Wolfram Language Paclet Repository

Community-contributed installable additions to the Wolfram Language

WolframAlphaMath/SpecialFunctionsAndCalculus

(1.7) current version: 1.15 »

Compute MeijerG forms of functions, fractional-order integro-derivatives, integral transforms and more

Contributed by: Paco Jain & Oleg Marichev

This paclet contains tools related to special functions and calculus. Operations include conversion off univariate function expressions into MeijerG forms, fractional-order differentiation, integral transforms and more.

Installation Instructions

To install this paclet in your Wolfram Language environment, evaluate this code:
PacletInstall["WolframAlphaMath/SpecialFunctionsAndCalculus"]


To load the code after installation, evaluate this code:
Needs["WolframAlphaMath`SpecialFunctionsAndCalculus`"]

Details

Authors: Paco Jain and Oleg Marichev

Examples

Basic Examples

Load the paclet in the current Wolfram session:

In[1]:=
Needs["WolframAlphaMath`SpecialFunctionsAndCalculus`"]

Get an example univariate function to work with:

In[2]:=
myFunc = InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.0025`], List[FaceForm[List[RGBColor[0.9607843137254902`, 0.5058823529411764`, 0.19607843137254902`], Opacity[1.`]]], FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[1, 3, 3], List[0, 1, 0], List[1, 3, 3], List[0, 1, 0], List[1, 3, 3], List[0, 1, 0], List[1, 3, 3], List[1, 3, 3], List[0, 1, 0], List[1, 3, 3], List[0, 1, 0], List[1, 3, 3]]], List[List[List[205.`, 22.863691329956055`], List[205.`, 212.31669425964355`], List[246.01799774169922`, 235.99870109558105`], List[369.0710144042969`, 307.0436840057373`], List[369.0710144042969`, 117.59068870544434`], List[205.`, 22.863691329956055`]], List[List[30.928985595703125`, 307.0436840057373`], List[153.98200225830078`, 235.99870109558105`], List[195.`, 212.31669425964355`], List[195.`, 22.863691329956055`], List[30.928985595703125`, 117.59068870544434`], List[30.928985595703125`, 307.0436840057373`]], List[List[200.`, 410.42970085144043`], List[364.0710144042969`, 315.7036876678467`], List[241.01799774169922`, 244.65868949890137`], List[200.`, 220.97669792175293`], List[158.98200225830078`, 244.65868949890137`], List[35.928985595703125`, 315.7036876678467`], List[200.`, 410.42970085144043`]], List[List[376.5710144042969`, 320.03370475769043`], List[202.5`, 420.53370475769043`], List[200.95300006866455`, 421.42667961120605`], List[199.04699993133545`, 421.42667961120605`], List[197.5`, 420.53370475769043`], List[23.428985595703125`, 320.03370475769043`], List[21.882003784179688`, 319.1406993865967`], List[20.928985595703125`, 317.4896984100342`], List[20.928985595703125`, 315.7036876678467`], List[20.928985595703125`, 114.70369529724121`], 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Out[2]=

Compute the (inactivated) MeijerG form of the function:

In[3]:=
InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.0025`], List[FaceForm[List[RGBColor[0.9607843137254902`, 0.5058823529411764`, 0.19607843137254902`], Opacity[1.`]]], FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[1, 3, 3], List[0, 1, 0], List[1, 3, 3], List[0, 1, 0], List[1, 3, 3], List[0, 1, 0], List[1, 3, 3], List[1, 3, 3], List[0, 1, 0], List[1, 3, 3], List[0, 1, 0], List[1, 3, 3]]], List[List[List[205.`, 22.863691329956055`], List[205.`, 212.31669425964355`], List[246.01799774169922`, 235.99870109558105`], List[369.0710144042969`, 307.0436840057373`], List[369.0710144042969`, 117.59068870544434`], List[205.`, 22.863691329956055`]], List[List[30.928985595703125`, 307.0436840057373`], List[153.98200225830078`, 235.99870109558105`], List[195.`, 212.31669425964355`], List[195.`, 22.863691329956055`], List[30.928985595703125`, 117.59068870544434`], List[30.928985595703125`, 307.0436840057373`]], List[List[200.`, 410.42970085144043`], List[364.0710144042969`, 315.7036876678467`], List[241.01799774169922`, 244.65868949890137`], List[200.`, 220.97669792175293`], List[158.98200225830078`, 244.65868949890137`], List[35.928985595703125`, 315.7036876678467`], List[200.`, 410.42970085144043`]], List[List[376.5710144042969`, 320.03370475769043`], List[202.5`, 420.53370475769043`], List[200.95300006866455`, 421.42667961120605`], List[199.04699993133545`, 421.42667961120605`], List[197.5`, 420.53370475769043`], List[23.428985595703125`, 320.03370475769043`], List[21.882003784179688`, 319.1406993865967`], List[20.928985595703125`, 317.4896984100342`], List[20.928985595703125`, 315.7036876678467`], List[20.928985595703125`, 114.70369529724121`], 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Out[3]=

