Bessel transform

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11: 10.22 Integrals

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10.22.57 $${\int }_0^{\mathrm{\infty }}\frac{J_{\mu }\left(at\right)J_{\nu }\left(at\right)}{t^{\lambda }}dt=\frac{{(\frac{1}{2}a)}^{\lambda -1}\mathrm{\Gamma }\left(\frac{1}{2}\mu +\frac{1}{2}\nu -\frac{1}{2}\lambda +\frac{1}{2}\right)\mathrm{\Gamma }\left(\lambda \right)}{2\mathrm{\Gamma }\left(\frac{1}{2}\lambda +\frac{1}{2}\nu -\frac{1}{2}\mu +\frac{1}{2}\right)\mathrm{\Gamma }\left(\frac{1}{2}\lambda +\frac{1}{2}\mu -\frac{1}{2}\nu +\frac{1}{2}\right)\mathrm{\Gamma }\left(\frac{1}{2}\lambda +\frac{1}{2}\mu +\frac{1}{2}\nu +\frac{1}{2}\right)},$$ $\mathrm{\Re }\left(\mu +\nu +1\right)>\mathrm{\Re }\lambda >0$.

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part and $\nu$: complex parameter

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/10.22.E57

Encodings:

pMML, png, TeX

See also:

Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10

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10.22.58 $${\int }_0^{\mathrm{\infty }}\frac{J_{\nu }\left(at\right)J_{\nu }\left(bt\right)}{t^{\lambda }}dt=\frac{{(ab)}^{\nu }\mathrm{\Gamma }\left(\nu -\frac{1}{2}\lambda +\frac{1}{2}\right)}{2^{\lambda }{(a^2+b^2)}^{\nu -\frac{1}{2}\lambda +\frac{1}{2}}\mathrm{\Gamma }\left(\frac{1}{2}\lambda +\frac{1}{2}\right)}𝐅(\frac{2\nu +1-\lambda }{4},\frac{2\nu +3-\lambda }{4};\nu +1;\frac{4a^2{b}^2}{{(a^2+b^2)}^2}),$$ $a\ne b$, $\mathrm{\Re }\left(2\nu +1\right)>\mathrm{\Re }\lambda >-1$.

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function and $\nu$: complex parameter

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/10.22.E58

Encodings:

pMML, png, TeX

See also:

Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10

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§10.22(v) Hankel Transform

►The Hankel transform (or Bessel transform) of a function $f(x)$ is defined as … ►

10.22.78 $$f(x)={\int }_0^{\mathrm{\infty }}{(xt)}^{\frac{1}{2}}\frac{J_{\nu }\left(xt\right)Y_{\nu }\left(at\right)-Y_{\nu }\left(xt\right)J_{\nu }\left(at\right)}{J_{\nu }^2\left(at\right)+Y_{\nu }^2\left(at\right)}{\int }_a^{\mathrm{\infty }}{(yt)}^{\frac{1}{2}}\left(J_{\nu }\left(yt\right)Y_{\nu }\left(at\right)-Y_{\nu }\left(yt\right)J_{\nu }\left(at\right)\right)f(y)dydt,$$ $a>0$.

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $dx$: differential of $x$ , $\int$: integral, $x$: real variable, $y$: real variable, $\nu$: complex parameter and $f(x)$: function

Sources:

Titchmarsh (1962a, p. 87); Watson (1944, §14.52)

Referenced by:

§1.18(ix), §1.18(vi), §10.22(vi), Erratum (V1.2.0) for Chapter 10 Additions

Permalink:

http://dlmf.nist.gov/10.22.E78

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §10.22(v), §10.22 and Ch.10

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12: 13.23 Integrals

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13.23.6 $$\frac{1}{\mathrm{\Gamma }\left(1+2\mu \right)2\pi \mathrm{i}}{\int }_{-\mathrm{\infty }}^{(0+)}{\mathrm{e}}^{zt+\frac{1}{2}{t}^{-1}}{t}^{\kappa }{M}_{\kappa ,\mu }\left(t^{-1}\right)dt=\frac{z^{-\kappa -\frac{1}{2}}}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)}{I}_{2\mu }\left(2\sqrt{z}\right),$$ $\mathrm{\Re }z>0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric 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, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{\Re }$: real part and $z$: complex variable

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/13.23.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §13.23(i), §13.23 and Ch.13

