FoxHReduce
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 all close allBasic Examples (2)
Represent Sin in terms of FoxH:
FoxHReduce[Sin[a x], x]Represent BesselJ having a parameter in its argument in terms of FoxH:
FoxHReduce[BesselJ[1, a x], x]Recover the original function using Activate:
Activate[%]Plot this function for different values of a:
Plot[Evaluate[Table[%, {a, 1, 4}]], {x, 0, 7}]Scope (18)
Elementary Functions (6)
Represent rational functions in terms of the FoxH function:
FoxHReduce[(1/x^2 + 1), x]FoxHReduce[(1/2 x + 3), x]Represent algebraic functions in terms of the FoxH function:
FoxHReduce[(1/Sqrt[x] + 1), x]FoxHReduce[(1/Sqrt[Sqrt[x] + Sqrt[1 + x]]), x]Represent trigonometric functions and their combinations in terms of the FoxH function:
FoxHReduce[Sin[a x], x]FoxHReduce[Cos[a x], x]FoxHReduce[2 Sin[x] + Cos[x], x]Represent hyperbolic functions and their combinations in terms of the FoxH function:
FoxHReduce[Sinh[b x], x]FoxHReduce[Cosh[b x], x]FoxHReduce[2 Sinh[x] + Cosh[x], x]Represent exponential and logarithmic functions in terms of the FoxH function:
FoxHReduce[E^c x, x]FoxHReduce[Log[x], x]Represent inverse trigonometric and hyperbolic functions in terms of the FoxH function:
FoxHReduce[ArcSin[x], x]FoxHReduce[ArcTanh[x], x]Special Functions (5)
FoxHReduce[AiryAi[x], x]FoxHReduce[AiryBi[x], x]FoxHReduce[BesselJ[n, x], x]FoxHReduce[BesselK[n, x], x]FoxHReduce[LegendreP[n, x], x]FoxHReduce[LegendreQ[n, x], x]FoxHReduce[Hypergeometric1F1[a, b, x], x]FoxHReduce[Hypergeometric2F1[a, b, c, x], x]FoxHReduce[EllipticK[m], m]FoxHReduce[EllipticE[m], m]Piecewise Functions (3)
FoxHReduce[UnitStep[x - 1], x]FoxHReduce[UnitBox[3x], x]Expressions involving UnitStep:
FoxHReduce[E^-x UnitStep[1 - Abs[x]], x]FoxHReduce[ConditionalExpression[E^-x, 1 < x < 2], x]Combinations of Special Functions (2)
Products of elementary functions:
FoxHReduce[Cosh[x] Sin[x], x]FoxHReduce[(ArcSin[x]/Sqrt[1 - x^2]), x]Representation for ExpIntegralEi with a monomial argument:
FoxHReduce[ExpIntegralEi[5 x^3], x, Assumptions -> x > 0]FoxHReduce[SinIntegral[12 x^1 / 3], x]General Functions (2)
The family of functions e-xaxb has nice and simple FoxH representation:
FoxHReduce[E^-x^a x^b, x]FoxHReduce[E^-x x^b, x]The family of Mittag–Leffler functions:
FoxHReduce[MittagLefflerE[3 / 2, x], x]Recover the original function using Activate:
Activate[%]Plot[{%}, {x, -40, 2}, PlotRange -> Automatic]Options (1)
Assumptions (1)
FoxHReduce returns a ConditionalExpression for this example:
FoxHReduce[MittagLefflerE[a, x], x]Use Assumptions to restrict conditions on the parameter:
FoxHReduce[MittagLefflerE[a, x], x, Assumptions -> a > 0]Applications (5)
FoxHReduce outputs the most general representation of special functions in terms of FoxH functions:
FoxHReduce[Sin[x], x]FoxHReduce[Hypergeometric1F1[a, b, x], x]The family of MittagLefflerE functions is FoxH representable:
FoxHReduce[MittagLefflerE[1 / 2, x], x]FoxHReduce[MittagLefflerE[a, b, x], x]However, these functions are not representable in terms of MeijerG:
MeijerGReduce[MittagLefflerE[a, x], x]For some families of special functions, the FoxH representation is simpler than the MeijerG one:
MeijerGReduce[E^-x x^b, x]FoxHReduce[E^-x x^b, x]In this case, MeijerGReduce generates a rather complicated output with two MeijerG functions:
MeijerGReduce[E^x^2 Erfc[-x], x]While representation via FoxHReduce is much more simpler:
FoxHReduce[E^x^2 Erfc[-x], x]For certain families, the FoxH representation is more intuitive than the MeijerG representation:
MeijerGReduce[BesselJ[n, x], x]FoxHReduce[BesselJ[n, x], x]Properties & Relations (6)
FoxHReduce returns FoxH representation of the function in Inactive form:
FoxHReduce[Sin[x], x]Use Activate to evaluate the result:
Activate[%]FoxHReduce maps over sums and products:
FoxHReduce[E^-x x^b + E^x^2 Erfc[-x], x]FoxHReduce[5 + 4 x^bMittagLefflerE[2, b, x], x]FoxHReduce takes lists and matrices as arguments:
FoxHReduce[{Sin[x], E^-x^2}, x]FoxHReduce[(| | |
| :------------ | :------------- |
| Cos[x] | HermiteH[n, x] |
| BesselK[n, x] | Exp[x] |), x]FoxHReduce may be regarded as the inverse of FoxH:
FoxH[{{{1 / 2, 1}}, {}}, {{{0, 1}, {1 / 3, 1}}, {}}, x]FoxHReduce[%, x]FoxHReduce may generate a ConditionalExpression:
FoxHReduce[MittagLefflerE[a, x], x]FoxHReduce may take an Inactive MeijerG as an input:
FoxHReduce[Inactive[MeijerG][{{}, {}}, {{(1/2)}, {0}}, (a x/2), (1/2)], x]Possible Issues (1)
Some advanced special functions are not represented in terms of FoxH:
FoxHReduce[HeunG[a, q, α, β, γ, δ, z], z]Neat Examples (1)
Create a gallery of FoxH representations for a set of elementary and special functions:
funclist = {E^-x, x^aE^-x^2, Sin[x], Cosh[Sqrt[x]], BesselJ[n, x], AiryAi[x], Hypergeometric0F1[a, x], Hypergeometric2F1[a, b, c, x], MeijerG[{{Subscript[a, 1]}, {Subscript[a, 2]}}, {{Subscript[b, 1]}, {Subscript[b, 2]}}, x], MeijerG[{{Subscript[a, 1], Subscript[a, 2]}, {Subscript[a, 2]}}, {{Subscript[b, 1]}, {Subscript[b, 2]}}, x, 3], MittagLefflerE[13, x], x MittagLefflerE[13, 2, x]};Grid[...]//TraditionalFormText
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