Compute the ath-order derivative of the function:

In[4]:=
InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.0025`], List[FaceForm[List[RGBColor[0.9607843137254902`, 0.5058823529411764`, 0.19607843137254902`], Opacity[1.`]]], FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[1, 3, 3], List[0, 1, 0], List[1, 3, 3], List[0, 1, 0], List[1, 3, 3], List[0, 1, 0], List[1, 3, 3], List[1, 3, 3], List[0, 1, 0], List[1, 3, 3], List[0, 1, 0], List[1, 3, 3]]], List[List[List[205.`, 22.863691329956055`], List[205.`, 212.31669425964355`], List[246.01799774169922`, 235.99870109558105`], List[369.0710144042969`, 307.0436840057373`], List[369.0710144042969`, 117.59068870544434`], List[205.`, 22.863691329956055`]], List[List[30.928985595703125`, 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Out[4]=

Compare with the known derivative in the case a→1:

In[5]:=
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Out[5]=
In[6]:=
D[myFunc, \[FormalZ]]
Out[6]=

Compute the Mellin transform of the function:

In[7]:=
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Style[Defer[PacletSymbol["WolframAlphaMath/SpecialFunctionsAndCalculus", "WolframAlphaMath`SpecialFunctionsAndCalculus`GenericIntegralTransform"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["WolframAlphaMath/SpecialFunctionsAndCalculus", "WolframAlphaMath`SpecialFunctionsAndCalculus`GenericIntegralTransform"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][myFunc, \[FormalZ], \[FormalT], "Mellin", GenerateConditions -> True]
Out[7]=

Confirm the above result using explicit Mellin integration:

In[8]:=
Integrate[\[FormalZ]^(\[FormalT] - 1) myFunc, {\[FormalZ], 0, Infinity}]
Out[8]=

Publisher

Wolfram|Alpha Math Team

Version History

  • 1.15 – 05 November 2024
  • 1.14.9 – 16 October 2024
  • 1.14.8 – 16 October 2024
  • 1.14.7 – 16 October 2024
  • 1.14.6 – 16 October 2024
  • 1.14.5 – 15 October 2024
  • 1.14.4 – 14 October 2024
  • 1.14.3 – 14 October 2024
  • 1.14.2 – 14 October 2024
  • 1.14.1 – 08 October 2024
  • 1.14 – 07 October 2024
  • 1.13.2 – 27 September 2024
  • 1.13.1 – 18 September 2024
  • 1.13 – 18 September 2024
  • 1.12.2 – 10 September 2024
  • 1.12.1 – 06 September 2024
  • 1.12 – 29 August 2024
  • 1.11.3 – 13 August 2024
  • 1.11.2 – 08 August 2024
  • 1.11.1 – 05 August 2024
  • 1.11 – 30 May 2024
  • 1.10.1 – 04 April 2024
  • 1.10 – 15 March 2024
  • 1.9 – 16 January 2024
  • 1.8.2 – 27 October 2023
  • 1.8.1 – 27 October 2023
  • 1.8 – 23 October 2023
  • 1.7.10 – 02 October 2023
  • 1.7.9 – 11 September 2023
  • 1.7.8 – 11 September 2023
  • 1.7.7 – 09 September 2023
  • 1.7.6 – 11 August 2023
  • 1.7.5 – 04 August 2023
  • 1.7.4 – 04 August 2023
  • 1.7.3 – 03 August 2023
  • 1.7.2 – 14 July 2023
  • 1.7.1 – 30 June 2023
  • 1.7 – 17 May 2023
  • 1.6.6 – 02 May 2023
  • 1.6.5 – 06 April 2023
  • 1.6.4 – 10 January 2023
  • 1.6.3 – 10 January 2023
  • 1.6.2 – 10 January 2023
  • 1.6.1 – 09 January 2023
  • 1.6 – 09 January 2023
  • 1.5.2 – 28 December 2022
  • 1.5.1 – 13 December 2022
  • 1.5 – 13 December 2022
  • 1.4 – 29 November 2022
  • 1.3.2 – 07 November 2022
  • 1.3.1 – 07 November 2022
  • 1.3.0 – 04 November 2022
  • 1.2.7 – 21 October 2022

License Information

MIT License

Paclet Source

Source Metadata

  • Source: https://stash.wolfram.com/users/pacoj/repos/specialfunctionsandcalculus/browse

See Also