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We can now revisit the title question, How to use polylogarithm function in c++? The point is worth making that there is no standard implementation of polylogarithm functions for C or even C++. If the goal is to avoid any additional library for your implementation, it pretty well sets you out to rolling your own routines, perhaps along the lines suggested by the David C. Wood paper that GertVdE’s Answer links to.

We can now revisit the title question, How to use polylogarithm function in c++? The point is worth making that there is no standard implementation of polylogarithm functions for C or even C++. If the goal is to avoid any additional library for your implementation, it pretty well sets you out to rolling your own routines, perhaps along the lines suggested by the David C. Wood paper that GertVdE’s Answer links to.

Alternatively you may wish to use other software to directly check (not referencing polylogarithms) the quadrature routines you wrote for your integral. As I sketch out in this Math.SE Question, the power series centered at the origin for the integral has limited convergence, but this can be mitigated by using a continued fraction expansion instead.

Alternatively you may wish to use other software to directly check (not referencing polylogarithms) the quadrature routines you wrote for your integral. As I sketch out in this Math.SE Question, the power series centered at the origin for the integral has limited convergence, but this can be mitigated by using a continued fraction expansion instead.

The GSL (GNU Scientific Library) apparently only has the dilogarithm function.

The GSL (GNU Scientific Library) apparently only has the dilogarithm function. However following a hint from @J.M. one finds the Debye function which gives the ulterior integral (up to a scalar multiple) implemented in double precision (see GSL 7.10 Debye Functions orders 1 through 6):

$$ D_n(x) = \frac{n}{x^n} \int_0^x \frac{t^n dt}{e^t - 1} $$