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FoxHReduce
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FoxHReduce

[Experimental]

✖
FoxHReduce[expr,x]

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

Details and Options

  • 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

open allclose all

Basic Examples  (2)Summary of the most common use cases

Represent Sin in terms of FoxH:

In[1]:=1
✖
https://wolfram.com/xid/05fgbpn4v2-6ksgj
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/05fgbpn4v2-mcjetp
Out[1]=1

Recover the original function using Activate:

In[2]:=2
✖
https://wolfram.com/xid/05fgbpn4v2-3tu4ld
Out[2]=2

Plot this function for different values of a:

In[3]:=3
✖
https://wolfram.com/xid/05fgbpn4v2-kaa8pi
Out[3]=3

Scope  (18)Survey of the scope of standard use cases

Elementary Functions  (6)

Represent rational functions in terms of the FoxH function:

In[1]:=1
✖
https://wolfram.com/xid/05fgbpn4v2-eznmdi
Out[1]=1
In[3]:=3
✖
https://wolfram.com/xid/05fgbpn4v2-pcwdtc
Out[3]=3

Represent algebraic functions in terms of the FoxH function:

In[1]:=1
✖
https://wolfram.com/xid/05fgbpn4v2-focqkz
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/05fgbpn4v2-cpdwb
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/05fgbpn4v2-cu8sz2
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/05fgbpn4v2-h1xpni
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/05fgbpn4v2-bbwv9
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/05fgbpn4v2-kyse2l
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/05fgbpn4v2-b5vkdo
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/05fgbpn4v2-h0z24u
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/05fgbpn4v2-jn34ur
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/05fgbpn4v2-b06ikj
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/05fgbpn4v2-gtzw0w
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/05fgbpn4v2-iah0or
Out[2]=2

Special Functions  (5)

Airy functions:

In[1]:=1
✖
https://wolfram.com/xid/05fgbpn4v2-cx03rz
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/05fgbpn4v2-ct4nx0
Out[2]=2

Bessel functions:

In[1]:=1
✖
https://wolfram.com/xid/05fgbpn4v2-c0kqix
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/05fgbpn4v2-gw6kjc
Out[2]=2

Legendre functions:

In[1]:=1
✖
https://wolfram.com/xid/05fgbpn4v2-k36ml6
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/05fgbpn4v2-dmozhs
Out[2]=2

Hypergeometric functions:

In[1]:=1
✖
https://wolfram.com/xid/05fgbpn4v2-gks69v
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/05fgbpn4v2-kvvorw
Out[2]=2

Elliptic functions:

In[1]:=1
✖
https://wolfram.com/xid/05fgbpn4v2-xuqss
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/05fgbpn4v2-q6oy7
Out[2]=2

Piecewise Functions  (3)

UnitStep:

In[1]:=1
✖
https://wolfram.com/xid/05fgbpn4v2-gg1lfd
Out[1]=1

UnitBox:

In[2]:=2
✖
https://wolfram.com/xid/05fgbpn4v2-e47g3c
Out[2]=2

Expressions involving UnitStep:

In[1]:=1
✖
https://wolfram.com/xid/05fgbpn4v2-bbc3ez
Out[1]=1

ConditionalExpression:

In[1]:=1
✖
https://wolfram.com/xid/05fgbpn4v2-ejuqu0
Out[1]=1

Combinations of Special Functions  (2)

Products of elementary functions:

In[1]:=1
✖
https://wolfram.com/xid/05fgbpn4v2-e2lofg
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/05fgbpn4v2-ouwfgj
Out[2]=2

Representation for ExpIntegralEi with a monomial argument:

In[1]:=1
✖
https://wolfram.com/xid/05fgbpn4v2-kf3v06
Out[1]=1

SinIntegral:

In[2]:=2
✖
https://wolfram.com/xid/05fgbpn4v2-hwkn8b
Out[2]=2

General Functions  (2)

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

In[1]:=1
✖
https://wolfram.com/xid/05fgbpn4v2-eg29bj
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/05fgbpn4v2-1mv978
Out[2]=2

The family of Mittag–Leffler functions:

In[1]:=1
✖
https://wolfram.com/xid/05fgbpn4v2-ef1rcy
Out[1]=1

Recover the original function using Activate:

In[2]:=2
✖
https://wolfram.com/xid/05fgbpn4v2-jq7ez
Out[2]=2

Plot this function:

In[3]:=3
✖
https://wolfram.com/xid/05fgbpn4v2-04ri77
Out[3]=3

Options  (1)Common values & functionality for each option

Assumptions  (1)

FoxHReduce returns a ConditionalExpression for this example:

In[1]:=1
✖
https://wolfram.com/xid/05fgbpn4v2-b4i3uy
Out[1]=1

Use Assumptions to restrict conditions on the parameter:

In[2]:=2
✖
https://wolfram.com/xid/05fgbpn4v2-u1y6a4
Out[2]=2

Applications  (5)Sample problems that can be solved with this function

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

In[4]:=4
✖
https://wolfram.com/xid/05fgbpn4v2-h67hgi
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/05fgbpn4v2-c3jxf4
Out[5]=5

The family of MittagLefflerE functions is FoxH representable:

In[1]:=1
✖
https://wolfram.com/xid/05fgbpn4v2-z1r8tw
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/05fgbpn4v2-r0cql5
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/05fgbpn4v2-5y6wik
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/05fgbpn4v2-6t9bu3
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/05fgbpn4v2-t1bn59
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/05fgbpn4v2-29v29i
Out[1]=1

While representation via FoxHReduce is much more simpler:

In[2]:=2
✖
https://wolfram.com/xid/05fgbpn4v2-gjiwz
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/05fgbpn4v2-fhax2v
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/05fgbpn4v2-kixyq5
Out[2]=2

Properties & Relations  (6)Properties of the function, and connections to other functions

FoxHReduce returns FoxH representation of the function in Inactive form:

In[1]:=1
✖
https://wolfram.com/xid/05fgbpn4v2-9j9jvv
Out[1]=1

Use Activate to evaluate the result:

In[2]:=2
✖
https://wolfram.com/xid/05fgbpn4v2-2jmlkj
Out[2]=2

FoxHReduce maps over sums and products:

In[1]:=1
✖
https://wolfram.com/xid/05fgbpn4v2-fk1u9d
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/05fgbpn4v2-edcaja
Out[2]=2

FoxHReduce takes lists and matrices as arguments:

In[1]:=1
✖
https://wolfram.com/xid/05fgbpn4v2-y27wiv
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/05fgbpn4v2-gil5u1
Out[2]=2

FoxHReduce may be regarded as the inverse of FoxH:

In[1]:=1
✖
https://wolfram.com/xid/05fgbpn4v2-l1sfu9
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/05fgbpn4v2-279bb
Out[2]=2

FoxHReduce may generate a ConditionalExpression:

In[1]:=1
✖
https://wolfram.com/xid/05fgbpn4v2-g3g3ds
Out[1]=1

FoxHReduce may take an Inactive MeijerG as an input:

In[1]:=1
✖
https://wolfram.com/xid/05fgbpn4v2-tua7fu
Out[1]=1

Possible Issues  (1)Common pitfalls and unexpected behavior

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

In[1]:=1
✖
https://wolfram.com/xid/05fgbpn4v2-9m4k6u
Out[1]=1

Neat Examples  (1)Surprising or curious use cases

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

In[1]:=1
✖
https://wolfram.com/xid/05fgbpn4v2-chq0g5
In[2]:=2
✖
https://wolfram.com/xid/05fgbpn4v2-70w5xz
Wolfram Research (2021), FoxHReduce, Wolfram Language function, https://reference.wolfram.com/language/ref/FoxHReduce.html.
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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.

✖
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.

✖
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

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

BibTeX

@misc{reference.wolfram_2025_foxhreduce, author="Wolfram Research", title="{FoxHReduce}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/FoxHReduce.html}", note=[Accessed: 27-March-2025 ]}

✖
@misc{reference.wolfram_2025_foxhreduce, author="Wolfram Research", title="{FoxHReduce}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/FoxHReduce.html}", note=[Accessed: 27-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_foxhreduce, organization={Wolfram Research}, title={FoxHReduce}, year={2021}, url={https://reference.wolfram.com/language/ref/FoxHReduce.html}, note=[Accessed: 27-March-2025 ]}

✖
@online{reference.wolfram_2025_foxhreduce, organization={Wolfram Research}, title={FoxHReduce}, year={2021}, url={https://reference.wolfram.com/language/ref/FoxHReduce.html}, note=[Accessed: 27-March-2025 ]}