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Hankel Transform (inttrans package)

withinttrans:

Introduction

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

s = converthankelft,t,s,v,int

s=0fttJvstⅆt

(1.1)

From this definition, it is clear that t0,, so this integral transform applies to complex functions of a real and nonnegative variable t. The Hankel transform is self-inversible provided that s0,, so that

ft=0ss Jvstⅆs

ft=0ssJvstⅆs

(1.2)

Thus the computation of the hankel transform of ft and the inverse transform assumes both that t0, and s0,.

 

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.

Query about the current status:

inttrans:-setupalternativehankeldefinition

alternativehankeldefinition=false

(1.3)

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

inttrans:-setupalternativehankeldefinition = true

alternativehankeldefinition=true

(1.4)

Check the integral form of this alternative definition

converthankelgt,t,s,v,int

0gtstJvstⅆt

(1.5)

Reset the definition to be as in (1.1)

inttrans:-setupalternativehankeldefinition = false

alternativehankeldefinition=false

(1.6)

From the integral forms (1.1) and (1.5), these two definitions are connected substituting in (1.1) ft=gtst, or substituting gt=ftts 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

hankel1α+t,t,s,0

πI0αs2α

(2.1)

hankelexpa2t22,t,s,0;

ⅇs22a2a2

(2.2)

hankellogtt,t,s,0;

ln2γss

(2.3)

hankelsinatt,t,s,1;

θsaasa2+s2

(2.4)

hankelcosatt,t,s,0;

θsaa2+s2

(2.5)

hankelarctant2t,t,s,1;

2kei0s

(2.6)

hankelexpatsinhbtt,t,s,0 assuming ab

absa22ab+b2+s2a2+2ab+b2+s2sa22ab+b2+s2a2+2ab+b2+s2a22ab+b2+s2+a2+2ab+b2+s2

(2.7)

Exponential, Sine, and Cosine Integral

hankelCiαt2,t,s,0

21+coss24αs2

(3.1)

hankelSsiαt2,t,s,0

2sins24αs2

(3.2)

hankelEiβt2,t,s,0

21+ⅇs24βs2

(3.3)

Error Integrals

hankelerfαtt,t,s,0

erfcs2αs

(4.1)

Hankel's Functions 1 and 2

hankel1HankelH1&mu;&comma;&alpha;tHankelH2&mu;&comma;&alpha;tt&comma;t&comma;s&comma;0 assuming 14<&mu;<14

16cosπμK2μ1+Iαs2K2μ1Iαs2sπ2

(5.1)

Bessel and Modified Bessel Functions

hankelBesselJ0&comma;&beta;t&alpha;&plus;t2&comma;t&comma;s&comma;0

θβsK0αβI0αs+θsβI0αβK0αs

(6.1)

hankelBesselJv&comma;&beta;t&alpha;&plus;t2&comma;t&comma;s&comma;vassuming v &gt; 12

θβsKvαβIvαs+θsβIvαβKvαs

(6.2)

hankelBesselY0&comma;&alpha;tt&comma;t&comma;s&comma;0

πY02αs2K02αssπ

(6.3)

hankelBesselY&mu;&comma;&alpha;tt&comma;t&comma;s&comma;&mu; assuming 12<μ<12

πY2μ2αs2K2μ2αssπ

(6.4)

hankelBesselK0&comma;&beta;t&comma;t&comma;s&comma;0

1β2+s2

(6.5)

hankelBesselKv&comma;&beta;t&comma;t&comma;s&comma;v

βvsvβ2+s2

(6.6)

hankel&ExponentialE;&alpha;t2BesselI0&comma;&beta;t&comma;t&comma;s&comma;0

&ExponentialE;β2s24αJ0βs2α2α

(6.7)

hankel&ExponentialE;&alpha;t2BesselIv&comma;&beta;t&comma;t&comma;s&comma;v

hankel&ExponentialE;αt2Ivβt&comma;t&comma;s&comma;v

(6.8)

 

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