All Questions
Tagged with polylogarithm logarithms
70
questions
1
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150
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Evaluating the indefinite integral $\int x^k \log (1-x) \log (x) \log (x+1) \, dx$
Recently I have calculated the long resisting indefinite integral $\int \frac{1}{x} \log (1-x) \log (x) \log (x+1) \, dx$ (https://math.stackexchange.com/a/3535943/198592).
Similar cases, but for ...
- 12.7k
2
votes
2
answers
44
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The equation $100\log(5x)\log(2x)+1 = 0$ has two distinct real roots $\alpha$ and $\beta$. Find the value of $\alpha\beta$.
The equation $100\log(5x)\log(2x)+1 = 0$ has two distinct real roots $\alpha$ and $\beta$. Find the value of $\alpha\beta$.
I'm having trouble with this because the answer key says $1/10$ as the ...
- 43
2
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2
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407
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Integral related to the softplus function
Let
$$
f(x) = \log(1+e^{2x+1}) - 2\log(1 + e^{2x}) + \log(1 + e^{2x-1}).
$$
According to Wolfram Alpha,
$$
\int_{-\infty}^\infty f(x)\,dx = \frac 12.\tag{$*$}
$$
$f(x)$ is a "bump function" built out ...
- 350
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182
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Generalizing Oksana's trilogarithm relation to $\text{Li}_3(\frac{n}8)$?
This was inspired by Oksana's post. Let, $$a = \ln 2 \quad\quad\\ b = \ln 3\quad\quad\\ c = \ln 5\quad\quad$$
then the following,
\begin{align}
A &= \text{Li}_3\left(\frac12\right)\\
B &= \...
- 54.9k
1
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Integrate $\int_{-\infty}^\infty [4(\log r_1 - \log r_2) - 2(x_1^2/r_1^2 - x_2^2/r_2^2)]^2 dx$
As the title suggests, I am having trouble evaluating the following definite integral:
$$\int_{-\infty}^\infty \left[4\left(\log r_1 - \log r_2\right) - 2\left(\frac{x_1^2}{r_1^2} - \frac{x_2^2}{r_2^...
- 11
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Upper-bounding $\exp \log^{d} \frac{\log n}{n}$
How would you upper-bound this expression?
$$f(n, d) = \exp \log^{d} \frac{\log n}{n}$$
If $d = 1 $ this woulld simplify to $\frac{\log n}{n}$.
Any suggestions on how to upperbound it?
Notation ...
- 2,670
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66
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Multi-logarithm generalisation with multipliers
I previously mentioned
a proposed "multi-stage logarithm" function, and managed to come up with a generalisation of the function.
Originally, the multi-logarithm was defined as:
$a_0^x+a_1^x+...+...
user301661
4
votes
2
answers
104
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"Multi-stage logarithm" series expansion (e.g. $a^x+b^x+c^x=d$)
As everyone knows, the solution to $a^x=b$ is $x=\log_a{b}$.
(Edit: Corrected from $x=\log_b{a}$.)
But what about $a^x+b^x=c$?
Let's define a "multilogarithm" function as:
$a_0^x+a_1^x+...+a_n^x=...
user301661
10
votes
4
answers
477
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Evaluate $\int_0^1 \frac{\mathrm{d}x}{\sqrt{1-x^2}}\frac{x }{1-k^2x^2}\log\left(\frac{1-x}{1+x}\right)$
I am trying to evaluate the following integral
$$I(k) = \int_0^1 \frac{\mathrm{d}x}{\sqrt{1-x^2}}\frac{x }{1-k^2x^2}\log\left(\frac{1-x}{1+x}\right)$$
with $0< k < 1$.
My attempt
By performing ...
- 168
0
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0
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135
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Validity of argument in dilogarithm identities on Wolfram
I've come across a series of identities existing between dilogarithms and powers of logarithms but I am not sure about when such equations are valid in terms of the restriction of the domain of the ...
- 2,850
1
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2
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495
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How to evaluate the integral $\int_0^2 \frac{\ln x}{\sqrt {x^2-2x+2}}dx$?
Yesterday's integral may be too difficult, I think the following integral should not be difficult.
$$I=\int_0^2 \frac{\ln x}{\sqrt {x^2-2x+2}}dx=\int_{-1}^1\frac{\ln(x+1)}{\sqrt {x^2+1}}dx=\int_{-1}^1\...
- 1,431
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5
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485
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Integral involving Dilogarithm $\int_{1/2}^{1} \frac{\mathrm{Li}_{2}\left(x\right)\ln\left(2x - 1\right)}{x}\,\mathrm{d}x$
I need your help in evaluating the following integral in closed form. $$\displaystyle\int\limits_{0.5}^{1}
\frac{\mathrm{Li}_{2}\left(x\right)\ln\left(2x - 1\right)}{x}\,\mathrm{d}x$$
Since the ...
- 407
0
votes
1
answer
122
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A dilogarithm identity (simplification/compaction) [duplicate]
I'm wondering if there is any compact expression to compute (or approximate):
$$\operatorname{Li}_2(pe^{-\alpha})-\operatorname{Li}_2(pe^{\alpha})$$
or
$$\operatorname{Re}\{\operatorname{Li}_2(pe^{-...
- 1,733
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1
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152
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A dilogarithm identity?
I'm wondering whether there any nice identities (or relationships) that can simplify or possibly compact the following expressions:
$$\operatorname{Li}_2(\beta e^{\alpha x}) - \operatorname{Li}_2(\...
- 1,733
-3
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How does one prove $f(g(x))=3f(x)$ in this case? [closed]
Prove $f(g(x))=3f(x)$ for:
$f(x)=\log_e \frac{1+x}{1-x}$ and $g(x)=\frac{3x+x^3}{1+3(x^2)}$
I have tried to use formula of $3\log_e x=\log_e x^3$
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