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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. |
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 |
G-transform | {"G", {{af1_List, ae1_List}, {bf1_List, be1_List}}} | |
Hankel transform | {"Hankel", {α_, ν_}} | |
Hilbert transform | {"Hilbert", α_} | |
Integrate transform | {"Integrate", α_} | put |
Laplace transform | {"Laplace", α_} | typeset |
Liouville transform | {"Liouville", α_} | definitions |
Meijer transform | {"Meijer", {ν_, α_}} | here |
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_} | |
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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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 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 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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This work is licensed under a Creative Commons Attribution 4.0 International License