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Integral Transforms with Modified Bessel Functions as Kernel

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Tables of Bessel Transforms
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Abstract

A representation of a given function f(x) by means of a double integral involving modified Bessel functions of order v is

$$ f(x) = {{(\pi {\text{i)}}}^{{ - 1}}}{\text{ }}\int\limits_{{c - i\infty }}^{{c + i\infty }} {{{I}_{v}}} (tx){{(tx)}^{{ \frac{1}{2} }}}dt\int\limits_{0}^{\infty } {{{k}_{v}}} (ut){{(ut)}^{{ \frac{1}{2} }}}f(u)du $$

or also

$$ f(x) = {{(2\pi {\text{i)}}}^{{ - 1}}}{\text{ }}\int\limits_{{c - i\infty }}^{{c + i\infty }} {[{{I}_{v}}} (tx) + {{I}_{{ - v}}}(tx)]{{(tx)}^{{ \frac{1}{2} }}}dt\int\limits_{0}^{\infty } {{{k}_{v}}} (ut){{(ut)}^{{ \frac{1}{2} }}}f(u)du $$

This is equivalent with the pair of inversion formulas

$$ g(y;v) = \int\limits_{0}^{\infty } f (x){{(xy)}^{{ \frac{1}{2} }}}{{k}_{v}}(xy)dx $$
((1))
$$ f(x) = {{(\pi {\text{i}})}^{{ - 1}}}\int\limits_{{c - i\infty }}^{{c + i\infty }} g (y,v){{(xy)}^{{ \frac{1}{2} }}}{{I}_{v}}(xy)dy $$
((2))

or

$$ g(y;v) = \int\limits_{0}^{\infty } f (x){{(xy)}^{{ \frac{1}{2} }}}_{v}(xy)dx $$
((3))
$$ f(x) = {{(2\pi {\text{i)}}}^{{ - 1}}}\int\limits_{{c - i\infty }}^{{c + i\infty }} g (y;v){{(xy)}^{{ \frac{1}{2} }}}[{{I}_{{ - v}}}(xy)]dy $$
((4))

Since for v = ± ½

$$c $$

the equations (3) and (4) become

$$ \begin{array}{*{20}{c}} {{{{\left( {2{{\pi }^{{ - 1}}}} \right)}}^{{ \frac{1}{2} }}}g\left( {y;\pm \frac{1}{2} } \right) = \int\limits_{0}^{\infty } {f\left( x \right)} {{e}^{{ - xy}}}dx} \\ {f\left( x \right) = {{{\left( {2\pi i} \right)}}^{{ - 1}}}\int\limits_{0}^{\infty } {\left( {2{{\pi }^{{ - 1}}}} \right)g\left( {y;\pm \frac{1}{2} } \right){{e}^{{xy}}}dy} n} \\ \end{array} $$

These are the Laplace transform formulas.

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© 1972 Springer-Verlag Berlin · Heidelberg

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Oberhettinger, F. (1972). Integral Transforms with Modified Bessel Functions as Kernel. In: Tables of Bessel Transforms. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-65462-6_2

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  • DOI: https://doi.org/10.1007/978-3-642-65462-6_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-05997-4

  • Online ISBN: 978-3-642-65462-6

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