Questions tagged [polylogarithm]

For questions about or related to polylogarithm functions.

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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 ...
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4 votes
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118 views

Evaluating $\int_0^1\frac{\ln^2(1+x)+2\ln(x)\ln(1+x^2)}{1+x^2}dx$

How to show that $$\int_0^1\frac{\ln^2(1+x)+2\ln(x)\ln(1+x^2)}{1+x^2}dx=\frac{5\pi^3}{64}+\frac{\pi}{16}\ln^2(2)-4\,\text{G}\ln(2)$$ without breaking up the integrand since we already know: $$\int_0^1\...
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2 votes
2 answers
235 views

Find close form for $\int_0^1 \frac{\log(x)\log(1+x^2)}{1+x^2}dx$

I am trying to find a closed form for the integral,$$\int_0^1 \frac{\log(x)\log(1+x^2)}{1+x^2}dx$$ I tried using the sub, $x=\frac{1-x}{1+x}$ but it was to no avail. I also tried the trig sub, $x=\tan(...
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  • 121
11 votes
0 answers
337 views

Is the closed form of $\int_0^1\frac{\text{Li}_{2a+1}(x)}{1+x^2}dx$ known in the literature?

Using $$\text{Li}_{2a+1}(x)-\text{Li}_{2a+1}(1/x)=\frac{i\,\pi\ln^{2a}(x)}{(2a)!}+2\sum_{k=0}^a \frac{\zeta(2a-2k)}{(2k+1)!}\ln^{2k+1}(x)\tag{1}$$ and $$\int_0^1x^{n-1}\operatorname{Li}_a(x)\mathrm{d}...
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1 vote
1 answer
29 views

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(...
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3 votes
0 answers
125 views

How can we prove a closed form for $\frac{1}{8} \text{Li}_2\left(\frac{2+\sqrt{3}}{4} \right)+\text{Li}_2\left(2+\sqrt{3}\right)$?

I have been working on a problem in number theory that I have reduced to the problem of showing that the two-term linear combination $$ \frac{1}{8} \text{Li}_2\left(\frac{2+\sqrt{3}}{4} \right) + \...
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0 votes
1 answer
42 views

How is $(2^a)^{\lg n} = n^a$? [closed]

I was learning from Introduction to Algorithms (Chapter 3 under the topic “Logarithms”) and came across this expression. $$ \lim_{ n \to 0 }{\frac{\lg^b n}{(2^a)^{\lg n}}} = \lim_{n \to 0} \frac{...
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  • 315
0 votes
0 answers
30 views

Do Polylogarithms Always Converge

I have been reading the Wikipedia page dedicated to polylogarithms to understand the following paper. I am trying to understand the first term in equation (6) (reproduced below): $$ P(\lambda) = \...
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  • 720
6 votes
2 answers
109 views

Series expansion of $\text{Li}_3(1-x)$ at $x \sim 0$

My question is simple, but maybe hard to answer. I would like to have a series expansion for $\text{Li}_3 (1-x)$ at $x \sim 0$ in the following form: $$\text{Li}_3 (1-x) = \sum_{n=0} c_n x^n + \log x \...
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  • 667
0 votes
1 answer
64 views

Closed form for $\rm{Li }_2\left( -{\frac {i\sqrt {3}}{3}} \right)$

In my personal research with Maple i find this closed form : $$\operatorname{Li }_2\left( -{\frac {i\sqrt {3}}{3}} \right)={\frac {{\pi}^{2}}{24}}+{\frac {\ln \left( 2 \right) \ln \left( 3 \right) }...
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  • 193
1 vote
0 answers
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Canonical reference for algebraic theory of polylogs?

I've noticed that there are lots of similar-looking (at least to my untrained eye) questions on Math Stackexchange involving polylogarithms, all equally delicious, in which the OP asks about how to ...
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1 vote
2 answers
115 views

Evaluating improper integral $\int_0^1 \frac{\log(x)}{x+\alpha}\; dx$ for small positive $\alpha$

Let $\alpha$ be a small positive real number. How do I obtain $$ I = \int_0^1 \frac{\log(x)}{x+\alpha}\; dx = -\frac{1}{2}(\log\alpha)^2 - \frac{\pi^2}{6} - \operatorname{Li}_2(-\alpha)$$? Maxima told ...
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  • 23
18 votes
0 answers
683 views

Are these generalizations known in the literature?

