# LOG#161. Polylogia flashes(III).

In the third post of this series I will write more fantastic identities related to our friends, the polylogs!

(1)

and by analytic continuation that equation can be extended to all . In fact

(2)

such as , since we define

(3)

and the equations

(4)

(5)

(6)

Duplication formula for the polylogarithm:

(7)

Connection with the Kummer’s function can be established

(8)

(9)

and thus

(10)

We also have

(11)

Some extra values of (negative) integer polylogs that are rational functions or logarithms

(12)

(13)

(14)

(15)

(16)

(17)

And more generally, we have the general formulae

(18)

(19)

and where are the Stirling numbers of the second kind.

(20)

Furthermore,

(21)

and where are the eulerian numbers.

We write now some interesting values of the polylog you will love too

and where

with .

The polylog and other functions can be also be related, as we have seen:

(22)

(23)

(24)

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

(25)

with , , . Moreover,

(26)

There is also a formula called inversion formula

(27)

and for

(28)

The expression

implies that

(29)

with

if and if . The following feynmanity (nullity) identity holds as well

(30)

and

(31)

for and

(32)

for . Polylogs and Clausen functions are related (we already saw this before)

(33)

The inverse tangent integral is related to polylogs

(34)

(35)

(36)

(37)

The Legendre chi function is also related to polylogs

(38)

The incomplete zeta function or Debye functions are polylogs as well

(39)

(40)

(41)

Now, we will write some polylog integrals

(42)

This last integral converges if and . It is the Bose-Einstein distribution!

The Fermi-Dirac integrals read

(43)

For and excepting and we have

and

(44)

with residue equal to

See you in the next polylog post!

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