Compute Definite Integrals Using G Reduction: New in Wolfram Language 11

Compute Definite Integrals Using G Reduction

Expressing functions in terms of MeijerG allows the computation of their product over the positive reals.

Create a rule to express the integral of a product of functions in terms of MeijerG functions.

In[1]:=

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

This integral can be expressed exactly in terms of a single MeijerG expression.

In[2]:=

IntegrateMeijerG[\[Alpha]_ Inactive[MeijerG][{a_, b_}, {c_, 
      d_}, \[Omega]_. z_] Inactive[MeijerG][{e_, f_}, {g_, 
      h_}, \[Eta]_. z_], {z_, 0, Infinity}] /; 
  FreeQ[{\[Alpha], \[Omega], \[Eta]}, 
   z] := \[Alpha] MeijerG[{Join[-c, e], Join[f, d]}, {Join[-a, g], 
    Join[h, -b]}, \[Eta]/\[Omega]]

Apply the scheme to evaluate .

In[3]:=

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

Out[3]=

In[4]:=

IntegrateMeijerG[(1 + z)^(-3/2) EllipticK[-2 z], {z, 0, Infinity}]

Out[4]=

Obtain the same result using Integrate.

In[5]:=

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

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Although the answer looks quite different, it is equivalent.

In[6]:=

IntegrateMeijerG[(1 + z)^(-3/2) EllipticK[-2 z], {z, 0, Infinity}];
Integrate[(1 + z)^(-3/2) EllipticK[-2 z], {z, 0, Infinity}];
FullSimplify[% == %%]

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