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Function Repository Resource:

FractionalIntegrate

Source Notebook

Calculate fractional integrals

Contributed by: Sami Yrjänheikki

ResourceFunction["FractionalIntegrate"][f,{x,α}]

gives the α-fractional integral of the function f with respect to variable x.

Details and Options

If α=1, ResourceFunction["FractionalIntegrate"][f,{x,α}] is equivalent to Integrate[f,x].
FractionalIntegrate uses the formula .

Examples

Basic Examples

Semi-integral of 2x:

In[1]:=
ResourceFunction["FractionalIntegrate"][2 x, {x, 1/2}]
Out[1]=

Taking the semi-integral twice yields the standard antiderivative:

In[2]:=
ResourceFunction["FractionalIntegrate"][
 ResourceFunction["FractionalIntegrate"][2 x, {x, 1/2}], {x, 1/2}]
Out[2]=

Scope

If α=1, the result is the usual integral:

In[3]:=
ResourceFunction["FractionalIntegrate"][Cos[x], {x, 1}] == Integrate[Cos[x], x]
Out[3]=

Semi-integral of elementary functions:

In[4]:=
ResourceFunction["FractionalIntegrate"][x^a, {x, 1/2}]
Out[4]=
In[5]:=
ResourceFunction["FractionalIntegrate"][Exp[x], {x, 1/2}]
Out[5]=
In[6]:=
ResourceFunction["FractionalIntegrate"][Log[x], {x, 1/2}]
Out[6]=

Semi-integral of trig functions:

In[7]:=
ResourceFunction["FractionalIntegrate"][Sin[x], {x, 1/2}]
Out[7]=
In[8]:=
ResourceFunction["FractionalIntegrate"][Cos[x], {x, 1/2}]
Out[8]=

Semi-integral of inverse trig functions:

In[9]:=
ResourceFunction["FractionalIntegrate"][ArcSin[x], {x, 1/2}]
Out[9]=
In[10]:=
ResourceFunction["FractionalIntegrate"][ArcCos[x], {x, 1/2}]
Out[10]=

More exotic fractional integrals:

In[11]:=
ResourceFunction["FractionalIntegrate"][
  FresnelC[x], {x, 2/3}] // FullSimplify
Out[11]=

Possible Issues

Not all integrals can be computed:

In[12]:=
ResourceFunction["FractionalIntegrate"][Tan[x], {x, 1/2}]
Out[12]=

Resource History

Related Resources

License Information