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13.23.7 $$\frac{1}{2\pi \mathrm{i}}{\int }_{-\mathrm{\infty }}^{(0+)}{\mathrm{e}}^{zt+\frac{1}{2}{t}^{-1}}{t}^{\kappa }{W}_{\kappa ,\mu }\left(t^{-1}\right)dt=\frac{2z^{-\kappa -\frac{1}{2}}}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)\mathrm{\Gamma }\left(\frac{1}{2}-\mu -\kappa \right)}{K}_{2\mu }\left(2\sqrt{z}\right),$$ $\mathrm{\Re }z>0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric 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, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{\Re }$: real part and $z$: complex variable

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/13.23.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §13.23(i), §13.23 and Ch.13

… ►For additional Hankel transforms and also other Bessel transforms see Erdélyi et al. (1954b, §8.18) and Oberhettinger (1972, §1.16 and 3.4.42–46, 4.4.45–47, 5.94–97). …

13: Bibliography C

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An algorithm for the Fourier-Bessel transform. Comput. Phys. Comm. 23 (4), pp. 343–353.

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External Links:

ISSN 0010-4655, Document, MathReview

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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Derivatives with respect to the degree and order of associated Legendre functions for $|z|>1$ using modified Bessel functions. Integral Transforms Spec. Funct. 21 (7-8), pp. 581–588.

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External Links:

ISSN 1065-2469, Document, FullText, MathReview (Dmitriy Karp), ZentralBlatt

Referenced by:

§14.11, §14.11.

Encodings:

BibTeX

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14: 10.43 Integrals

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10.43.26 $${\int }_0^{\mathrm{\infty }}\frac{K_{\mu }\left(at\right)J_{\nu }\left(bt\right)}{t^{\lambda }}dt=\frac{b^{\nu }\mathrm{\Gamma }\left(\frac{1}{2}\nu -\frac{1}{2}\lambda +\frac{1}{2}\mu +\frac{1}{2}\right)\mathrm{\Gamma }\left(\frac{1}{2}\nu -\frac{1}{2}\lambda -\frac{1}{2}\mu +\frac{1}{2}\right)}{2^{\lambda +1}{a}^{\nu -\lambda +1}}𝐅(\frac{\nu -\lambda +\mu +1}{2},\frac{\nu -\lambda -\mu +1}{2};\nu +1;-\frac{b^2}{a^2}),$$ $\mathrm{\Re }\left(\nu +1-\lambda \right)>|\mathrm{\Re }\mu |,\mathrm{\Re }a>|\mathrm{\Im }b|$.

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\mathrm{\Im }$: imaginary part, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{\Re }$: real part, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function and $\nu$: complex parameter

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/10.43.E26

Encodings:

pMML, png, TeX

See also:

Annotations for §10.43(iv), §10.43 and Ch.10

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10.43.27 $${\int }_0^{\mathrm{\infty }}{t}^{\mu +\nu +1}{K}_{\mu }\left(at\right)J_{\nu }\left(bt\right)dt=\frac{{(2a)}^{\mu }{(2b)}^{\nu }\mathrm{\Gamma }\left(\mu +\nu +1\right)}{{(a^2+b^2)}^{\mu +\nu +1}},$$ $\mathrm{\Re }\left(\nu +1\right)>|\mathrm{\Re }\mu |,\mathrm{\Re }a>|\mathrm{\Im }b|$.

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\mathrm{\Im }$: imaginary part, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{\Re }$: real part and $\nu$: complex parameter

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/10.43.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §10.43(iv), §10.43 and Ch.10

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§10.43(v) Kontorovich–Lebedev Transform

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10.43.30 $$f(y)=\frac{2y}{{\pi }^2}\sinh \left(\pi y\right){\int }_0^{\mathrm{\infty }}\frac{g(x)}{x}{K}_{\mathrm{i}y}\left(x\right)dx.$$

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Defines:

$f(x)$: Kontorovich–Lebedev transform of $g(x)$ (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $x$: real variable, $y$: real variable and $g(x)$: function

Referenced by:

§10.43(v)

Permalink:

http://dlmf.nist.gov/10.43.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §10.43(v), §10.43 and Ch.10

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10.43.31 $$g(x)={\int }_0^{\mathrm{\infty }}f(y)K_{\mathrm{i}y}\left(x\right)dy,$$

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Symbols:

$dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $x$: 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

Encodings:

pMML, png, TeX

See also:

Annotations for §10.43(v), §10.43 and Ch.10

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15: 1.14 Integral Transforms

