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Trilogarithm

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

Posts: 357
I am looking for an integral representaion of

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

Of course I know the recursive integral equation using the dilogarithm .
Wanna learn what we discuss , see Book

Post 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

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

???

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

Posts: 357
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 \(\displaystyle \text{Li}_s(z)\).
Wanna learn what we discuss , see Book


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