^2)}^{\frac{1}{2}}\right)=\frac{1}{\pi \mathrm{i}}{\mathrm{e}}^{-\frac{1}{2}\nu \pi \mathrm{i}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{\mathrm{i}z\cosh t+\mathrm{i}\zeta \sinh t-\nu t}dt,$$
$\mathrm{\Im }\left(z\pm \zeta \right)>0$,
ⓘ Symbols: $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel 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, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit, $\int$: integral, $z$: complex variable and $\nu$: complex parameter Referenced by: §10.9(i) Permalink: http://dlmf.nist.gov/10.9.E15 Encodings: TeX, pMML, png See also: Annotations for §10.9(i), §10.9(i), §10.9 and Ch.10

10.9.16		$${\left(\frac{z+\zeta }{z-\zeta }\right)}^{\frac{1}{2}\nu }{H}_{\nu }^{(2)}\left({(z^2-{\zeta }^2)}^{\frac{1}{2}}\right)=-\frac{1}{\pi \mathrm{i}}{\mathrm{e}}^{\frac{1}{2}\nu \pi \mathrm{i}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-\mathrm{i}z\cosh t-\mathrm{i}\zeta \sinh t-\nu t}dt,$$
.
ⓘ Symbols: $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel 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, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit, $\int$: integral, $z$: complex variable and $\nu$: complex parameter Referenced by: §10.9(i) Permalink: http://dlmf.nist.gov/10.9.E16 Encodings: TeX, pMML, png See also: Annotations for §10.9(i), §10.9(i), §10.9 and Ch.10

When ,

10.9.17		$$J_{\nu }\left(z\right)=\frac{1}{2\pi \mathrm{i}}{\int }_{\mathrm{\infty }-\pi \mathrm{i}}^{\mathrm{\infty }+\pi \mathrm{i}}{\mathrm{e}}^{z\sinh t-\nu t}dt,$$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $z$: complex variable and $\nu$: complex parameter Permalink: http://dlmf.nist.gov/10.9.E17 Encodings: TeX, pMML, png See also: Annotations for §10.9(ii), §10.9(ii), §10.9 and Ch.10

and

10.9.18		$H_{\nu }^{(1)}\left(z\right)$	$={\displaystyle \frac{1}{\pi \mathrm{i}}}{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }+\pi \mathrm{i}}}{\mathrm{e}}^{z\sinh t-\nu t}dt,$
		$H_{\nu }^{(2)}\left(z\right)$	$=-{\displaystyle \frac{1}{\pi \mathrm{i}}}{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }-\pi \mathrm{i}}}{\mathrm{e}}^{z\sinh t-\nu t}dt.$
ⓘ Symbols: $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $z$: complex variable and $\nu$: complex parameter A&S Ref: 9.1.25 Referenced by: §10.74(iii) Permalink: http://dlmf.nist.gov/10.9.E18 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.9(ii), §10.9(ii), §10.9 and Ch.10

10.9.19		$$J_{\nu }\left(z\right)=\frac{{(\frac{1}{2}z)}^{\nu }}{2\pi \mathrm{i}}{\int }_{-\mathrm{\infty }}^{(0+)}\exp \left(t-\frac{z^2}{4t}\right)\frac{dt}{t^{\nu +1}},$$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $\int$: integral, $z$: complex variable and $\nu$: complex parameter Permalink: http://dlmf.nist.gov/10.9.E19 Encodings: TeX, pMML, png See also: Annotations for §10.9(ii), §10.9(ii), §10.9 and Ch.10

where the integration path is a simple loop contour, and $t^{\nu +1}$ is continuous on the path and takes its principal value at the intersection with the positive real axis.

In (10.9.20) and (10.9.21) the integration paths are simple loop contours not enclosing $t=-1$. Also, ${(t^2-1)}^{\nu -\frac{1}{2}}$ is continuous on the path, and takes its principal value at the intersection with the interval $(1,\mathrm{\infty })$.