By using $$\int_0^\infty\frac{\ln^{2n}(x)}{1+x^2}dx=|E_{2n}|\left(\frac{\pi}{2}\right)^{2n+1}\tag{a}$$ and $$\text{Li}_{a}(-z)+(-1)^a\text{Li}_{a}(-1/z)=-2\sum_{k=0}^{\lfloor{a/2}\rfloor }\frac{\eta(...
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0 votes
1 answer
127 views

Evaluate: ${{\int_{0}^{1}\frac{\ln(1+x)^5}{x+2}dx-\int_{0}^{1}\frac{\ln(1+x)^5}{x+3}dx+5\ln2\int_{0}^{1}\frac{\ln(1+x)^4}{x+3}dx}}$

Evaluate: $${{I=\int_{0}^{1}\frac{\ln(1+x)^5}{x+2}dx-\int_{0}^{1}\frac{\ln(1+x)^5}{x+3}dx+5\ln2\int_{0}^{1}\frac{\ln(1+x)^4}{x+3}dx.}}$$ The answer is given below: $$ I=-\frac{7}{12}\pi^4\ln^2(2)-\...
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1 vote
1 answer
150 views

Can this formula for $\zeta(3)$ be proven or simplified further?

This question is related to the equivalence of formulas (1) and (2) below where formula (1) is from a post on the Harmonic Series Facebook group and formula (2) is based on evaluation of the integral ...
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9 votes
2 answers
262 views

Proving $\int_0^{1/2}\frac{\text{Li}_2(-x)}{1-x}dx=-\text{Li}_3\left(-\frac12\right)-\frac{13}{24}\zeta(3)$

By comparing some results, I found that $$\int_0^{\frac12}\frac{\text{Li}_2(-x)}{1-x}dx=-\text{Li}_3\left(-\frac12\right)-\frac{13}{24}\zeta(3).\tag{1}$$ I tried to prove it starting with applying IBP:...
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1 vote
0 answers
102 views

Closed-form for $\int_0^{a^2} \mathrm{Ei} (-s) \frac{1 - e^s}{s} ds$

In my partial answer to this question: Integral involving polylogarithm and an exponential, I arrive at the integral $$ \int_0^{a^2} \mathrm{Ei} (-s) \frac{1 - e^s}{s} ds , ~~~~ (\ast) $$ where $a \in ...
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1 vote
1 answer
94 views

Integral involving product of dilogarithm and an exponential

I am interested in the integral \begin{equation} \int_0^1 \mathrm{Li}_2 (u) e^{-a^2 u} d u , ~~~~ (\ast) \end{equation} where $\mathrm{Li}_2$ is the dilogarithm. This integral arose in my attempt to ...
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3 votes
1 answer
28 views

Sum of Fermi-Dirac integrals with opposite chemical potentials: closed form (Le Bellac eq. 1.13)

I am trying to reproduce the result of eq. (1.13) in Le Bellac's Thermal Field Theory book to compute the grand canonical potential of a gas of massless fermions: $$ Ω = - \frac{V T^4}{6 π^2} \int_0^\...
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  • 31
12 votes
1 answer
469 views

General expressions for $\mathcal{L}(n)=\int_{0}^{\infty}\operatorname{Ci}(x)^n\text{d}x$

Define $$\operatorname{Ci}(x)=-\int_{x}^{ \infty} \frac{\cos(y)}{y}\text{d}y.$$ It is easy to show $$ \mathcal{L}(1)=\int_{0}^{\infty}\operatorname{Ci}(x)\text{d}x=0 $$ and $$\mathcal{L}(2)=\int_{0}^{\...
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6 votes
0 answers
144 views

A generalized "Rare" integral involving $\operatorname{Li}_3$

In my previous post, it can be shown that $$\int_{0}^{1} \frac{\operatorname{Li}_2(-x)- \operatorname{Li}_2(1-x)+\ln(x)\ln(1+x)+\pi x\ln(1+x) -\pi x\ln(x)}{1+x^2}\frac{\text{d}x}{\sqrt{1-x^2} } =\...
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11 votes
1 answer
490 views

A rare integral involving $\operatorname{Li}_2$

A rare but interesting integral problem: $$\int_{0}^{1} \frac{\operatorname{Li}_2(-x)- \operatorname{Li}_2(1-x)+\ln(x)\ln(1+x)+\pi x\ln(1+x) -\pi x\ln(x)}{1+x^2}\frac{\text{d}x}{\sqrt{1-x^2} } =\...
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7 votes
2 answers
322 views

