Wolfram Function Repository

Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

 PolyLogToIntegral

Source Notebook

Convert polylogarithmic functions and related zeta functions to inactive integral representations

Contributed by: Jayanta Phadikar

ResourceFunction["PolyLogToIntegral"][expr]

converts expr, such as a PolyLog,MultiplePolyLog,Zeta,HurwitzZeta,MultipleZeta,HarmonicPolyLog or GeneralizedPolyLog expression, to an inactive Integrate representation.

Details

ResourceFunction["PolyLogToIntegral"] returns an inactive integral representation of a polylogarithmic or zeta-type expression.
The result is built with Inactive[Integrate], so it displays as an integral but is not evaluated automatically.
Use Activate on the result to turn the inactive integral into an ordinary Integrate expression.
The function supports ordinary PolyLog expressions and the related forms shown in the examples, including Zeta, HurwitzZeta, MultipleZeta, MultiplePolyLog, HarmonicPolyLog and GeneralizedPolyLog.
Higher-order and higher-depth inputs generally produce iterated integrals. The number of integration variables grows with the weight or depth of the expression.
Formal integration variables, such as x and subscripted formal variables, are used to reduce clashes with variables already present in the input expression.
The returned integral should be interpreted on a domain where the integral representation is valid, or as a formal inactive representation when additional analytic-continuation assumptions are required.

Examples

Basic Examples (2) 

Convert a classical polylogarithm to an iterated inactive integral:

In[1]:=
ResourceFunction["PolyLogToIntegral"][PolyLog[3, 1/2]]
Out[1]=
In[2]:=
Activate[%] == PolyLog[3, 1/2]
Out[2]=

Convert a harmonic polylogarithm with indices -1, 0 and 1 to an iterated integral:

In[3]:=
ResourceFunction["PolyLogToIntegral"][HarmonicPolyLog[{-1, 0, 1}, 1/2]]
Out[3]=
In[4]:=
Activate[%] == HarmonicPolyLog[{-1, 0, 1}, 1/2] // Simplify
Out[4]=

Scope (9) 

Convert a Riemann zeta value to its standard iterated integral form:

In[5]:=
ResourceFunction["PolyLogToIntegral"][Zeta[7]]
Out[5]=
In[6]:=
Activate[%] == Zeta[7]
Out[6]=

Convert a Hurwitz zeta value; the shift parameter appears in the resulting integrand:

In[7]:=
ResourceFunction["PolyLogToIntegral"][HurwitzZeta[3, 6]]
Out[7]=
In[8]:=
Activate[%] == HurwitzZeta[3, 6] // FullSimplify
Out[8]=

Convert a depth-two multiple zeta value to a nested integral representation:

In[9]:=
ResourceFunction["PolyLogToIntegral"][MultipleZeta[{2, 6}]]
Out[9]=
In[10]:=
Activate[%] == MultipleZeta[{2, 6}]
Out[10]=

Complex polylogarithm arguments are carried into the integral representation:

In[11]:=
ResourceFunction["PolyLogToIntegral"][PolyLog[3, I]]
Out[11]=
In[12]:=
Activate[%] == PolyLog[3, I]
Out[12]=

A three-argument PolyLog expression is also converted to an inactive integral:

In[13]:=
ResourceFunction["PolyLogToIntegral"][PolyLog[3, 2, I]]
Out[13]=
In[14]:=
Activate[%] == PolyLog[3, 2, I]
Out[14]=

Convert a multiple polylogarithm with rational colors to an iterated integral:

In[15]:=
ResourceFunction["PolyLogToIntegral"][ MultiplePolyLog[{2, 3}, {1/5, 1/7}]]
Out[15]=
In[16]:=
Activate[%] == MultiplePolyLog[{2, 3}, {1/5, 1/7}]
Out[16]=

Harmonic polylogarithms with larger signed indices are converted using the corresponding kernel factors:

In[17]:=
ResourceFunction["PolyLogToIntegral"][HarmonicPolyLog[{-3, 2, 1}, 1/2]]
Out[17]=
In[18]:=
Activate[%] == HarmonicPolyLog[{-3, 2, 1}, 1/2]
Out[18]=

Convert a generalized polylogarithm to its inactive integral representation:

In[19]:=
ResourceFunction["PolyLogToIntegral"][ GeneralizedPolyLog[{1, 2, 3, 4}, 1/2]]
Out[19]=
In[20]:=
Activate[%] == GeneralizedPolyLog[{1, 2, 3, 4}, 1/2]
Out[20]=

Inactive GeneralizedPolyLog input can also be converted, while the returned representation remains an inactive integral:

In[21]:=
ResourceFunction["PolyLogToIntegral"][ Inactive[GeneralizedPolyLog][{1, 2, 3, 4}, 1/2]]
Out[21]=
In[22]:=
Activate[%] == GeneralizedPolyLog[{1, 2, 3, 4}, 1/2]
Out[22]=

Publisher

Jayanta Kumar Phadikar

Version History

  • 1.0.0 – 03 July 2026

License Information