 Hankel Transform (inttrans package) - Maple Programming Help

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Hankel Transform (inttrans package)

 > $\mathrm{with}\left(\mathrm{inttrans}\right):$

Introduction

The hankel transform, sometimes referred to as the Bessel transform, has uses in particular types of differential equations.

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 ${ℍ}{}\left({s}\right){=}{{\int }}_{{0}}^{{\mathrm{\infty }}}{f}{}\left({t}\right){}{t}{}{{J}}_{{v}}{}\left({s}{}{t}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{t}$ (1.1)

From this definition, it is clear that $t\in \left[0,\mathrm{\infty }\right)$, so this integral transform applies to complex functions of a real and nonnegative variable t. The Hankel transform is self-inversible provided that $s\in \left[0,\mathrm{\infty }\right)$, so that

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 ${f}{}\left({t}\right){=}{{\int }}_{{0}}^{{\mathrm{\infty }}}{ℍ}{}\left({s}\right){}{s}{}{{J}}_{{v}}{}\left({s}{}{t}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{s}$ (1.2)

Thus the computation of the hankel transform of $f\left(t\right)$ and the inverse transform assumes both that $t\in \left[0,\mathrm{\infty }\right)$ and $s\in \left[0,\mathrm{\infty }\right)$.

NOTE: since Maple 2020, the two definitions frequently found in the literature are implemented and you can compute with any of them by changing the inttrans:-setup accordingly. The default definition is as shown above. The alternative definition, that was the default one in previous releases of the Maple system. can be seen and set as follows.

 > $\mathrm{inttrans}:-\mathrm{setup}\left(\mathrm{alternativehankeldefinition}\right)$
 ${\mathrm{alternativehankeldefinition}}{=}{\mathrm{false}}$ (1.3)

Set the alternative definition to be the one in use (as it was in previous releases of the Maple system)

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 ${\mathrm{alternativehankeldefinition}}{=}{\mathrm{true}}$ (1.4)

Check the integral form of this alternative definition

 > $\mathrm{convert}\left(\mathrm{hankel}\left(g\left(t\right),t,s,v\right),\mathrm{int}\right)$
 ${{\int }}_{{0}}^{{\mathrm{\infty }}}{g}{}\left({t}\right){}\sqrt{{s}{}{t}}{}{{J}}_{{v}}{}\left({s}{}{t}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{t}$ (1.5)

Reset the definition to be as in (1.1)

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 ${\mathrm{alternativehankeldefinition}}{=}{\mathrm{false}}$ (1.6)

From the integral forms (1.1) and (1.5), these two definitions are connected substituting in (1.1) $f\left(t\right)=\frac{g\left(t\right)\sqrt{s}}{\sqrt{t}}$, or substituting $g\left(t\right)=\frac{f\left(t\right)\sqrt{t}}{\sqrt{s}}$ in (1.4).

The next sections are written for the definition (1.1) but the input output can be related for the alternative definition using the substituting equations just mentioned.

Algebraic, Exponential, Logarithmic, Trigonometric, Inverse Trigonometric, and Hyperbolic Functions