Finding $\int_{1}^{\infty} \frac{1}{1+x^2} \frac{\operatorname{Li}_2\left ( \frac{1-x}{2} \right ) }{\pi^2+\ln^2\left(\frac{x-1}{2}\right)}\text{d}x$

Prove the integral $$\int_{1}^{\infty} \frac{1}{1+x^2} \frac{\operatorname{Li}_2\left ( \frac{1-x}{2} \right ) }{ \pi^2+\ln^2\left ( \frac{x-1}{2} \right ) }\text{d}x =\frac{96C\ln2+7\pi^3}{12(\pi^2+...
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1 vote
0 answers
45 views

Difference of polylogarithms of complex conjugate arguments

I have the expression $$\tag{1} \operatorname{Li}_{1/2}(z)-\operatorname{Li}_{1/2}(z^*) $$ Where $\operatorname{Li}$ is the polylogarithm and $^*$ denotes complex conjugation. The expression is ...
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  • 3,368
3 votes
1 answer
70 views

Identity involving PolyLog functions

The following is a common identity between dilogarithms: $$ \text{Li}_2(y)+\text{Li}_2\left(\frac{y}{y-1}\right)+\frac{1}{2} \log ^2(1-y)=0\quad \text{with}\quad 0<y<1\,. $$ Similarly, with some ...
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  • 351
1 vote
0 answers
26 views

Asymptotics of Geometric Distribution Moments

It is known (e.g., see Wikipedia) that the $k$th moment of a Geometric distribution with success probability $p$ is $$\mathbb{E}\left[X^k\right] = \sum_{j = 0}^\infty j^k \cdot p(1-p)^j = p \cdot\text{...
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  • 6,218
9 votes
1 answer
258 views

Different ways to evaluate $\sum_{n=1}^\infty\frac{H_nH_n^{(2)}}{(n+1)(n+2)(n+3)}$

The following question: How to compute the harmonic series $$\sum_{n=1}^\infty\frac{H_nH_n^{(2)}}{(n+1)(n+2)(n+3)}$$ where $H_n=\sum_{k=1}^n\frac{1}{k}$ and $H_n^{(2)}=\sum_{k=1}^n\frac{1}{k^2}$, was ...
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0 votes
0 answers
33 views

Closed form of integral $\mathcal{P}\int_{-\infty}^{+\infty}dy\frac{ \text{Li}_{k}(\exp(-(y-\frac{\xi }{2})^2))}{\exp (2 \xi y)-1}$

How could one calculate the closed form solution of this integral: $\mathcal{P}\int_{-\infty}^{+\infty}dy\frac{ \text{Li}_{k}(\exp(-(y-\frac{\xi }{2})^2))}{\exp (2 \xi y)-1}$ Here the integral is ...
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0 votes
1 answer
59 views

How to solve a Non-algebraic equation?

While working with exponential growth and decay, I encountered a problem where I need to solve an equation involving logarithm. I could not separate or could not make it explicit. $y=\frac{\ln((1+r)(1-...
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  • 168
0 votes
0 answers
130 views

When are $\sqrt{-\log x}^{1+\varepsilon}\operatorname{Li}_{1/2}(x)$ and $\sqrt{-\log x}\operatorname{Li}_\nu(x)$ strictly increasing?

[Cross-posted on MathOverflow] This problem (posed by a Discord user) originates from the study of the fire-diffuse-fire model of calcium dynamics. Briefly, I was told that the goal was to prove there ...
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  • 25.4k
6 votes
1 answer
408 views

Is the closed form of $\int_0^1 \frac{x\ln^a(1+x)}{1+x^2}dx$ known in the literature?

