All Questions
Tagged with polylogarithm special-functions
119
questions
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39
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Relations between Dilogarithms and Imaginary part of Hurwitz-Zeta function
I'm working through a paper that involves a problem concerning the calculation of the Imaginary part of the derivative of the Hurwitz-Zeta function $\zeta_H(z,a)$ with respect to $z$, evaluated at a ...
1
vote
1
answer
57
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Connection between the polylogarithm and the Bernoulli polynomials.
I have been studying the polylogarithm function and came across its relation with Bernoulli polynomials, as Wikipedia site asserts:
For positive integer polylogarithm orders $s$, the Hurwitz zeta ...
- 800
2
votes
0
answers
68
views
Help verifying expression involving dilogarithms.
I need help verifying that the following equality holds:
$$Li_2(-2-2\sqrt2)+Li_2(3-2\sqrt2)+Li_2(\frac{1}{\sqrt2})-Li_2(-\frac{1}{\sqrt2})-Li_2(2-\sqrt2)-Li_2(-1-\sqrt2)-2Li_2(-3+2\sqrt2)$$
$$=$$
$$\...
- 359
4
votes
0
answers
83
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Closed form of dilogarithm fucntion involving many arctangents
I am trying to find closed form for this expression:
$$ - 2{\text{L}}{{\text{i}}_2}\left( {\frac{1}{3}} \right) - {\text{L}}{{\text{i}}_2}\left( {\frac{1}{6}\left( {1 + i\sqrt 2 } \right)} \right) - {\...
- 2,702
5
votes
1
answer
193
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Evaluating $\int_{0}^{1}\mathrm{d}x\,\frac{\operatorname{arsinh}{(ax)}\operatorname{arsinh}{(bx)}}{x}$ in terms of polylogarithms
Define the function $\mathcal{I}:\mathbb{R}^{2}\rightarrow\mathbb{R}$ by the definite integral
$$\mathcal{I}{\left(a,b\right)}:=\int_{0}^{1}\mathrm{d}x\,\frac{\operatorname{arsinh}{\left(ax\right)}\...
- 30.7k
2
votes
0
answers
68
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Evaluating $\int\frac{\log(x+a)}{x}\,dx$ in terms of dilogarithms
As per the title, I evaluated
$$\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
5
votes
1
answer
286
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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
4
votes
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answers
81
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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?
$$\operatorname{Li}_n(z)+(-1)^n\operatorname{Li}_n(1/z)=-\frac{(2\pi i)^n}{n!}B_n\!...
- 6,662
1
vote
0
answers
94
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Series with power of generalized harmonic number $\displaystyle\sum_{k=1}^{\infty}\left(H_k^{(s)}\right)^n x^k$
It's possible to generalize these series?
$$\sum_{k=1}^{\infty}H_k^{(s)}x^k=\frac{\operatorname{Li}_s(x)}{1-x}$$
$$\sum_{k=1}^{\infty}H_k^2 x^k=\frac{\ln(1-x)^2+\operatorname{Li}_2(x)}{1-x}$$
Where:
$$...
2
votes
2
answers
93
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Finding a recurrence relation to evaluate $\int_{a}^{1}\mathrm{d}x\,\frac{x^{n}}{\sqrt{1-x^{2}}}\ln{\left(\frac{x+a}{x-a}\right)}$
For each $n\in\mathbb{Z}_{\ge0}$, define the function $\mathcal{J}_{n}:(0,1)\rightarrow\mathbb{R}$ via the doubly improper integral
$$\mathcal{J}_{n}{\left(a\right)}:=\int_{a}^{1}\mathrm{d}x\,\frac{x^{...
- 30.7k
1
vote
2
answers
97
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Asymptotics of an integral involving the exponential integral
Consider the integral:
$$
I(a)=\int_a^\infty e^x E_1(x)\dfrac{dx}{x},
$$
where $a>0$ and $E_1(x)$ is the exponential integral function.
I would like to better understand the behavior of $I(a)$ for $...
- 765
2
votes
1
answer
173
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Calculate the integral of the given polylogarithm function? $\int_0^1\frac{\operatorname{Li}_ 4(x)}{1+x}dx=?$ [closed]
$$\int_0^1 \frac{\operatorname{Li}_2(-x)\operatorname{Li}_2(x)}{x}\,\mathrm dx=?$$
where $$\operatorname{Li}_2(-x)=\sum_{k=1}^{\infty}\frac{(-x)^k}{k^2}$$ for $$|x|>1$$
actually my goal is to edit ...
- 71
3
votes
1
answer
242
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How to solve $\int\frac{x\arctan x}{x^4+1}dx$ in a practical way
I need to evaluate the following indefinite integral for some other definite integral
$$\int\frac{x\arctan x}{x^4+1}dx$$
I found that
$$\int_o^\infty\arctan{(e^{-x})}\arctan{(e^{-2x})}dx=\frac{\pi G}{...
- 2,370
3
votes
1
answer
92
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Why is $B_{2n}(\frac12+ix)\in\mathbb R$ whenever $x\in\mathbb R$?
I just noticed that $B_{2n}(\frac12+ix)\in\mathbb R$, where:
$x\in\mathbb R$, $n\in\mathbb N$, and $B_n(x)$ is the $n$th Bernoulli Polynomial.
Why? Is there a simple, slick proof? Does it follow from ...
- 6,662
1
vote
1
answer
52
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Proving $-\text{Li}_2(x^{-1}-1)+\text{Li}_2(1-x^{-1})-\text{Li}_2(2 x)+\text{Li}_2(2-2 x)+\text{Li}_2(2) = i \pi \log(x)$ for $x>1/2$.
While working on a physics problem, I have stumbled upon the following identity. For $x>\frac{1}{2}$ note
$$
-\text{Li}_2\left(\frac{1}{x}-1\right)+\text{Li}_2\left(1-\frac{1}{x}\right)-\text{Li}_2(...
- 2,777