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Trilogarithm

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Post Tue Jul 30, 2013 7:19 pm
zaidalyafey User avatar
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I am looking for an integral representaion of

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

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


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Post Tue Sep 10, 2013 5:33 pm

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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 User avatar
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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


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