We know how hard these integrals $$\int_0^1 \frac{x\ln(1+x)}{1+x^2}dx; \int_0^1 \frac{x\ln^2(1+x)}{1+x^2}dx; \int_0^1 \frac{x\ln^3(1+x)}{1+x^2}dx; ...$$ can be. So I decided to come up with a ...
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1 vote
0 answers
90 views

Conjectured closed form for ${\it {Li_2}} \left( 1-{\frac {\sqrt {2}}{2}}-i \left( 1-{\frac {\sqrt { 2}}{2}} \right) \right)$

With Maple i find this closed form: ${\it {Li_2}} \left( 1-{\frac {\sqrt {2}}{2}}-i \left( 1-{\frac {\sqrt { 2}}{2}} \right) \right)$=$-{\frac {{\pi}^{2}}{64}}-{\frac { \left( \ln \left( 1+\sqrt {2} ...
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  • 193
0 votes
1 answer
104 views

Evaluate $\int_{{\frac {\pi}{8}}}^{{\frac {7\,\pi}{8}}}\!{\frac {\ln \left( 1- \cos \left( t \right) \right) }{\sin \left( t \right) }}\,{\rm d}t$

I'm interested in this integral: $\int_{{\frac {\pi}{8}}}^{{\frac {7\,\pi}{8}}}\!{\frac {\ln \left( 1- \cos \left( t \right) \right) }{\sin \left( t \right) }}\,{\rm d}t$ I found this particular ...
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  • 193
8 votes
1 answer
309 views

Evaluating $\int_0^1\frac{\operatorname{Li}_2(x)\ln(1+x)}x\,dx$

Well, I've been trying to solve the following integral: \begin{equation*} \int_0^1\frac{\text{Li}_3(x)}{1+x}\mathrm dx, \end{equation*} where by integration by parts, making $u=\text{Li}_3(x)$ and $\...
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  • 461
5 votes
2 answers
263 views

Evaluating $\int_0^\infty\frac{\tan^{-1}av\cot^{-1}av}{1+v^2}\,dv$

The Weierstrass substitution stuck in my head after I used it to prove the rigidity of the braced hendecagon (and tridecagon). Thus I had another look at this question which I eventually answered in a ...
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  • 89.9k
4 votes
2 answers
142 views

Prove $\int_0^1\frac{\text{Li}_2(-x^2)}{\sqrt{1-x^2}}\,dx=\pi\int_0^1\frac{\ln\left(\frac{2}{1+\sqrt{1+x}}\right)}{x}\,dx$

I managed here to prove $$\int_0^1\frac{\text{Li}_2(-x^2)}{\sqrt{1-x^2}}\,dx=\pi\int_0^1\frac{\ln\left(\frac{2}{1+\sqrt{1+x}}\right)}{x}\,dx$$ but what I did was converting the LHS integral to a ...
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4 votes
1 answer
100 views

Closed form evaluation of a trigonometric integral in terms of polylogarithms

Define the function $\mathcal{K}:\mathbb{R}\times\mathbb{R}\times\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\times\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\rightarrow\mathbb{R}$ via the definite ...
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  • 27.1k
6 votes
0 answers
315 views

Evaluate two integrals involving $\operatorname{Li}_3,\operatorname{Li}_4$

I need to evaluate $$\int_{1}^{\infty} \frac{\displaystyle{\operatorname{Re}\left ( \operatorname{Li}_3\left ( \frac{1+x}{2} \right ) \right ) \ln^2\left ( \frac{1+x}{2} \right ) }}{x(1+x^2)} \...
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12 votes
4 answers
416 views

How to evaluate$J(k) = \int_{0}^{1} \frac{\ln^2x\ln\left ( \frac{1-x}{1+x} \right ) }{(x-1)^2-k^2(x+1)^2}\text{d}x$

I am trying evaluating this $$J(k) = \int_{0}^{1} \frac{\ln^2x\ln\left ( \frac{1-x}{1+x} \right ) }{(x-1)^2-k^2(x+1)^2}\ \text{d}x.$$ For $k=1$, there has $$J(1)=\frac{\pi^4}{96}.$$ Maybe $J(k)$ ...
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3 votes
0 answers
269 views

Evaluate $\int_{1}^{\infty}\frac{\operatorname{Li}_3(-x)\ln(x-1)}{1+x^2}\text{d}x$

Using $$ \operatorname{Li}_3(-x) =-\frac{x}{2}\int_{0}^{1}\frac{\ln^2t}{1+tx} \text{d}t $$ It might be $$ -\frac{1}{2}\int_{0}^{1}\ln^2t \int_{1}^{\infty}\frac{x\ln(x-1)}{(1+tx)(1+x^2)}\text{d}x\text{...
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6 votes
2 answers
311 views

evaluate $\int_{0}^{1}\frac{\text{Li}_2(\frac{x^2-1}{4})}{1-x^2}dx$

I came across this integral: $$\int_{0}^{1}\frac{\text{Li}_2(\frac{x^2-1}{4})}{1-x^2}dx=\frac{1}{2}\int_{-1}^{1}\frac{\text{Li}_2(\frac{x^2-1}{4})}{1-x^2}dx$$ One way to evaluate is to start with the ...
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8 votes
1 answer
349 views