§1.14 Integral Transforms

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16: 18.17 Integrals

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18.17.18 $${\int }_0^1{(1-x^2)}^{\lambda -\frac{1}{2}}{C}_{2n+1}^{(\lambda )}\left(x\right)\sin \left(xy\right)dx=\frac{{(-1)}^n\pi \mathrm{\Gamma }\left(2n+2\lambda +1\right)J_{2n+\lambda +1}\left(y\right)}{(2n+1)!\mathrm{\Gamma }\left(\lambda \right){(2y)}^{\lambda }}.$$

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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 $x$ , $!$: factorial (as in $n!$), $\int$: integral, $\sin z$: sine function, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $y$: real variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.17(v)

Permalink:

http://dlmf.nist.gov/18.17.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

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17: Bibliography P

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Bounds for ratios of modified Bessel functions. Integral Transform. Spec. Funct. 9 (4), pp. 293–298.

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External Links:

ISSN 1065-2469, Document, MathReview, ZentralBlatt

Referenced by:

§10.37.

Encodings:

BibTeX

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18: 3.5 Quadrature

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Example. Laplace Transform Inversion

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19: 10.42 Zeros

… ►Properties of the zeros of $I_{\nu }\left(z\right)$ and $K_{\nu }\left(z\right)$ may be deduced from those of $J_{\nu }\left(z\right)$ and $H_{\nu }^{(1)}\left(z\right)$, respectively, by application of the transformations (10.27.6) and (10.27.8). …

20: 10.9 Integral Representations

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10.9.22 $$J_{\nu }\left(x\right)=\frac{1}{2\pi \mathrm{i}}{\int }_{-\mathrm{i}\mathrm{\infty }}^{\mathrm{i}\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(-t\right){(\frac{1}{2}x)}^{\nu +2t}}{\mathrm{\Gamma }\left(\nu +t+1\right)}dt,$$ $\mathrm{\Re }\nu >0$, $x>0$,

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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:

pMML, png, TeX

See also:

Annotations for §10.9(ii), §10.9(ii), §10.9 and Ch.10

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10.9.23 $$J_{\nu }\left(z\right)=\frac{1}{2\pi \mathrm{i}}{\int }_{-\mathrm{\infty }-\mathrm{i}c}^{-\mathrm{\infty }+\mathrm{i}c}\frac{\mathrm{\Gamma }\left(t\right)}{\mathrm{\Gamma }\left(\nu -t+1\right)}{(\frac{1}{2}z)}^{\nu -2t}dt,$$

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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:

pMML, png, TeX

See also:

Annotations for §10.9(ii), §10.9(ii), §10.9 and Ch.10

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10.9.24 $$H_{\nu }^{(1)}\left(z\right)=-\frac{{\mathrm{e}}^{-\frac{1}{2}\nu \pi \mathrm{i}}}{2{\pi }^2}{\int }_{c-\mathrm{i}\mathrm{\infty }}^{c+\mathrm{i}\mathrm{\infty }}\mathrm{\Gamma }\left(t\right)\mathrm{\Gamma }\left(t-\nu \right){(-\frac{1}{2}\mathrm{i}z)}^{\nu -2t}dt,$$ ,

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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:

pMML, png, TeX

See also:

Annotations for §10.9(ii), §10.9(ii), §10.9 and Ch.10

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10.9.25 $$H_{\nu }^{(2)}\left(z\right)=\frac{{\mathrm{e}}^{\frac{1}{2}\nu \pi \mathrm{i}}}{2{\pi }^2}{\int }_{c-\mathrm{i}\mathrm{\infty }}^{c+\mathrm{i}\mathrm{\infty }}\mathrm{\Gamma }\left(t\right)\mathrm{\Gamma }\left(t-\nu \right){(\frac{1}{2}\mathrm{i}z)}^{\nu -2t}dt,$$ .

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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:

pMML, png, TeX

See also:

Annotations for §10.9(ii), §10.9(ii), §10.9 and Ch.10

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10.9.29 $$J_{\mu }\left(x\right)J_{\nu }\left(x\right)=\frac{1}{2\pi \mathrm{i}}{\int }_{-\mathrm{i}\mathrm{\infty }}^{\mathrm{i}\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(-t\right)\mathrm{\Gamma }\left(2t+\mu +\nu +1\right){(\frac{1}{2}x)}^{\mu +\nu +2t}}{\mathrm{\Gamma }\left(t+\mu +1\right)\mathrm{\Gamma }\left(t+\nu +1\right)\mathrm{\Gamma }\left(t+\mu +\nu +1\right)}dt,$$ $x>0$,

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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:

pMML, png, TeX

See also:

Annotations for §10.9(iii), §10.9(iii), §10.9 and Ch.10

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