 > $\mathrm{hankel}\left(\frac{1}{\mathrm{α}+t},t,s,0\right)$
 $\frac{{\mathrm{\pi }}{}{{I}}_{{0}}{}\left({\mathrm{\alpha }}{}{s}\right)}{{2}{}{\mathrm{\alpha }}}$ (2.1)
 > $\mathrm{hankel}\left(\mathrm{exp}\left(-\frac{{a}^{2}\cdot {t}^{2}}{2}\right),t,s,0\right);$
 $\frac{{{ⅇ}}^{{-}\frac{{{s}}^{{2}}}{{2}{}{{a}}^{{2}}}}}{{{a}}^{{2}}}$ (2.2)
 > $\mathrm{hankel}\left(\frac{\mathrm{log}\left(t\right)}{t},t,s,0\right);$
 ${-}\frac{{\mathrm{ln}}{}\left({2}{}{\mathrm{\gamma }}{}{s}\right)}{{s}}$ (2.3)
 > $\mathrm{hankel}\left(\frac{\mathrm{sin}\left(a\cdot t\right)}{t},t,s,1\right);$
 $\frac{{\mathrm{\theta }}{}\left({s}{-}{a}\right){}{a}}{{s}{}\sqrt{{-}{{a}}^{{2}}{+}{{s}}^{{2}}}}$ (2.4)
 > $\mathrm{hankel}\left(\frac{\mathrm{cos}\left(a\cdot t\right)}{t},t,s,0\right);$
 $\frac{{\mathrm{\theta }}{}\left({s}{-}{a}\right)}{\sqrt{{-}{{a}}^{{2}}{+}{{s}}^{{2}}}}$ (2.5)
 > $\mathrm{hankel}\left(\frac{\mathrm{arctan}\left({t}^{2}\right)}{t},t,s,1\right);$
 ${-}{2}{}{{\mathrm{kei}}}_{{0}}{}\left({s}\right)$ (2.6)
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 $\frac{\sqrt{{a}{}{b}{}{s}}{}\sqrt{\sqrt{{{a}}^{{2}}{-}{2}{}{a}{}{b}{+}{{b}}^{{2}}{+}{{s}}^{{2}}}{-}\sqrt{{{a}}^{{2}}{+}{2}{}{a}{}{b}{+}{{b}}^{{2}}{+}{{s}}^{{2}}}}}{\sqrt{{s}}{}\sqrt{{{a}}^{{2}}{-}{2}{}{a}{}{b}{+}{{b}}^{{2}}{+}{{s}}^{{2}}}{}\sqrt{{{a}}^{{2}}{+}{2}{}{a}{}{b}{+}{{b}}^{{2}}{+}{{s}}^{{2}}}{}\sqrt{\sqrt{{{a}}^{{2}}{-}{2}{}{a}{}{b}{+}{{b}}^{{2}}{+}{{s}}^{{2}}}{+}\sqrt{{{a}}^{{2}}{+}{2}{}{a}{}{b}{+}{{b}}^{{2}}{+}{{s}}^{{2}}}}}$ (2.7)

Exponential, Sine, and Cosine Integral

 > $\mathrm{hankel}\left(\mathrm{Ci}\left(\mathrm{α}{t}^{2}\right),t,s,0\right)$
 ${-}\frac{{2}{}\left({-}{1}{+}{\mathrm{cos}}{}\left(\frac{{{s}}^{{2}}}{{4}{}{\mathrm{\alpha }}}\right)\right)}{{{s}}^{{2}}}$ (3.1)
 > $\mathrm{hankel}\left(\mathrm{Ssi}\left(\mathrm{α}{t}^{2}\right),t,s,0\right)$
 ${-}\frac{{2}{}{\mathrm{sin}}{}\left(\frac{{{s}}^{{2}}}{{4}{}{\mathrm{\alpha }}}\right)}{{{s}}^{{2}}}$ (3.2)
 > $\mathrm{hankel}\left(\mathrm{Ei}\left(\mathrm{β}{t}^{2}\right),t,s,0\right)$
 $\frac{{2}{}\left({-}{1}{+}{{ⅇ}}^{\frac{{{s}}^{{2}}}{{4}{}{\mathrm{\beta }}}}\right)}{{{s}}^{{2}}}$ (3.3)

Error Integrals

 > $\mathrm{hankel}\left(\frac{\mathrm{erf}\left(\mathrm{α}t\right)}{t},t,s,0\right)$
 $\frac{{\mathrm{erfc}}{}\left(\frac{{s}}{{2}{}{\mathrm{\alpha }}}\right)}{{s}}$ (4.1)

Hankel's Functions 1 and 2

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 $\frac{{16}{}{\mathrm{cos}}{}\left({\mathrm{\pi }}{}{\mathrm{\mu }}\right){}{{K}}_{{2}{}{\mathrm{\mu }}}{}\left(\left({1}{+}{I}\right){}\sqrt{{\mathrm{\alpha }}}{}\sqrt{{s}}{}\sqrt{{2}}\right){}{{K}}_{{2}{}{\mathrm{\mu }}}{}\left(\left({1}{-}{I}\right){}\sqrt{{\mathrm{\alpha }}}{}\sqrt{{s}}{}\sqrt{{2}}\right)}{{s}{}{{\mathrm{\pi }}}^{{2}}}$ (5.1)

