How can I insert a polylogarithm function to FORTRAN?

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In summary, the conversation discusses the insertion of a polylogarithm function into a FORTRAN code. The speaker is seeking help with defining the function and a reference is provided. They also offer to share their own code for calculating dilogarithm and trilogarithm functions.
  • #1
bulash
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Hi everyone,

Anyone knows how can I insert a polylogarithm function to FORTRAN? I am trying to put Li_2(z) polylogarithm function to my equation.

Thank you in advance.

bulash.
 
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  • #2
You'd have to write the function, if you do not have it in a library of functions.
If you need help with the definition of the function, check the following site:
http://en.wikipedia.org/wiki/Polylogarithm
If you have difficulties with writing the function, you can post what you've got, and we will then know how we can help.
 
  • #3
OK. Thank you very much!
 
  • #4
A good reference for a polylogarithm function algorithm is the following:
Note on fast polylogarithm computation
File Format: PDF/Adobe Acrobat - View as HTML
Abstract: The polylogarithm function Li ... assumed that −π < arg z ≤ π, whence the analytic continuation with proper branch cut ...
people.reed.edu/~crandall/papers/Polylog.pdf - Similar pages
by RE Crandall -

If you are interested please send me an email (rashore1@yahoo.com) and I will then send you copies of the FORTRAN codes I wrote to calculate the dilogarithm and trilogarithm functions of complex argument based on Crandall's algorithm. I've checked them using Mathematica and they work quite well.
 

What is the Polylogarithm function?

The Polylogarithm function, denoted as Lis(x), is a special function in mathematics that is defined as the sum of the powers of x, raised to the s-th power, divided by their respective exponent. It is a generalization of the natural logarithm function and is used to solve various problems in complex analysis, number theory, and physics.

What is the domain and range of the Polylogarithm function?

The Polylogarithm function is defined for all complex numbers x with |x| < 1 and s ≥ 1. So, its domain is the unit disk in the complex plane. The range of the function depends on the value of s. For s = 1, the range is the real line, for s = 2, the range is the set of complex numbers with |arg(z)| < π, and for s > 2, the range is the entire complex plane.

What are some properties of the Polylogarithm function?

Some important properties of the Polylogarithm function include:

  • It is an analytic function in its domain.
  • For s = 1, it reduces to the natural logarithm function, i.e. Li1(x) = -ln(1-x).
  • For s = 2, it is related to the dilogarithm function, i.e. Li2(x) = -Li2(1-x).
  • It satisfies the functional equation Lis(x) + xLis(1/x) = x for all x in its domain.
  • For integer values of s, it has explicit formulas in terms of Bernoulli numbers and the Riemann zeta function.

How is the Polylogarithm function used in physics?

The Polylogarithm function is used in various areas of physics, including quantum field theory, statistical mechanics, and string theory. It appears in the calculation of Feynman diagrams, which are used to represent the interactions between subatomic particles. It also arises in the study of black holes, where it is used to calculate the entropy of a black hole.

What are some applications of the Polylogarithm function?

The Polylogarithm function has a wide range of applications in mathematics, physics, and engineering. Some examples include:

  • In number theory, it is used to study the distribution of prime numbers and solve diophantine equations.
  • In complex analysis, it is used to evaluate integrals and solve differential equations.
  • In signal processing, it is used to model the behavior of filters and amplifiers.
  • In computer science, it is used in the analysis of algorithms and the design of efficient data structures.

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