All Questions
Tagged with logarithms closed-form
125
questions
3
votes
1
answer
62
views
Euler Sums of Weight 6
For the past couple of days I have been looking at Euler Sums, and I happened upon this particular one:
$$
\sum_{n=1}^{\infty}\left(-1\right)^{n}\,
\frac{H_{n}}{n^{5}}
$$
I think most people realize ...
- 203
1
vote
2
answers
81
views
Is there a closed form for the integral $\int_{0}^{\infty}\ln(z)z^{\lambda - 1}\exp\left(-\frac{w}{2}\left(z + \frac{1}{z}\right)\right)\mathrm{d}z$?
As the title says, I would like to know if there is a closed form for the integral:
\begin{align*}
\int_{0}^{\infty}\ln(z)z^{\lambda - 1}\exp\left(-\frac{w}{2}\left(z + \frac{1}{z}\right)\right)\...
- 31
8
votes
1
answer
177
views
how to integrate $\int_0^1 \ln^4(1+x) \ln(1-x) \, dx$?
I'm trying to evaluate the integral $$\int_0^1 \ln^4(1+x) \ln(1-x) \, dx,$$ and I'd like some help with my approach and figuring out the remaining steps.
or is it possible to evaluate $$\int_0^1 \ln^n(...
3
votes
1
answer
146
views
how to evaluate this integral $\int_0^1 \frac{\ln x \, \text{Li}_2(1-x)}{2+x} \, dx$
Question statement: how to evaluate this integral $$\int_0^1 \frac{\ln x \, \text{Li}_2(1-x)}{2+x} \, dx$$
I don't know if there is a closed form for this integral or not.
Here is my attempt to solve ...
3
votes
0
answers
186
views
how to find closed form for $\int_0^1 \frac{x}{x^2+1} \left(\ln(1-x) \right)^{n-1}dx$?
here in my answer I got real part for polylogarithm function at $1+i$ for natural $n$
$$ \Re\left(\text{Li}_n(1+i)\right)=\left(\frac{-1}{4}\right)^{n+1}A_n-B_n $$
where
$$ B_n=\sum_{k=0}^{\lfloor\...
- 1,637
3
votes
0
answers
129
views
evaluate $\int_{0}^{1} \log\left(\frac{(\sqrt{1 - x} - 1)(\sqrt{1 - x^2} + 1)}{(\sqrt{1 - x} + 1)(\sqrt{1 - x^2} - 1)}\right) \arcsin(x^2) \, dx$
$$\int_{0}^{1} \log\left(\frac{(\sqrt{1 - x} - 1)(\sqrt{1 - x^2} + 1)}{(\sqrt{1 - x} + 1)(\sqrt{1 - x^2} - 1)}\right) \arcsin(x^2) \, dx = \frac{\pi^3}{12} - \pi\left(6\sqrt{2} + 2 + \log\left(\frac{3}...
2
votes
2
answers
144
views
Is there a closed form solution to $\sum_{n=1}^{\infty}\frac{\ln\left(\frac{n+1}{n}\right)}{n}$ [duplicate]
While coming up with an idea for another way to milk the integral in my previous question, I got stuck at this summation: $$\sum_{n=1}^{\infty}\frac{\ln\left(\frac{n+1}{n}\right)}{n}$$
I do not know ...
- 1,688
1
vote
3
answers
262
views
Can the "simple" equation $e^x=\log(x)$ be solved using algebra?
I came across this really simple-looking yet astonishingly hard problem to solve:
$$e^x=\log(x).$$
I tried to use Lambert-W function, but I cannot get it to the required standard form. Even Wolfram ...
- 13
2
votes
2
answers
133
views
Does the second positive solution (besides $x = 1$) of the equation $e^{x^2-1}=x^3-x\ln x$ have a closed form?
Does the equation $$e^{x^2-1}=x^3-x\ln x$$ have a closed form solution ?
The given equation has $2$ positive real roots. Graphically
It is not hard to see that $x=1$ is a rational solution. The ...
- 644
8
votes
5
answers
611
views
Evaluation of $\int_0^1\frac{\log x\,dx}{\sqrt{x(1-x)(1-cx)}}$
Assume $c$ is a small real number.
QUESTION. What is the value of this integral in terms of the complete elliptic function $K(k)$?
$$\int_0^1\frac{\log x}{\sqrt{x(1-x)(1-cx)}}\,dx.$$
I got as far as ...
- 339
1
vote
2
answers
186
views
Closed form for solution of $\ln(x)\ln(x+1)=1 $ [closed]
Is it possible to find a close form for the solution of this equation (maybe with the use of Lambert W function)?
$$ \ln(x)\ln(x+1)=1 $$
- 843
3
votes
8
answers
386
views
Solving $\log_2(x+1)+x=2$
Here's the equation I'd like to solve:
$$\log_2\left(x+1\right)+x=2$$
Now I am aware that there's only one solution to the equation by graphing $y=-x+2$ and $y=\log_2\left(x+1\right)$.
The question is:...
- 405
2
votes
2
answers
116
views
Finding (approximate) intersection of logarithm and quadratic functions
I am trying to find a closed form solution to an equation of the form $$x^2+ax+b=c\log x$$
I found some resources that express solutions in terms of the Lambert W function if the LHS is linear but I ...
- 81
-1
votes
1
answer
199
views
How to solve for $x$ from $x + \ln(x) = \ln(c)$?
How do I solve this equation for P? For everything I've tried, P ends up trapped in an exponent or another natural log.
$$ \ln\left(\frac{GC}{a}\right) = hP + \ln(P) $$
- 1
5
votes
5
answers
228
views
Is there an algebraic solution to $\log_{\sqrt2}{\left(x\right)} = (\sqrt2)^x$?
I’m trying to solve
$$\log_{\sqrt2}{\left(x\right)} = (\sqrt2)^x$$
My next step is
$$\ln{x}= (\sqrt2)^x\ \cdot\ \ln\sqrt2$$
EDIT:
I’m only up to high school math.
- 123