Bessel and Modified Bessel Functions

 > $\mathrm{hankel}\left(\frac{\mathrm{BesselJ}\left(0,\mathrm{β}t\right)}{\mathrm{α}+{t}^{2}},t,s,0\right)$
 ${\mathrm{\theta }}{}\left({\mathrm{\beta }}{-}{s}\right){}{{K}}_{{0}}{}\left(\sqrt{{\mathrm{\alpha }}}{}{\mathrm{\beta }}\right){}{{I}}_{{0}}{}\left(\sqrt{{\mathrm{\alpha }}}{}{s}\right){+}{\mathrm{\theta }}{}\left({s}{-}{\mathrm{\beta }}\right){}{{I}}_{{0}}{}\left(\sqrt{{\mathrm{\alpha }}}{}{\mathrm{\beta }}\right){}{{K}}_{{0}}{}\left(\sqrt{{\mathrm{\alpha }}}{}{s}\right)$ (6.1)
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 ${\mathrm{\theta }}{}\left({\mathrm{\beta }}{-}{s}\right){}{{K}}_{{v}}{}\left(\sqrt{{\mathrm{\alpha }}}{}{\mathrm{\beta }}\right){}{{I}}_{{v}}{}\left(\sqrt{{\mathrm{\alpha }}}{}{s}\right){+}{\mathrm{\theta }}{}\left({s}{-}{\mathrm{\beta }}\right){}{{I}}_{{v}}{}\left(\sqrt{{\mathrm{\alpha }}}{}{\mathrm{\beta }}\right){}{{K}}_{{v}}{}\left(\sqrt{{\mathrm{\alpha }}}{}{s}\right)$ (6.2)
 > $\mathrm{hankel}\left(\frac{\mathrm{BesselY}\left(0,\frac{\mathrm{α}}{t}\right)}{t},t,s,0\right)$
 $\frac{{\mathrm{\pi }}{}{{Y}}_{{0}}{}\left({2}{}\sqrt{{\mathrm{\alpha }}}{}\sqrt{{s}}\right){-}{2}{}{{K}}_{{0}}{}\left({2}{}\sqrt{{\mathrm{\alpha }}}{}\sqrt{{s}}\right)}{{s}{}{\mathrm{\pi }}}$ (6.3)
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 $\frac{{\mathrm{\pi }}{}{{Y}}_{{2}{}{\mathrm{\mu }}}{}\left({2}{}\sqrt{{\mathrm{\alpha }}}{}\sqrt{{s}}\right){-}{2}{}{{K}}_{{2}{}{\mathrm{\mu }}}{}\left({2}{}\sqrt{{\mathrm{\alpha }}}{}\sqrt{{s}}\right)}{{s}{}{\mathrm{\pi }}}$ (6.4)
 > $\mathrm{hankel}\left(\mathrm{BesselK}\left(0,\mathrm{β}t\right),t,s,0\right)$
 $\frac{{1}}{{{\mathrm{\beta }}}^{{2}}{+}{{s}}^{{2}}}$ (6.5)
 > $\mathrm{hankel}\left(\mathrm{BesselK}\left(v,\mathrm{β}t\right),t,s,v\right)$
 $\frac{{{\mathrm{\beta }}}^{{-}{v}}{}{{s}}^{{v}}}{{{\mathrm{\beta }}}^{{2}}{+}{{s}}^{{2}}}$ (6.6)
 > $\mathrm{hankel}\left({ⅇ}^{\mathrm{α}{t}^{2}}\mathrm{BesselI}\left(0,\mathrm{β}t\right),t,s,0\right)$
 ${-}\frac{{{ⅇ}}^{{-}\frac{{{\mathrm{\beta }}}^{{2}}{-}{{s}}^{{2}}}{{4}{}{\mathrm{\alpha }}}}{}{{J}}_{{0}}{}\left(\frac{{\mathrm{\beta }}{}{s}}{{2}{}{\mathrm{\alpha }}}\right)}{{2}{}{\mathrm{\alpha }}}$ (6.7)
 > $\mathrm{hankel}\left({ⅇ}^{\mathrm{α}{t}^{2}}\mathrm{BesselI}\left(v,\mathrm{β}t\right),t,s,v\right)$
 ${\mathrm{hankel}}{}\left({{ⅇ}}^{{\mathrm{\alpha }}{}{{t}}^{{2}}}{}{{I}}_{{v}}{}\left({\mathrm{\beta }}{}{t}\right){,}{t}{,}{s}{,}{v}\right)$ (6.8)

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