FoxHReduce

FoxHReduce[expr,x]

attempts to reduce expr to a single FoxH object as a function of x.

Details

  • FoxH representations of mathematical functions are widely used in the areas of symbolic integration, integral transforms, statistics and others.
  • FoxHReduce will attempt to represent any expression as a FoxH object.
  • FoxHReduce returns results in an inert form Inactive[FoxH][].
  • The original function can be recovered from the result by using Activate. »
  • FoxHReduce automatically threads over lists.
  • Assumptions on parameters may be specified using the Assumptions option.
  • FoxHReduce has properties similar to MeijerGReduce, but it is able to generate a FoxH representation for different functions not representable in terms of MeijerGReduce.

Examples

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Basic Examples  (2)

Represent Sin in terms of FoxH:

Represent BesselJ having a parameter in its argument in terms of FoxH:

Recover the original function using Activate:

Plot this function for different values of a:

Scope  (18)

Elementary Functions  (6)

Represent rational functions in terms of the FoxH function:

Represent algebraic functions in terms of the FoxH function:

Represent trigonometric functions and their combinations in terms of the FoxH function:

Represent hyperbolic functions and their combinations in terms of the FoxH function:

Represent exponential and logarithmic functions in terms of the FoxH function:

Represent inverse trigonometric and hyperbolic functions in terms of the FoxH function:

Special Functions  (5)

Airy functions:

Bessel functions:

Legendre functions:

Hypergeometric functions:

Elliptic functions:

Piecewise Functions  (3)

UnitStep:

UnitBox:

Expressions involving UnitStep:

ConditionalExpression:

Combinations of Special Functions  (2)

Products of elementary functions:

Representation for ExpIntegralEi with a monomial argument:

SinIntegral:

General Functions  (2)

The family of functions e-xaxb has nice and simple FoxH representation:

The family of MittagLeffler functions:

Recover the original function using Activate:

Plot this function:

Options  (1)

Assumptions  (1)

FoxHReduce returns a ConditionalExpression for this example:

Use Assumptions to restrict conditions on the parameter:

Applications  (5)

FoxHReduce outputs the most general representation of special functions in terms of FoxH functions:

The family of MittagLefflerE functions is FoxH representable:

However, these functions are not representable in terms of MeijerG:

For some families of special functions, the FoxH representation is simpler than the MeijerG one:

In this case, MeijerGReduce generates a rather complicated output with two MeijerG functions:

While representation via FoxHReduce is much more simpler:

For certain families, the FoxH representation is more intuitive than the MeijerG representation:

Properties & Relations  (6)

FoxHReduce returns FoxH representation of the function in Inactive form:

Use Activate to evaluate the result:

FoxHReduce maps over sums and products:

FoxHReduce takes lists and matrices as arguments:

FoxHReduce may be regarded as the inverse of FoxH:

FoxHReduce may generate a ConditionalExpression:

FoxHReduce may take an Inactive MeijerG as an input:

Possible Issues  (1)

Some advanced special functions are not represented in terms of FoxH:

Neat Examples  (1)

Create a gallery of FoxH representations for a set of elementary and special functions:

Wolfram Research (2021), FoxHReduce, Wolfram Language function, https://reference.wolfram.com/language/ref/FoxHReduce.html.

Text

Wolfram Research (2021), FoxHReduce, Wolfram Language function, https://reference.wolfram.com/language/ref/FoxHReduce.html.

CMS

Wolfram Language. 2021. "FoxHReduce." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FoxHReduce.html.

APA

Wolfram Language. (2021). FoxHReduce. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FoxHReduce.html

BibTeX

@misc{reference.wolfram_2022_foxhreduce, author="Wolfram Research", title="{FoxHReduce}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/FoxHReduce.html}", note=[Accessed: 03-July-2022 ]}

BibLaTeX

@online{reference.wolfram_2022_foxhreduce, organization={Wolfram Research}, title={FoxHReduce}, year={2021}, url={https://reference.wolfram.com/language/ref/FoxHReduce.html}, note=[Accessed: 03-July-2022 ]}