HankelTransform

The order- $n$ Hankel transform $F(s) = \int_0^\infty f(r)\,J_n(sr)\,r\,dr$ ( $n=0$ by default), as a rule table over the standard pairs.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageHankelTransform!

HankelTransform(f, r, s)the order-0 Hankel transform of f,

\int_0^\infty f(r)\,J_0(sr)\,r\,dr

.

HankelTransform(f, r, s, n)the order-

n

Hankel transform.

Typed signatures

Type	Package	Applies
`(expression, expression, expression, number?) -> expression`	`analytic`	always

Domain: Transforms

Details

Order 0 (each pair, and its strip Re(a) > 0 where one applies, confirmed against wolframscript's HankelTransform[..., GenerateConditions -> True], matching its normalisation exactly): $e^{-ar}\to a/(a^2+s^2)^{3/2}$ , $e^{-ar^2}\to \frac{1}{2a}e^{-s^2/(4a)}$ , $1/r\to 1/s$ , $1/\sqrt{r^2+a^2}\to e^{-as}/s$ .
Two further orders Wolfram's own examples state explicitly, also covered: order 1's $e^{-ar}\to s/(a^2+s^2)^{3/2}$ , and $1/r\to 1/s$ at ANY order n (n > -1/2, an identity independent of the order).
Four more order-0 pairs from the tables, each confirmed against wolframscript: $\operatorname{erfc}(kr)/r\to\operatorname{erf}(s/2k)/s$ and $E_1(kr)/r\to\operatorname{arsinh}(s/k)/s$ ( $k>0$ ), $(1-e^{-mr})/r^2\to\operatorname{arsinh}(m/s)$ ( $m>0$ ; for $m<0$ the integral diverges, so a symbolic $m$ is declined), and $\ln(1+a^2/r^2)\to 2(1-|a|sK_1(|a|s))/s^2$ .
Two conditionally convergent order-0 pairs that jump at $s=|a|$ , confirmed against wolframscript (whose MeijerG answer agrees on each side) and an oscillatory NIntegrate: $\sin(ar)/r\to\operatorname{sgn}(a)/\sqrt{a^2-s^2}$ for $s<|a|$ and $0$ above, and $\cos(ar)/r\to 0$ for $s<|a|$ and $1/\sqrt{s^2-a^2}$ above. A symbolic $s$ stays a Piecewise; $s=|a|$ diverges and is held, as Wolfram holds it.
Declined: any other order for $e^{-ar}$ / $e^{-ar^2}$ / $1/\sqrt{r^2+a^2}$ (Wolfram's own closed forms there involve Hypergeometric2F1Regularized/Hypergeometric1F1Regularized — not elementary, not chased), and an unknown-sign a.

Examples

Basic #

$H_0\{e^{-ar}\}(s) = a/(a^2+s^2)^{3/2}$

In

\mathrm{Assuming}(a\gt0, \mathrm{HankelTransform}(\exp(-(ar)), r, s))

Out

\frac{a}{\sqrt{a^2+s^2}^{3}}

$H_0\{1/r\}(s) = 1/s$

In

\mathrm{HankelTransform}(\frac{1}{r}, r, s)

Out

\frac{1}{s}

$H_0\{1/\sqrt{r^2+a^2}\}(s) = e^{-as}/s$

In

\mathrm{Assuming}(a\gt0, \mathrm{HankelTransform}(\frac{1}{\sqrt{r^2+a^2}}, r, s))

Out

\frac{\exp(-(as))}{s}

In

\mathrm{HankelTransform}(\exp(-r), r, s)

Out

\sqrt{s^2+1}^{-3}

In

\mathrm{HankelTransform}(\frac{\cos(r)}{r^2+1}, r, s)

Out

\mathrm{HankelTransform}(\frac{\cos({{r}})}{{{r}}^2+1}, r, s)

Scope #

order 1 — $H_1\{e^{-ar}\}(s) = s/(a^2+s^2)^{3/2}$

In

\mathrm{Assuming}(a\gt0, \mathrm{HankelTransform}(\exp(-(ar)), r, s, 1))

Out

\frac{s}{\sqrt{a^2+s^2}^{3}}

$H_0\{\sin(r)/r\}(s) = 1/\sqrt{1-s^2}$ for $s<1$ : at $s=1/3$ , $3/(2\sqrt2)$

In

\mathrm{HankelTransform}(\frac{\sin(r)}{r}, r, \frac{1}{3})

Out

\frac{3\sqrt{2}}{4}

A symbolic $s$ stays a Piecewise: $1/\sqrt{1-s^2}$ below 1, 0 above; $s=1$ diverges

In

\mathrm{HankelTransform}(\frac{\sin(r)}{r}, r, s)

Out

\mathrm{Piecewise}(\bigl\lbrack\bigl\lbrack\frac{1}{(1-s^2)^{1/2}}, s\lt1\bigr\rbrack, \bigl\lbrack0, 1\lt s\bigr\rbrack\bigr\rbrack, \operatorname{Indeterminate})

$H_0\{\cos(r)/r\}(s) = 1/\sqrt{s^2-1}$ for $s>1$ (and 0 below): at $s=3$ , $1/(2\sqrt2)$

In

\mathrm{HankelTransform}(\frac{\cos(r)}{r}, r, 3)

Out

\frac{\sqrt{2}}{4}

In

\mathrm{HankelTransform}(\frac{1}{re^{r}}, r, s)

Out

\frac{1}{\sqrt{s^2+1}}

In

\mathrm{HankelTransform}((1-\exp(-(mr)))/r^2, r, s)

Out

\mathrm{HankelTransform}(\frac{1-e^{-(m{{r}})}}{{{r}}^2}, r, s)

In

\mathrm{HankelTransform}(\frac{1}{r\exp(a^2r^2)}, r, s)

Out

\frac{\operatorname{I}_{0}(\frac{s^2}{8a^2})\exp(-\frac{s^2}{8a^2})\sqrt{π}}{2\vert a\vert}

In

\mathrm{HankelTransform}(\ln(\frac{a^2}{r^2}+1), r, s)

Out

(2(1-s\vert a\vert\operatorname{K}_{1}(s\vert a\vert)))/s^2

In

\mathrm{HankelTransform}(\frac{\operatorname{erfc}(r)}{r}, r, s)

Out

\frac{1}{s}(\operatorname{erf}(\frac{s}{2}))

In

\mathrm{HankelTransform}(\frac{\mathrm{ExpIntegralE}(1, r)}{r}, r, s)

Out

\frac{\mathrm{arsinh}(s)}{s}

$H_0\{(1-e^{-2r})/r^2\}(s) = \operatorname{arsinh}(2/s)$

In

\mathrm{HankelTransform}((1-\exp(-2r))/r^2, r, s)

Out

\mathrm{arsinh}(\frac{2}{s})

About

Examples	18
Checked against	Wolfram Language 14 agree4 differ
Crosswalk	Wolfram Language

Implementation

primitive · kernela kernel — an algorithm over mutable state that would not be readable as a tree

nativeenginetypescriptpackages/symbols/analysis/analytic/src/hankel-transform.ts

See also: MellinTransform, LaplaceTransform