10.9.20		$$J_{\nu }\left(z\right)=\frac{\mathrm{\Gamma }\left(\frac{1}{2}-\nu \right){(\frac{1}{2}z)}^{\nu }}{{\pi }^{\frac{3}{2}}\mathrm{i}}{\int }_0^{(1+)}\cos \left(zt\right){(t^2-1)}^{\nu -\frac{1}{2}}dt,$$
$\nu \ne \frac{1}{2},\frac{3}{2},\mathrm{\dots }$.
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\mathrm{i}$: imaginary unit, $\int$: integral, $z$: complex variable and $\nu$: complex parameter Referenced by: §10.9(ii) Permalink: http://dlmf.nist.gov/10.9.E20 Encodings: TeX, pMML, png See also: Annotations for §10.9(ii), §10.9(ii), §10.9 and Ch.10

10.9.21		$H_{\nu }^{(1)}\left(z\right)$	$={\displaystyle \frac{\mathrm{\Gamma }\left(\frac{1}{2}-\nu \right){(\frac{1}{2}z)}^{\nu }}{{\pi }^{\frac{3}{2}}\mathrm{i}}}{\displaystyle {\int }_{1+\mathrm{i}\mathrm{\infty }}^{(1+)}}{\mathrm{e}}^{\mathrm{i}zt}{(t^2-1)}^{\nu -\frac{1}{2}}dt,$
		$H_{\nu }^{(2)}\left(z\right)$	$={\displaystyle \frac{\mathrm{\Gamma }\left(\frac{1}{2}-\nu \right){(\frac{1}{2}z)}^{\nu }}{{\pi }^{\frac{3}{2}}\mathrm{i}}}{\displaystyle {\int }_{1-\mathrm{i}\mathrm{\infty }}^{(1+)}}{\mathrm{e}}^{-\mathrm{i}zt}{(t^2-1)}^{\nu -\frac{1}{2}}dt,$
	.
	ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{ph}$: phase, $z$: complex variable and $\nu$: complex parameter Referenced by: §10.9(ii) Permalink: http://dlmf.nist.gov/10.9.E21 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.9(ii), §10.9(ii), §10.9 and Ch.10

10.9.22		$$J_{\nu }\left(x\right)=\frac{1}{2\pi \mathrm{i}}{\int }_{-\mathrm{i}\mathrm{\infty }}^{\mathrm{i}\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(-t\right){(\frac{1}{2}x)}^{\nu +2t}}{\mathrm{\Gamma }\left(\nu +t+1\right)}dt,$$
$\mathrm{\Re }\nu >0$, $x>0$,
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable and $\nu$: complex parameter Keywords: Mellin transform A&S Ref: 9.1.26 Referenced by: §10.9(ii) Permalink: http://dlmf.nist.gov/10.9.E22 Encodings: TeX, pMML, png See also: Annotations for §10.9(ii), §10.9(ii), §10.9 and Ch.10

where the integration path passes to the left of $t=0,1,2,\mathrm{\dots }$.

10.9.23		$$J_{\nu }\left(z\right)=\frac{1}{2\pi \mathrm{i}}{\int }_{-\mathrm{\infty }-\mathrm{i}c}^{-\mathrm{\infty }+\mathrm{i}c}\frac{\mathrm{\Gamma }\left(t\right)}{\mathrm{\Gamma }\left(\nu -t+1\right)}{(\frac{1}{2}z)}^{\nu -2t}dt,$$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{i}$: imaginary unit, $\int$: integral, $z$: complex variable, $\nu$: complex parameter and $c$: constant Keywords: Mellin transform Permalink: http://dlmf.nist.gov/10.9.E23 Encodings: TeX, pMML, png See also: Annotations for §10.9(ii), §10.9(ii), §10.9 and Ch.10

where $c$ is a positive constant and the integration path encloses the points $t=0,-1,-2,\mathrm{\dots }$.

