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Compute a variety of integral transforms on input expressions
ResourceFunction["GenericIntegralTransform"][f,z,t,transform] gives the integral transform 𝒯transform[f(z)](t) of the input function f(z), in terms of new variable t. | |
ResourceFunction["GenericIntegralTransform"][] prints a table of all available transforms and their definitions. | |
ResourceFunction["GenericIntegralTransform"][patt] prints a table of available transforms whose names match the string pattern patt. | |
ResourceFunction["GenericIntegralTransform"][…, form] prints a table of transforms with definitions displayed in the specified form. |
GenerateConditions | False | whether to provide conditions under which the given result is valid |
"FoxHForm" | False | whether to provide results (where possible) in terms of the FoxH function |
Fourier transform | "Fourier" | "FourierExp" | |
Fourier cosine transform | "FourierCos" | |
Fourier sine transform | "FourierSin" | |
G-transform | ||
Hankel transform | {"Hankel",ν} | |
Hankel transform with Power | {"Hankel",ν,α} | |
Hartley tranform | "Hartley" | |
Hilbert transform | "Hilbert" | |
Hilbert transform with Power | {"Hilbert",α} | |
Integrate transform | "Integrate" | |
Integrate transform with Power | {"Integrate",α} | |
Laplace transform | "Laplace" | |
Laplace transform with Power | {"Laplace",α} | |
Mellin transform | "Mellin" | |
Mellin transform with Power | {"Mellin",α} | |
Neumann transform | {"Neumann”, ν} | |
Neumann transform with Power | {"Neumann”, ν,α} | |
Riesz transform | {"Riesz”, α} | with Re[α]>0 |
Stieltjes transform | {"Stieltjes”, ρ} | |
Struve transform | {“Struve”, ν} | |
Struve transform with Power | {"Struve”, ν,α} | |
Weyl transform | {“Weyl”, α} |
Compute the Mellin transform of a Bessel function:
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Compute the Mellin transform of an exponential function:
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Get the conditions of convergence for when the result above is valid:
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Generically, the result returned is a ConditionalExpression. The conditions under which this result holds can be expanded by clicking the “+“, followed by “Uniconize”, or simply by examining the InputForm:
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Compute the Laplace transform of a cosine. By default, GenericIntegralTransform gives results in terms of an Inactive MeijerG function:
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Use Activate to allow the Inactive[MeijerG] to evaluate to elementary functions:
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Compute the Hankel transform of an arctangent:
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To evaluate in terms of simpler functions, use Activate and FunctionExpand:
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Some results are available in terms of the FoxH function. To return this form where possible, use the "FoxHForm" option:
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Get a table of all available transforms, with definitions in TraditionalForm:
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List all transforms starting with the character "H", with definitions given in InputForm:
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Compute the G-transform of a sine:
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Get the result in terms of FoxH:
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Compute the Hankel transform of a sine:
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Get the result in terms of FoxH:
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Compute the Hilbert transform of a sine:
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Get the result in terms of FoxH:
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Compute the "Integrate transform" of a sine:
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Get the result in terms of FoxH:
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Compute the Laplace transform of a sine:
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Get the result in terms of FoxH:
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Compute the Liousville transform of a sine:
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Get the result in terms of FoxH:
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Compute the Meijer transform of a sine:
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Get the result in terms of FoxH:
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Compute the Mellin transform of a sine:
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A result in terms of FoxH is not available:
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Compute the Mellin transform of BesselJ:
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Compute the Neumann transform of a sine:
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Get the result in terms of FoxH:
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Compute the Riesz transform of a sine:
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Get the result in terms of FoxH:
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Compute the Stieltjes transform of a sine:
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Get the result in terms of FoxH:
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Compute the Struve transform of a sine:
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Get the result in terms of FoxH:
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Compute a G-transform of a BesselY function:
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Get the result instead in terms of FoxH:
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The default setting GenerateConditions→False returns a result only, without regard to conditions of convergence:
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With GenerateConditions→True, the result can be a ConditionalExpression whose second part gives the conditions of convergence:
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Wolfram Language 13.0 (December 2021) or above
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