Integral $\int _0^1\frac{\ln \left(x\right)\text{Li}_2\left(x\right)}{x\left(4\pi ^2+\ln ^2\left(x\right)\right)}\:\mathrm{d}x$

A comrade sent me this conjecture $$\int _0^1\frac{\ln \left(x\right)\operatorname{Li}_2\left(x\right)}{x\left(4\pi ^2+\ln ^2\left(x\right)\right)}\:\mathrm{d}x=3\zeta (2)\left(4\ln \left(A\right)-1\...
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2 votes
1 answer
151 views

Finding a closed-form for the sum $\sum_{n=1}^{\infty}\frac{\left(-1\right)^{n}H_{2n}}{n^{4}}$

Let $\mathcal{S}$ denote the sum of the following alternating series: $$\mathcal{S}:=\sum_{n=1}^{\infty}\frac{\left(-1\right)^{n}H_{2n}}{n^{4}}\approx-1.392562725547,$$ where $H_{n}$ denotes the $n$-...
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  • 27.1k
3 votes
1 answer
202 views

Generating function of the polylogarithm.

Let $\operatorname{Li}_s(z)$ denote the polylogarithm function $$\operatorname{Li}_s(z) = \sum_{k=1}^\infty \frac{z^k}{k^s}.$$ Does there exists a closed form or a known function which generates the ...
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  • 710
2 votes
2 answers
184 views

How can I evaluate $\int _0^1\frac{\operatorname{Li}_2\left(-x^2\right)}{\sqrt{1-x^2}}\:\mathrm{d}x$

I've been trying to find and prove that: $$\int _0^1\frac{\operatorname{Li}_2\left(-x^2\right)}{\sqrt{1-x^2}}\:\mathrm{d}x=\pi \operatorname{Li}_2\left(\frac{1-\sqrt{2}}{2}\right)-\frac{\pi }{2}\left(\...
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2 votes
1 answer
76 views

Closed form evaluation of a class of inverse hyperbolic integrals

Define the function $\mathcal{I}:\mathbb{R}_{>0}^{2}\rightarrow\mathbb{R}$ via the definite integral $$\mathcal{I}{\left(a,b\right)}:=\int_{0}^{1}\mathrm{d}x\,\frac{\operatorname{arsinh}{\left(ax\...
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  • 27.1k
4 votes
3 answers
179 views

Arctan integral $ \int_{0}^{\infty}\frac{\arctan(x)}{x^{2}+k^{2}}$

Is there a closed form for the integral $$ \int_{0}^{\infty}\frac{\arctan(x)}{x^{2}+k^{2}}$$ for $\forall k \ge 1 $? Well, I was able to get the closed form for the case where $|k|\le1$, and it is of ...
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  • 119
7 votes
1 answer
163 views

Iterated integral involving polylogarithms

To establish notation the polylogarithm Li$_n(x)$ has the power series expansion $$ \text{Li}_n(x)= \sum_{k=1}^\infty \frac{x^k}{k^n} $$ and the Riemann zeta can be considered the special value $\zeta(...
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  • 6,503
0 votes
1 answer
52 views

The rate of convergence of the remainder of the power series for the Polylog function [closed]

Let $0<p<1$ be a positive real number strictly smaller than one and $q>0$ be a positive real number. Consider the series $$ \mathsf{Li}_{-q}(p) = \sum_{\ell=1}^{+\infty}\ell^{q}p^{\ell} $$ ...
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9 votes
1 answer
182 views

Is there a closed-form for $\sum _{k=1}^{\infty }\frac{\operatorname{Si}\left(k\right)}{k^2}$?

So far I've got this: $$\sum _{k=1}^{\infty }\frac{\operatorname{Si}\left(k\right)}{k^2}=\int _0^1\left(\sum _{k=1}^{\infty }\frac{\sin \left(kx\right)}{k^2}\right)\frac{1}{x}\:dx$$ $$=\int _0^1\frac{\...
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