All Questions
Tagged with polylogarithm real-analysis
9
questions with no upvoted or accepted answers
8
votes
0
answers
404
views
Powerful Integral $\int_0^x\frac{t\ln(1-t)}{1+t^2}\ dt$
This integral can be found in Cornel's book, (Almost) Impossible Integral, Sums and Series page $97$ where he showed that
$$\int_0^x\frac{t\ln(1-t)}{1+t^2}\ dt=\frac14\left(\frac12\ln^2(1+x^2)-2\...
- 25.8k
6
votes
1
answer
285
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Calculate $\int _0^1\frac{\arcsin ^2\left(x\right)\ln \left(x\right)\ln \left(1-x\right)}{x}\:\mathrm{d}x$
this integral got posted on a mathematics group by a friend
$$I=\int _0^1\frac{\arcsin ^2\left(x\right)\ln \left(x\right)\ln \left(1-x\right)}{x}\:\mathrm{d}x$$
I tried seeing what I'd get from ...
- 231
3
votes
0
answers
316
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Two tough integrals with logarithms and polylogarithms
The following two integrals are given in (Almost) Impossible Integrals, Sums, and Series (see Sect. $\textbf{1.55}$, page $35$),
$$i) \int_0^{\pi/2} \cot (x) \log (\cos (x)) \log ^2(\sin (x)) \...
- 5,495
3
votes
0
answers
106
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Proving that swapping the order of this summation is justified
I'm unsure if this has been discovered already, but it's heavily related to my current research, particularly to this question of mine (this conjecture was also originally posted at the beginning of ...
- 856
2
votes
1
answer
86
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Is there a nice way to represent $\sum_{n=1}^\infty \frac{(-1)^{n+1}H_n}{n+m+1}$?
Here, $H_n$ denotes the harmonic number. More colloquially, is there any way to represent $$\int_0^1 x^{n-1}\log^2\left(1+x\right)\ \mathrm{d}x$$ in a nice way? The latter is corollary to the original ...
- 125
2
votes
0
answers
368
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Upper bound the Polylogarithm $\sum_{n=1}^\infty \frac{x^n}{n^2}$
Let $x \in (0,1)$ be some real number, we can then consider the Polylogarithm:
$$\operatorname{L}_2(x)=\sum_{n=1}^\infty \frac{x^n}{n^2}$$
It is not hard to see that the following upper bound holds:
$$...
- 1,135
1
vote
0
answers
119
views
Generalized form of this Harmonic Number series $\sum_{n=1}^{\infty} \frac{{H_n}x^{n+1}}{(n+1)^3}$
i've tried to Evaluate $$\int_{0}^{\frac{\pi}{6}}x\ln^2(2\sin(x))dx$$ without using Contour integral
first i changed $2\sin(x)$ into polar form ,and i got $$\int_{0}^{\frac{\pi}{6}}x\ln^2(2\sin(x))dx ...
- 347
0
votes
1
answer
148
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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:
$$
I=-\frac{7}{12}\pi^4\ln^2(2)-\...
- 5,370
0
votes
0
answers
42
views
Upper-bounding $\exp \log^{d} \frac{\log n}{n}$
How would you upper-bound this expression?
$$f(n, d) = \exp \log^{d} \frac{\log n}{n}$$
If $d = 1 $ this woulld simplify to $\frac{\log n}{n}$.
Any suggestions on how to upperbound it?
Notation ...
- 2,670