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MeijerGReduce[expr,x]

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

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Elementary Functions  
Special Functions  
Piecewise Functions  
General Functions  
Options  
Assumptions  
Applications  
Properties & Relations  
Neat Examples  
See Also
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MeijerGReduce

MeijerGReduce[expr,x]

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

Details and Options

  • MeijerG representations of mathematical functions are widely used in the areas of symbolic integration, integral transforms, statistics and others.
  • MeijerGReduce will attempt to represent any expression as a MeijerG object.
  • MeijerGReduce returns results in an inert form Inactive[MeijerG][…].
  • MeijerGReduce automatically threads over lists.
  • Assumptions on parameters may be specified using the Assumptions option.

Examples

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Basic Examples  (2)

Represent Sin in terms of MeijerG:

Wolfram Language code: MeijerGReduce[Sin[x], x]

Represent BesselJ in terms of MeijerG:

Wolfram Language code: MeijerGReduce[BesselJ[1, a x], x]

Recover the original function using Activate:

Wolfram Language code: Activate[%]
Wolfram Language code: Plot[Evaluate[Table[%, {a, 1, 4}]], {x, 0, 7}]

Scope  (18)

Elementary Functions  (6)

Rational functions:

Wolfram Language code: MeijerGReduce[(1/x^2 + 1), x]
Wolfram Language code: MeijerGReduce[(1/2 x + 3), x]

Algebraic functions:

Wolfram Language code: MeijerGReduce[(1/Sqrt[x] + 1), x]
Wolfram Language code: MeijerGReduce[(1/Sqrt[Sqrt[x] + Sqrt[1 + x]]), x]

Trigonometric functions:

Wolfram Language code: MeijerGReduce[Sin[a x], x]
Wolfram Language code: MeijerGReduce[Cos[a x], x]

Linear combination of trigonometric functions:

Wolfram Language code: MeijerGReduce[2 Sin[x] + Cos[x], x]

Hyperbolic functions:

Wolfram Language code: MeijerGReduce[Sinh[b x], x]
Wolfram Language code: MeijerGReduce[Cosh[b x], x]

Linear combination of hyperbolic functions:

Wolfram Language code: MeijerGReduce[2 Sinh[x] + Cosh[x], x]

Exponential and logarithmic functions:

Wolfram Language code: MeijerGReduce[E^c x, x]
Wolfram Language code: MeijerGReduce[Log[x], x]

Inverse trigonometric and hyperbolic functions:

Wolfram Language code: MeijerGReduce[ArcSin[x], x]
Wolfram Language code: MeijerGReduce[ArcTanh[x], x]

Special Functions  (5)

Airy functions:

Wolfram Language code: MeijerGReduce[AiryAi[x], x]
Wolfram Language code: MeijerGReduce[AiryBi[x], x]

Bessel functions:

Wolfram Language code: MeijerGReduce[BesselJ[n, x], x]
Wolfram Language code: MeijerGReduce[BesselK[n, x], x]

Legendre functions:

Wolfram Language code: MeijerGReduce[LegendreP[n, x], x]
Wolfram Language code: MeijerGReduce[LegendreQ[n, x], x]

Hypergeometric functions:

Wolfram Language code: MeijerGReduce[Hypergeometric1F1[a, b, x], x]
Wolfram Language code: MeijerGReduce[Hypergeometric2F1[a, b, c, x], x]

Elliptic functions:

Wolfram Language code: MeijerGReduce[EllipticK[m], m]
Wolfram Language code: MeijerGReduce[EllipticE[m], m]

Piecewise Functions  (3)

UnitStep:

Wolfram Language code: MeijerGReduce[UnitStep[x - 1], x]

UnitBox:

Wolfram Language code: MeijerGReduce[UnitBox[3x], x]

Expressions involving UnitStep:

Wolfram Language code: MeijerGReduce[x Log[x]UnitStep[1 - Abs[x]], x]
Wolfram Language code: MeijerGReduce[ArcCsc[x]UnitStep[Abs[x] - 1], x]

ConditionalExpression:

Wolfram Language code: MeijerGReduce[ConditionalExpression[E^-x, 1 < x < 2], x]

General Functions  (4)

Products of elementary functions:

Wolfram Language code: MeijerGReduce[Cosh[x] Sin[x], x]
Wolfram Language code: MeijerGReduce[(ArcSin[x]/Sqrt[1 - x^2]), x]

Product of exponential and Airy functions:

Wolfram Language code: MeijerGReduce[E^xAiryBi[((3/2))^2 / 3 x^2 / 3], x]

Products involving Bessel functions:

Wolfram Language code: MeijerGReduce[BesselJ[2ν, x] BesselJ[2μ, x], x]
Wolfram Language code: MeijerGReduce[E^-xBesselJ[2ν, x] BesselJ[2μ, x] Sin[x], x]

