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 GenericIntegralTransform (1.2.0) current version: 3.1.4 »

Source Notebook

Compute a variety of integral transforms on input expressions

Contributed by: Paco Jain & Oleg Marichev

ResourceFunction["GenericIntegralTransform"][f,z,t,"type"]

gives the integral transform 𝒯type[f](t) corresponding to the input function f(z), in terms of the new variable t.

Details and Options

ResourceFunction["GenericIntegralTransform"] supports the following options:
GenerateConditionsTruewhether to provide conditions under which the given result is valid
"FoxHForm"Falsewhether to provide results (where possible) in terms of the FoxH function
Currently supported values for "transform" include the following:
G-transform{"G", {{af1_List, ae1_List}, {bf1_List, be1_List}}}
Hankel transform{"Hankel", {α_, ν_}}
Hilbert transform{"Hilbert", α_}
Integrate transform{"Integrate", α_}
Laplace transform{"Laplace", α_}
Liouville transform{"Liouville", α_}
Meijer transform{"Meijer", {ν_, α_}}
Mellin transform"Mellin"
Neumann transform{"Neumann", {ν_, α_}}
Riesz transform{"Riesz", α_}
Stieltjes transform{{"Stieltjes", ρ_}}
Struve transform{"Struve", {ν_, α_}}
Weyl transform{"Weyl", α_}

Examples

Basic Examples (2) 

Compute the Mellin transform of a Bessel function:

In[1]:=
ResourceFunction["GenericIntegralTransform"][ BesselJ[\[Nu], z], z, t, "Mellin"]
Out[1]=

Compute the Mellin transform of an exponential function. Generically, the result returned is a ConditionalExpression:

In[2]:=
ResourceFunction["GenericIntegralTransform"][E^(-a x), x, s, "Mellin"]
Out[2]=

The conditions under which this result holds can be expanded by clicking the "+", followed by "Uniconize":

In[3]:=
ConditionalExpression[ a^-s Gamma[ s], (Abs[Arg[a]] < \[Pi]/2 && 0 < Re[s] < \[Infinity]) || (Abs[Arg[a]] == \[Pi]/2 && 0 < Re[s] < \[Infinity] && -(1/2) + Re[s] < 1/2)]

Scope

G-transform

In[4]:=
ResourceFunction["GenericIntegralTransform"][ Sin[z], z, t, {"G", {{{}, {\[Alpha]}}, {{0}, {}}}}]
Out[4]=

Hankel transform

In[5]:=
ResourceFunction["GenericIntegralTransform"][ Sin[z], z, t, {"Hankel", {\[Alpha], \[Nu]}}]
Out[5]=

Mellin transform

In[6]:=
ResourceFunction["GenericIntegralTransform"][ BesselJ[\[Nu], z], z, t, "Mellin"]
Out[6]=

Stieltjes transform

In[7]:=
ResourceFunction["GenericIntegralTransform"][ Sin[z], z, t, {"Stieltjes", \[Rho]}]
Out[7]=

Struve transform

In[8]:=
ResourceFunction["GenericIntegralTransform"][ Sin[z], z, t, {"Struve", {\[Nu], \[Alpha]}}]
Out[8]=

Options (2) 

Using the option setting GenerateConditions→False to suppress convergence conditions from transform results:

In[9]:=
ResourceFunction["GenericIntegralTransform"][ BesselJ[\[Nu], z], z, t, "Mellin", GenerateConditions -> False]
Out[9]=

Contrast with the default setting GenerateConditions→True:

In[10]:=
ResourceFunction["GenericIntegralTransform"][ BesselJ[\[Nu], z], z, t, "Mellin"]
Out[10]=

Publisher

Wolfram|Alpha Math Team

Version History

  • 3.1.4 – 13 September 2023
  • 3.1.3 – 13 September 2023
  • 3.1.2 – 11 September 2023
  • 3.1.1 – 11 September 2023
  • 3.1.0 – 11 September 2023
  • 3.0.0 – 11 September 2023
  • 2.0.3 – 07 November 2022
  • 2.0.2 – 19 October 2022
  • 2.0.1 – 19 October 2022
  • 2.0.0 – 19 October 2022
  • 1.2.0 – 19 October 2022
  • 1.1.0 – 10 October 2022
  • 1.0.0 – 07 October 2022

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