Board index Special Functions Trilogarithm

## Trilogarithm

Post your questions related to Special Functions here.

Moderator: Shobhit

### Trilogarithm

Tue Jul 30, 2013 7:19 pm
zaidalyafey
Global Moderator

Posts: 354
I am looking for an integral representaion of

$\displaystyle \text{Li}_3(z)=$

Of course I know the recursive integral equation using the dilogarithm .
$\displaystyle \sum_{k\geq 0}\frac{f^{(k)} (s)}{(s+1)_k} (-s)^{k}= \int^{1}_0x^{s} f(s x) \left( \frac{x}{s}\right) \, dx$

Wanna learn what we discuss , see tutorials

### Re: Trilogarithm

Tue Sep 10, 2013 5:33 pm

Posts: 138
Location: North Londinium, UK
zaidalyafey wrote:
I am looking for an integral representaion of

$\displaystyle \text{Li}_3(z)=$

Of course I know the recursive integral equation using the dilogarithm .

Sorry if I'm being daft here, but are you excluding the generalized representation

$\displaystyle \text{Li}_m(z)=\frac{(-1)^{m-1}}{(m-2)!}\int_0^1\frac{(\log x)^{m-2}\log(1-z\,x)}{x}\,dx$

???

### Re: Trilogarithm

Tue Sep 10, 2013 5:51 pm
zaidalyafey
Global Moderator

Posts: 354
I think the following equation served my needs

$\displaystyle \text{Li}_{n+1}(z)= \frac{z}{(-1)^n n!}\int^1_0 \frac{\log^{n}(t) }{1-zt} \, dt$

Of course we can extend this equation for polylgoarithms of non-integral parts $\displaystyle \text{Li}_s(z)$.
$\displaystyle \sum_{k\geq 0}\frac{f^{(k)} (s)}{(s+1)_k} (-s)^{k}= \int^{1}_0x^{s} f(s x) \left( \frac{x}{s}\right) \, dx$

Wanna learn what we discuss , see tutorials