Representation for ExpIntegralEi with a monomial argument:

Wolfram Language code: MeijerGReduce[ExpIntegralEi[5 x^3], x, Assumptions -> x > 0]

SinIntegral:

Wolfram Language code: MeijerGReduce[SinIntegral[12 x^1 / 3], x]

Options  (1)

Assumptions  (1)

MeijerGReduce returns a ConditionalExpression for this example:

Wolfram Language code: MeijerGReduce[AiryAi[a x], x]

Use Assumptions to restrict conditions on the parameter:

Wolfram Language code: MeijerGReduce[AiryAi[a x], x, Assumptions -> a > 0]

Applications  (1)

Build a simple scheme for integration over the interval , using MeijerGReduce:

Wolfram Language code: IntegrateMeijerG[f_ g_, {z_, 0, Infinity}] /; FreeQ[{f, g}, MeijerG] := IntegrateMeijerG[MeijerGReduce[f, z]MeijerGReduce[g, z], {z, 0, Infinity}]

Add a rule for integrating products of inactive MeijerG expressions:

Wolfram Language code: IntegrateMeijerG[α_ Inactive[MeijerG][{a_, b_}, {c_, d_}, ω_. z_] Inactive[MeijerG][{e_, f_}, {g_, h_}, η_.z_], {z_, 0, Infinity}] /; FreeQ[{α, ω, η}, z] := α MeijerG[{Join[-c, e], Join[f, d]}, {Join[-a, g], Join[h, -b]}, η / ω]

Apply the scheme to evaluate int_0^infty(z+1)^(-3/2) TemplateBox[{{-, z}}, EllipticK]dz:

Wolfram Language code: Plot[(1/(1 + z)^3 / 2)EllipticK[-z], {z, 0, 10}, Filling -> Axis, PlotRange -> All]
Wolfram Language code: IntegrateMeijerG[(1/(1 + z)^3 / 2)EllipticK[-z], {z, 0, Infinity}]

Obtain the same result using Integrate:

Wolfram Language code: Integrate[(1/(1 + z)^3 / 2)EllipticK[-z], {z, 0, Infinity}]

Properties & Relations  (4)

MeijerGReduce returns results in Inactive form to prevent evaluation of MeijerG:

Wolfram Language code: MeijerGReduce[E^-x, x]

Use Activate to evaluate the result:

Wolfram Language code: Activate[%]

MeijerGReduce maps over sums and products:

Wolfram Language code: MeijerGReduce[E^-x + Cos[x], x]
Wolfram Language code: MeijerGReduce[5 E^-x, x]

MeijerGReduce may be regarded as the inverse of MeijerG:

Wolfram Language code: MeijerG[{{(1/2), (1/2), (1/2)}, {}}, {{0}, {-(3/2), -(3/2)}}, -x]
Wolfram Language code: MeijerGReduce[%, x]

The result from MeijerGReduce can be used in MellinTransform:

Wolfram Language code: MeijerGReduce[E^-x, x]
Wolfram Language code: MellinTransform[%, x, s]

Obtain the Mellin transform directly:

Wolfram Language code: MellinTransform[E^-x, x, s]

Neat Examples  (1)

Create a gallery of MeijerG representations:

Wolfram Language code: funclist = {E^-x, Sin[x], Cos[x], Sinh[x], Cosh[x], BesselJ[n, x], BesselK[n, x], AiryAi[x], AiryBi[x], Hypergeometric2F1[a, b, c, x], UnitStep[x - 1], UnitBox[x]};
Wolfram Language code: Grid[Join[{{f[x], Text["MeijerG representation"]}}, Transpose[{funclist, HoldForm /@ (MeijerGReduce[#1, x]&) /@ funclist /.  Inactive[MeijerG] -> MeijerG}]], IconizedObject[«Grid options»]]//TraditionalForm
Wolfram Research (2016), MeijerGReduce, Wolfram Language function, https://reference.wolfram.com/language/ref/MeijerGReduce.html.

Text

Wolfram Research (2016), MeijerGReduce, Wolfram Language function, https://reference.wolfram.com/language/ref/MeijerGReduce.html.

CMS

Wolfram Language. 2016. "MeijerGReduce." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/MeijerGReduce.html.

APA

Wolfram Language. (2016). MeijerGReduce. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MeijerGReduce.html

BibTeX

@misc{reference.wolfram_2026_meijergreduce, author="Wolfram Research", title="{MeijerGReduce}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/MeijerGReduce.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_meijergreduce, organization={Wolfram Research}, title={MeijerGReduce}, year={2016}, url={https://reference.wolfram.com/language/ref/MeijerGReduce.html}, note=[Accessed: 12-June-2026]}