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Tagged with closed-form calculus
70
questions with no upvoted or accepted answers
16
votes
0
answers
818
views
Juantheron-like integral
While seeing this post, the following integral is just struck me
\begin{equation}
\int_0^\infty \frac{dx}{(1+x^2)(1+\tan x)}\tag1
\end{equation}
I have tried like what user @OlivierOloa did in his ...
- 2,387
10
votes
0
answers
262
views
Integrals involving powers and beta (or hypergeometric) function
I have the three following integrals, very similar the one to the others,
$$I_1^{(p)}(N)\equiv\frac{1}{2^{N+p}}\int_0^1(1+t)^{N-1}(1-t)^pB\left(\frac{1}{t+1};N+p+1,N\right)\text{d}t$$
$$I_2^{(p)}(...
- 299
10
votes
0
answers
2k
views
Exact values of error function
The error function is defined as
$$\operatorname{erf}(z)=\frac{2}{\sqrt{\pi}} \int_0^z e^{-t^2} \, dt.$$
We know that the Gaussian integral is
$$\int_{-\infty}^{\infty} e^{-x^2}\,dx=\sqrt{\pi}.$$
...
- 12.4k
9
votes
0
answers
444
views
Why do these two integrals use roots of reciprocal polynomials?
There is the nice integral by V. Reshetnikov,
$$\int_0^1\frac{dx}{\sqrt[3]x\ \sqrt[6]{1-x}\ \sqrt{1-x\,\alpha^2}}=\frac{1}{N}\,\frac{2\pi}{\sqrt{3}\;\alpha}\tag1$$
also discussed in this post. By ...
- 54.9k
9
votes
0
answers
387
views
Closed form of $\sum _{n=0}^{\infty} \frac{\left(-\pi ^2\right)^n \cos \left(2^nb\right)}{(2 n)!}$
Is it possible to calculate the sum
$$
\sum _{n=0}^{\infty} \frac{\left(-\pi ^2\right)^n \cos \left(2^nb\right)}{(2 n)!}
$$
in closed form?
Formal naive argument gives
$$
\sum _{n=0}^{\infty} \...
- 1,695
8
votes
0
answers
413
views
More on the log sine integral $\int_0^{\pi }\theta ^{3}\log^{3}\left ( 2\sin\frac{\theta }{2} \right )\mathrm{d}\theta$
I. In this post, the OP asks about the particular log sine integral,
$$\mathrm{Ls}_{7}^{\left ( 3 \right )} =-\int_{0}^{\pi }\theta ^{3}\log^{3}\left ( 2\sin\frac{\theta }{2} \right )\,\mathrm{d}\...
- 54.9k
8
votes
0
answers
143
views
More on $\sum_{n=1}^\infty\frac{(4n)!}{\Gamma\left(\frac23+n\right)\,\Gamma\left(\frac43+n\right)\,n!^2\,(-256)^n}$
Let,
$$\alpha=2\sqrt[3]{1+\sqrt2}-\frac2{\sqrt[3]{1+\sqrt2}}$$
In this post, it was asked if,
$$\sum_{n=1}^\infty\frac{(4\,n)!}{\Gamma\left(\frac23+n\right)\,\Gamma\left(\frac43+n\right)\,n!^2\,(-256)^...
- 54.9k
8
votes
0
answers
415
views
Improper Integral $\int_0^\infty\tan\left(\frac x{\sqrt{x^3+x^2}}\right)\frac{\ln(1+\sqrt x)}xdx$
This integral is from integral
Find
$$\int_0^\infty\tan\left(\frac x{\sqrt{x^3+x^2}}\right)\frac{\ln(1+\sqrt x)}xdx$$
I have get
$$\int_0^\infty\tan\left(\frac x{\sqrt{x^3+x^2}}\right)\frac{\ln(1+\...
- 93.6k
7
votes
1
answer
295
views
how to evaluate $\int_0^{\infty} \frac{x \ln ^2\left(1-e^{-2 \pi x}\right)}{e^{\frac{\pi x}{2}}+1} d x$
Question: how to evaluate $$\int_0^{\infty} \frac{x \ln ^2\left(1-e^{-2 \pi x}\right)}{e^{\frac{\pi x}{2}}+1} d x$$
MY try to evaluate the integral
$$
\begin{aligned}
& I=\int_0^{\infty} \frac{x \...
user1313975
7
votes
0
answers
276
views
Evaluate $\int_{0}^{1} \frac{K(k)E(k)^2-\frac{\pi^3}{8} }{k} \text{d}k$ and $\int_{0}^{1} \frac{E(k)^3-\frac{\pi^3}{8} }{k} \text{d}k$
Let $K(k),E(k)$ be the complete elliptic integral of the first kind and second kind respectively, where $k$ is the elliptic modulus. Consider four integrals,
$$\begin{aligned}
&I_1=\int_{0}^{1} \...
- 5,370
7
votes
0
answers
302
views
Closed-form of integrals containing double exponentials
Are there closed forms for the following integrals?
$$\begin{align}
I_1(w) & = \int_{-\infty}^{\infty} \frac{\exp(-we^y)}{y^2+\pi^2} dy, \\
I_2(w) & = \int_{-\infty}^{\infty} \frac{\exp(-we^y+...
- 12.4k
6
votes
0
answers
104
views
Please help me identify any errors in my solution to the following DE: $xf(x)-f'(x)=0$, $f(0)=1$
Context/background:
I am self-studying series, first in the context of generating functions and now in the context of functional/differential equations. As such, I like to set myself practise problems,...
- 1,632
6
votes
0
answers
172
views
How to evaluate $\int_0^1 \dfrac{\operatorname{Li}_2\left(\frac{x}{4}\right)}{4-x}\,\log\left(\dfrac{1+\sqrt{1-x}}{1-\sqrt{1-x}}\right)\,dx$
crossposted: https://mathoverflow.net/q/464839
How to evaluate $$\int_0^1 \dfrac{\operatorname{Li}_2\left(\frac{x}{4}\right)}{4-x}\,\log\left(\dfrac{1+\sqrt{1-x}}{1-\sqrt{1-x}}\right)\,dx=\dfrac{\pi^...
6
votes
0
answers
270
views
Integral of a product of five Bessel functions of order $0$
Does the following integral have a closed form?
$$ \mathcal{J}(2,3,5,7,11) = \int_0^\infty x J_0(x\sqrt{2})J_0(x\sqrt{3})J_0(x\sqrt{5})J_0(x\sqrt{7})J_0(x\sqrt{11})\,dx. $$
I know that some similar ...
- 14.6k
5
votes
0
answers
326
views
Converting $ \int_0^{\infty} \frac{e^{-\varepsilon s} \, (s+s^2)^{\beta}}{\log^{\gamma}((1+s)/s)} \, ds$ to a sum?
Could anyone shed some light on how to convert the following integral to a sum?
$$ I=\int_0^{\infty} \frac{e^{-\varepsilon s} \, (s+s^2)^{\beta}}{\log^{\gamma}((1+s)/s)} \, ds; \qquad\,\,\varepsilon,\...
- 1,691