In the third post of this series I will write more fantastic identities related to our friends, the polylogs!
and by analytic continuation that equation can be extended to all . In fact
such as , since we define
and the equations
Duplication formula for the polylogarithm:
Connection with the Kummer’s function can be established
We also have
Some extra values of (negative) integer polylogs that are rational functions or logarithms
And more generally, we have the general formulae
and where are the Stirling numbers of the second kind.
and where are the eulerian numbers.
We write now some interesting values of the polylog you will love too
The polylog and other functions can be also be related, as we have seen:
where is the Dirichlet beta function. The complete Fermi-Dirac integral is also polylogarithmic
The incomplete polylog is also interesting
and the incomplete gamma function is defined by
with , , , . Moreover,
There is also a formula called inversion formula
if and if . The following feynmanity (nullity) identity holds as well
for . Polylogs and Clausen functions are related (we already saw this before)
The inverse tangent integral is related to polylogs
The Legendre chi function is also related to polylogs
The incomplete zeta function or Debye functions are polylogs as well
Now, we will write some polylog integrals
This last integral converges if and . It is the Bose-Einstein distribution!
The Fermi-Dirac integrals read
For and excepting and we have
with residue equal to
See you in the next polylog post!