In the complex plane has a branch point at . The principal branch has a cut along the interval and agrees with (25.12.1) when ; see also §4.2(i). The remainder of the equations in this subsection apply to principal branches.
When , , (25.12.1) becomes
The cosine series in (25.12.7) has the elementary sum
The right-hand side is called Clausen’s integral.
For real or complex and the polylogarithm is defined by
For each fixed complex the series defines an analytic function of for . The series also converges when , provided that . For other values of , is defined by analytic continuation.
Further properties include
The Fermi–Dirac and Bose–Einstein integrals are defined by
In terms of polylogarithms
For a uniform asymptotic approximation for see Temme and Olde Daalhuis (1990).