Questions tagged [polylogarithm]
For questions about or related to polylogarithm functions.
546
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Evaluating $\int\frac{\log(x+a)}{x}\,dx$ in terms of dilogarithms
As per the title, I evaluated
$$\int\frac{\log(x+a)}{x}\,dx$$
And wanted to make sure my solution is correct, and if not, where I went wrong in my process. Here is my work.
$$\int\frac{\log(x+a)}{x}\,...
- 1,123
1
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1
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Verification of the generalized polylogarithm formula
Here I posted a generalized formula for the polylogarithm I had discovered. However, for $x=\frac{1}{2}$, $z=\frac{1}{2}$, $p=1$ wolfram alpha yields a result different than what the double integral ...
- 1,039
6
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2
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How to show $\int_0^1\frac{\operatorname{Li}_2\left(\frac{1+x^2}{2}\right)}{1+x^2}dx=\ln(2)G$
I am trying to prove that
$$\int_0^1\frac{\operatorname{Li}_2\left(\frac{1+x^2}{2}\right)}{1+x^2}dx=\ln(2)G,$$
where $G$ is the Catalan constant and $\operatorname{Li}_2(x)$ is the dilogarithm ...
- 25.8k
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Efficient calculation for Lerch Transcendent Expression
I've encountered:
$$\Phi(z, s, \alpha) = \sum_{k=0}^\infty \frac { z^k} {(k+\alpha)^s}.$$
When trying to compute:
$$\frac{1}{x}\sum_{p=0}^m \frac{2}{(2p-1)\ x^{2p-1}}\ s.t. x\in\mathbb{N} =\ ???$$ ...
- 301
3
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141
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Prove that $-\int_{0}^{1}\frac{{\rm Li}_2(-x(1-x))}{x}\ dx=\frac{2}{5}\zeta(3)$
Prove that $$-\int_{0}^{1}\frac{{\rm Li}_2(-x(1-x))}{x}\ dx=\frac{2}{5}\zeta(3)$$
where ${\rm Li}_2(x)$ is the Poly Logarithm function and $\zeta(s)$ is the Riemann zeta function
Let $$I=-\int_{0}^{1}...
- 862
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Evaluate $\sum\limits_{n=0}^\infty\operatorname W(e^{e^{an}})x^n$ with Lambert W function
$\def\W{\operatorname W} \def\Li{\operatorname{Li}} $
Interested by $\sum_\limits{n=1}^\infty\frac{\W(n^2)}{n^2}$, here is an example where Lagrange reversion applies to a Lambert W sum:
$$\W(x)=\ln(...
- 12.4k
5
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1
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Closed forms of the integral $ \int_0^1 \frac{\mathrm{Li}_n(x)}{(1+x)^n} d x $
(This is related to this question).
How would one find the closed forms the integral
$$ \int_0^1 \frac{\mathrm{Li}_n(x)}{(1+x)^n} d x?
$$
I tried using Nielsen Generalized Polylogarithm as mentioned ...
- 105
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1
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What's a better time complexity, $O(\log^2(n))$, or $O(n)$?
(By $O(\log^2(n))$, I mean $O((\log n)^2)$ rather than $O(\log(\log(n)) )$
I know $O(\log(n))$ is better than $O(\log^2(n))$ which itself is better than $O(\log^3(n))$ etc. But how do these compare to ...
0
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1
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33
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Proving $2^{\log^{1-\varepsilon}n}\in \omega(\operatorname{polylog}(n))$
Let $\varepsilon \in (0,1)$. I wish to show that $2^{\log^{1-\varepsilon}(n)}\in \omega(\operatorname{polylog}(n))$. I attempted to turn this into a function and use L'Hospital's rule but that got me ...
- 2,885
1
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1
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Show that $\int_0^1 \frac{Li_{1 - 2m}(1 - 1/x)}{x} dx = 0$.
I would like to show that,for $m \geq 2$,
$$I_m := \int_0^1 \frac{\operatorname{Li}_{1 - 2m}(1 - 1/x)}{x} dx = 0$$ where $\operatorname{Li}_{1 - 2m}$ is the $1-2m$ polylogarithm (https://en.wikipedia....
- 2,043
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How to derive this polylogarithm identity (involving Bernoulli polynomials)?
How can one derive the following identity, found here, relating the polylogarithm functions to Bernoulli polynomials?
$$\operatorname{Li}_n(z)+(-1)^n\operatorname{Li}_n(1/z)=-\frac{(2\pi i)^n}{n!}B_n\!...
- 6,662
2
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1
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255
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Generalized formula for the polylogarithm
Some time ago, I discovered the formula for repeated application of $z\frac{d}{dz}$ here. Recently, I thought about taking the function to which this would be applied to be the integral representation ...
- 1,039
5
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3
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258
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How to find the exact value of $\sum_{n=1}^{\infty} \frac{\sin \left(\frac{n \pi}{4}\right)}{n^2 \cdot 2^{\frac{n}{2}}} $?
Once I met the identity
$$
\boxed{S_0=\sum_{n=1}^{\infty} \frac{\sin \left(\frac{n \pi}{4}\right)}{2^{\frac{n}{2}}}=1},
$$
I first tried to prove it by $e^{xi}=\cos x+i\sin x$.
$$
\begin{aligned}
\...
- 22.1k
2
votes
1
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Function upper-bounding polylogarithm function when $0 < z < 1$
The polylogarithm function (aka Jonquière's function) is defined as $Li_s(z) = \sum_{k=1}^{\infty} z^k k^{-s}$.
Is there a closed-form upper bound for this function when $s \leq -1$ and is real, and $...
- 91
6
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Calculate $\int _0^1\frac{\arcsin ^2\left(x\right)\ln \left(x\right)\ln \left(1-x\right)}{x}\:\mathrm{d}x$
this integral got posted on a mathematics group by a friend
$$I=\int _0^1\frac{\arcsin ^2\left(x\right)\ln \left(x\right)\ln \left(1-x\right)}{x}\:\mathrm{d}x$$
I tried seeing what I'd get from ...
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