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inttrans

  

hankel

  

Hankel transform

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

hankel(expr, t, s, nu)

Parameters

expr

-

expression, equation, or set of expressions and/or equations to be transformed

t

-

variable expr is transformed with respect to t

s

-

parameter of transform

nu

-

order of the transform

opt

-

option to run this under (optional)

Description

• 

The hankel function computes the Hankel transform (F[nu](s)) of expr (f(t)), a linear transformation C0,C0, defined by:

Fνs=0ftstJνstⅆt

• 

The function Fs returned is defined on the positive real axis only.

• 

Expressions involving exponentials, products involving powers of t, trigonometrics (sin, cos) with linear arguments, and a variety of other functions can all be transformed.

• 

The Hankel transform is self-inverting for nu >12 .

• 

The hankel function can transform Bessel's operator, and if nu0, derivatives of functions as well, and can be used in the solution of ODEs and PDEs.

• 

The hankel function attempts to simplify an expression according to a set of heuristics and then match the result with a table of patterns.  Entries can be added to this table by addtable(hankel, f(t), F(s), t, s), where F(s) is the transform of f(t), which may have an arbitrary number of parameters.

• 

hankel  recognizes the Dirac-delta (or unit-impulse) function as Dirac(t) and Heaviside's unit step function as Heaviside(t).

• 

If the option opt is set to 'NO_INT', then the program will not resort to integration of the original problem if all other methods fail.  This will increase the speed at which the transform will run.

• 

The command with(inttrans,hankel) allows the use of the abbreviated form of this command.

Examples

withinttrans:

assume0<a&colon;

assume12<&nu;&colon;additionally&nu;0&colon;

Fhankelgx&comma;x&comma;y&comma;&nu;

F:=hankelgx&comma;x&comma;y&comma;&nu;~

(1)

hankelF&comma;y&comma;z&comma;&nu;

gz

(2)

hankeltft&comma;t&comma;s&comma;&nu;

hankelft&comma;t&comma;s&comma;&nu;~&plus;1&nu;~&plus;shankelft&comma;t&comma;s&comma;&nu;~&plus;1s&plus;hankelft&comma;t&comma;s&comma;&nu;~&plus;1s

(3)

hankelftt&comma;t&comma;s&comma;&nu;

12shankelft&comma;t&comma;s&comma;&nu;~1&plus;hankelft&comma;t&comma;s&comma;&nu;~&plus;1&nu;~

(4)

hankelDft&comma;t&comma;s&comma;&nu;

12shankelft&comma;t&comma;s&comma;&nu;~1&nu;~hankelft&comma;t&comma;s&comma;&nu;~&plus;1&nu;~&plus;hankelft&comma;t&comma;s&comma;&nu;~1&plus;hankelft&comma;t&comma;s&comma;&nu;~&plus;1&nu;~

(5)

Bessel's differential equation

pde2z2fr&comma;z&plus;2r2fr&comma;z&plus;rfr&comma;zr&nu;r2fr&comma;z&equals;0

pde:=2z2fr&comma;z&plus;2r2fr&comma;z&plus;rfr&comma;zr&nu;~2fr&comma;zr2&equals;0

(6)

hankelpde&comma;r&comma;p&comma;&nu;

2z2hankelfr&comma;z&comma;r&comma;p&comma;&nu;~p2hankelfr&comma;z&comma;r&comma;p&comma;&nu;~&equals;0

(7)

hankelt12&comma;t&comma;s2&comma;1

1s23&sol;2

(8)

hankel&ExponentialE;at&comma;t&comma;s&comma;&nu;

&Gamma;&nu;~&plus;32hypergeom12&nu;~&plus;34&comma;54&plus;12&nu;~&comma;&nu;~&plus;1&comma;s2a~2s&nu;~&plus;122&nu;~a~&nu;~32&Gamma;&nu;~&plus;1

(9)

Adding to the table

addtablehankel&comma;myfunct&comma;MyFuncs&comma;&nu;&comma;t&comma;s&comma;hankel&equals;&nu;::Range&infin;&comma;&infin;

hankelmyfuncx&comma;x&comma;y&comma;&nu;

MyFuncy&comma;&nu;~

(10)

See Also

dsolve

inttrans

inttrans[addtable]

 


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