Function Repository Resource:

# ProductIntegrate

Evaluate the product integral of a function

Contributed by: Jan Mangaldan
 ResourceFunction["ProductIntegrate"][f,x] gives the indefinite product integral . ResourceFunction["ProductIntegrate"][f,{x,xmin,xmax}] gives the definite product integral .

## Details

The product integral is also known as the multiplicative integral or geometric integral.
ResourceFunction["ProductIntegrate"] uses the "type II" definition, .
ResourceFunction["ProductIntegrate"] supports the same options as Integrate.

## Examples

### Basic Examples (2)

Indefinite product integral of an exponential function:

 In[1]:=
 Out[1]=

Definite product integral of an exponential function:

 In[2]:=
 Out[2]=

### Scope (3)

Evaluate the indefinite product integral of a power function:

 In[3]:=
 Out[3]=

Use Assuming to get a simpler expression:

 In[4]:=
 Out[4]=

This is the same as using the Assumptions option:

 In[5]:=
 Out[5]=

### Options (2)

#### Assumptions (2)

By default, conditions are generated on parameters that guarantee convergence:

 In[6]:=
 Out[6]=

With Assumptions, a result valid under the given assumptions is given:

 In[7]:=
 Out[7]=

### Properties and Relations (3)

Evaluate the indefinite product integral of a linear function:

 In[8]:=
 Out[8]=

Use the fundamental theorem of product calculus:

 In[9]:=
 Out[9]=

This is the same as directly evaluating a definite product integral:

 In[10]:=
 Out[10]=

ProductIntegrate is the inverse of the resource function ProductD, under certain conditions:

 In[11]:=
 Out[11]=
 In[12]:=
 Out[12]=

ProductIntegrate uses Integrate internally, and if the underlying Integrate fails to evaluate, the expression is left unevaluated:

 In[13]:=
 Out[13]=

## Version History

• 1.0.0 – 08 March 2021