In (10.9.24) and (10.9.25) $c$ is any constant exceeding $\max (\mathrm{\Re }\nu ,0)$.

10.9.24		$H_{\nu }^{(1)}\left(z\right)$	$=-{\displaystyle \frac{{\mathrm{e}}^{-\frac{1}{2}\nu \pi \mathrm{i}}}{2{\pi }^2}}{\displaystyle {\int }_{c-\mathrm{i}\mathrm{\infty }}^{c+\mathrm{i}\mathrm{\infty }}}\mathrm{\Gamma }\left(t\right)\mathrm{\Gamma }\left(t-\nu \right){(-\frac{1}{2}\mathrm{i}z)}^{\nu -2t}dt,$
,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{ph}$: phase, $z$: complex variable, $\nu$: complex parameter and $c$: constant Keywords: Mellin transform Referenced by: §10.9(ii) Permalink: http://dlmf.nist.gov/10.9.E24 Encodings: TeX, pMML, png See also: Annotations for §10.9(ii), §10.9(ii), §10.9 and Ch.10
10.9.25		$H_{\nu }^{(2)}\left(z\right)$	$={\displaystyle \frac{{\mathrm{e}}^{\frac{1}{2}\nu \pi \mathrm{i}}}{2{\pi }^2}}{\displaystyle {\int }_{c-\mathrm{i}\mathrm{\infty }}^{c+\mathrm{i}\mathrm{\infty }}}\mathrm{\Gamma }\left(t\right)\mathrm{\Gamma }\left(t-\nu \right){(\frac{1}{2}\mathrm{i}z)}^{\nu -2t}dt,$
.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{ph}$: phase, $z$: complex variable, $\nu$: complex parameter and $c$: constant Keywords: Mellin transform Referenced by: §10.9(ii), §10.9(ii) Permalink: http://dlmf.nist.gov/10.9.E25 Encodings: TeX, pMML, png See also: Annotations for §10.9(ii), §10.9(ii), §10.9 and Ch.10

For (10.9.22)–(10.9.25) and further integrals of this type see Paris and Kaminski (2001, pp. 114–116).

10.9.29		$$J_{\mu }\left(x\right)J_{\nu }\left(x\right)=\frac{1}{2\pi \mathrm{i}}{\int }_{-\mathrm{i}\mathrm{\infty }}^{\mathrm{i}\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(-t\right)\mathrm{\Gamma }\left(2t+\mu +\nu +1\right){(\frac{1}{2}x)}^{\mu +\nu +2t}}{\mathrm{\Gamma }\left(t+\mu +1\right)\mathrm{\Gamma }\left(t+\nu +1\right)\mathrm{\Gamma }\left(t+\mu +\nu +1\right)}dt,$$
$x>0$,
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{i}$: imaginary unit, $\int$: integral, $x$: real variable and $\nu$: complex parameter Keywords: Mellin transform Permalink: http://dlmf.nist.gov/10.9.E29 Encodings: TeX, pMML, png See also: Annotations for §10.9(iii), §10.9(iii), §10.9 and Ch.10

where the path of integration separates the poles of $\mathrm{\Gamma }\left(-t\right)$ from those of $\mathrm{\Gamma }\left(2t+\mu +\nu +1\right)$. See Paris and Kaminski (2001, p. 116) for related results.

10.9.30		$$J_{\nu }^2\left(z\right)+Y_{\nu }^2\left(z\right)=\frac{8}{{\pi }^2}{\int }_0^{\mathrm{\infty }}\cosh \left(2\nu t\right)K_0\left(2z\sinh t\right)dt,$$
.
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: 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, $\mathrm{ph}$: phase, $z$: complex variable and $\nu$: complex parameter Permalink: http://dlmf.nist.gov/10.9.E30 Encodings: TeX, pMML, png See also: Annotations for §10.9(iii), §10.9(iii), §10.9 and Ch.10

For the function $K_0$ see §10.25(ii).