The polylogarithm function (or Jonquière's function) of index and argument is a special function, defined in the complex plane for and by analytic continuation otherwise. It can be plotted for complex values ; for example, along the celebrated critical line for Riemann's zeta function . The polylogarithm function appears in the Fermi–Dirac and Bose–Einstein distributions and also in quantum electrodynamics calculations for Feynman diagrams. The 2D plot shows the function , and the 3D plot shows .
The polylogarithm function is defined as
For , it is equivalent to the natural logarithm, . For and , it is called the dilogarithm and the trilogarithm; the integral of a polylogarithm is itself a polylogarithm
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