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Function Repository Resource:

 MellinBarnesIntegrate

Source Notebook

Integrate a mathematical function using Mellin–Barnes integration

Contributed by: Paco Jain & Oleg Marichev (Wolfram Research)

ResourceFunction["MellinBarnesIntegrate"][f,x]

gives the indefinite integral using the Mellin-Barnes method.

ResourceFunction["MellinBarnesIntegrate"][f,{x,0,Infinity}]

gives the definite integral ⅆx .

ResourceFunction["MellinBarnesIntegrate"][f,{x,-Infinity,0}]

gives the definite integral ⅆx .

ResourceFunction["MellinBarnesIntegrate"][f,{x,-Infinity,Infinity}]

gives the definite integral ⅆx .

Details

MellinBarnesIntegrate returns its results in terms of inactivated FoxH and MeijerG functions. These can often be re-expressed in terms of more basic functions by applying Activate and, in some cases, FunctionExpand.
Mellin-Barnes integration is also refered to as Slater convolution. It was heavily developed by Marichev and Adamchik.
ResourceFunction["MellinBarnesIntegrate"] may give results over a finite range. It is primarily intended for infinite ranges.

Examples

Basic Examples (2) 

Compute an indefinite integral:

In[1]:=
ResourceFunction["MellinBarnesIntegrate"][Sin[a x], x]
Out[1]=

Activate the MeijerG function to obtain an answer in terms of elementary functions:

In[2]:=
Activate[%]
Out[2]=

Compute an definite integral:

In[3]:=
ResourceFunction["MellinBarnesIntegrate"][ 1/(x^3 + 1), {x, 0, Infinity}]
Out[3]=

Use FunctionExpand to evaluate the gamma functions:

In[4]:=
FunctionExpand[%]
Out[4]=

Confirm the result via Integrate:

In[5]:=
Integrate[1/(x^3 + 1), {x, 0, Infinity}]
Out[5]=

Scope (13) 

Basic Uses (5) 

Compute an indefinite integral:

In[6]:=
int = ResourceFunction["MellinBarnesIntegrate"][1/(x^3 + 1), x]
Out[6]=

Verify the answer by differentiation:

In[7]:=
D[int, x] // Activate
Out[7]=

Compute a definite integral over an infinite interval:

In[8]:=
ResourceFunction["MellinBarnesIntegrate"][ BesselJ[2, x]/(1 + x^2), {x, 0, \[Infinity]}]
Out[8]=

Use FunctionExpand and Activate to obtain the result in terms of more basic functions:

In[9]:=
FunctionExpand[Activate[%]]
Out[9]=

Compute an integral over a doubly-infinite interval:

In[10]:=
ResourceFunction["MellinBarnesIntegrate"][ Sin[x]/x, {x, -\[Infinity], \[Infinity]}]
Out[10]=

Compute an integral over the negative real line:

In[11]:=
ResourceFunction["MellinBarnesIntegrate"][ Exp[-x^2], {x, -\[Infinity], 0}]
Out[11]=

An integrand containing symbolic parameters:

In[12]:=
res = ResourceFunction["MellinBarnesIntegrate"][ integrand = BesselJ[1, a x]/(x^2 + c), {x, 0, Infinity}]
Out[12]=

Compare with the results of NIntegrate:

In[13]:=
Block[{a = RandomReal[], c = RandomReal[]}, {N[Activate[res]], NIntegrate[integrand, {x, 0, Infinity}]} ]
Out[13]=

Elementary Functions (4) 

Integrate rational functions:

In[14]:=
ResourceFunction["MellinBarnesIntegrate"][1/(x^2 + 1), x]
Out[14]=
In[15]:=
ResourceFunction["MellinBarnesIntegrate"][1/(2 x + 3), x]
Out[15]=
In[16]:=
ResourceFunction["MellinBarnesIntegrate"][ x/(x^3 + b), {x, 0, \[Infinity]}]
Out[16]=

Integrate algebraic functions:

In[17]:=
ResourceFunction["MellinBarnesIntegrate"][1/(Sqrt[x] + 1), x]
Out[17]=

Integrate trigonometric functions:

In[18]:=
ResourceFunction["MellinBarnesIntegrate"][Sin[a x], x]
Out[18]=
In[19]:=
ResourceFunction["MellinBarnesIntegrate"][Cos[a x + d], x]
Out[19]=
In[20]:=
ResourceFunction["MellinBarnesIntegrate"][Cot[a x], x] // FunctionExpand
Out[20]=

Integrate inverse trigonometric and hyperbolic functions:

In[21]:=
ResourceFunction["MellinBarnesIntegrate"][ArcSin[x], x]
Out[21]=
In[22]:=
ResourceFunction["MellinBarnesIntegrate"][ArcTanh[x], x]
Out[22]=

Special Functions (4) 

Integrate Airy functions:

In[23]:=
ResourceFunction["MellinBarnesIntegrate"][AiryAi[x], x]
Out[23]=
In[24]:=
ResourceFunction["MellinBarnesIntegrate"][AiryBi[x], x]
Out[24]=

Integrate Bessel functions:

In[25]:=
ResourceFunction["MellinBarnesIntegrate"][BesselJ[n, x], x]
Out[25]=
In[26]:=
ResourceFunction["MellinBarnesIntegrate"][BesselK[n, x], x]
Out[26]=

Integrate hypergeometric functions:

In[27]:=
ResourceFunction["MellinBarnesIntegrate"][ Hypergeometric2F1[a, b, c, x], x]
Out[27]=

Integrate elliptic integrals:

In[28]:=
ResourceFunction["MellinBarnesIntegrate"][EllipticK[m], m]
Out[28]=
In[29]:=
ResourceFunction["MellinBarnesIntegrate"][EllipticE[m], m]
Out[29]=

Publisher

Wolfram|Alpha Math Team

Version History

  • 1.0.0 – 20 October 2025

Source Metadata

  • Citation:
    • V. S. Adamchik and O. I. Marichev. 1990. The algorithm for calculating integrals of hypergeometric type functions and its realization in REDUCE system. In Proceedings of the international symposium on Symbolic and algebraic computation (ISSAC '90). Association for Computing Machinery, New York, NY, USA, 212–224. https://doi.org/10.1145/96877.96930
    • Slater, Lucy Joan (1966). Generalized Hypergeometric Functions. Cambridge, UK: Cambridge University Press. ISBN 0-521-06483-X.

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