All Questions
Tagged with closed-form calculus
835
questions
37
votes
4
answers
2k
views
Closed form for $\sum_{n=-\infty}^{\infty}\frac{1}{(n-a)^2+b^2}$.
Find the closed form of $$\sum_{n=-\infty}^{\infty}\frac{1}{(n-a)^2+b^2}.$$
We can use the Fourier series of $e^{-bx}$ ($|x|<\pi$) to find $$\sum_{n=-\infty}^{\infty}\frac{1}{n^2+b^2}.$$ But here ...
- 45.3k
6
votes
3
answers
269
views
Unforeseen issue in my MastersThesis: Is there a "closed form" solution?
I'm an electrical engineer and I recently came across an unforeseen issue in my masters thesis because I lack a deeper mathematical education.
I want to know for which positive real $x$ the following ...
- 143
14
votes
6
answers
638
views
How to find $\int_0^\pi (\log(1 - 2a \cos(x) + a^2))^2 \mathrm{d}x, \quad a>1$?
Integration by parts is of no success. What else to try?
$$\int_0^\pi (\log(1 - 2a \cos(x) + a^2))^2 \mathrm{d}x, \quad a>1$$
- 149
2
votes
3
answers
24k
views
Finding the sum of series help: $\displaystyle\sum\limits_{n=1}^{\infty} (-9)^nx^n. $
I have a question that I am kind of stuck on. I am good with series stuff, but this one kind of threw me off
$$\sum\limits_{n=1}^{\infty} (-9)^nx^n. $$
This is the question and I was supposed to find ...
- 886
23
votes
6
answers
1k
views
Evaluate $\int_{0}^{\pi/4}x\ln^{2}(\sin(x))dx$
Here is a some what challenging log sine integral.
$$I=\int_{0}^{\pi/4}x\ln^{2}(\sin(x))dx$$
The upper limit of integration is $\frac{\pi}{4}$ instead of the usual $\frac{\pi}{2}$.
I tried the ...
- 14.1k
71
votes
5
answers
4k
views
Show that $\int_{0}^{\pi/2}\frac {\log^2\sin x\log^2\cos x}{\cos x\sin x}\mathrm{d}x=\frac14\left( 2\zeta (5)-\zeta(2)\zeta (3)\right)$
Show that :
$$
\int_{0}^{\Large\frac\pi2}
{\ln^{2}\left(\vphantom{\large A}\cos\left(x\right)\right)
\ln^{2}\left(\vphantom{\large A}\sin\left(x\right)\right)
\over
\cos\left(x\right)\sin\left(x\...
- 3,955
18
votes
4
answers
1k
views
How to integrate $\int_{0}^{\infty }{\frac{\sin x}{\cosh x+\cos x}\cdot \frac{{{x}^{n}}}{n!}\ \text{d}x} $?
I have done one with $\displaystyle\int_0^{\infty}\frac{x-\sin x}{x^3}\ \text{d}x$, but I have no ideas with these:
$$\begin{align*}
I&=\int_{0}^{\infty }{\frac{\sin x}{\cosh x+\cos x}\cdot \frac{{...
- 2,251
7
votes
2
answers
2k
views
Functions with no closed-form derivative
There are many well-known functions (such as $x^x$) which have no closed-form integral. However, are there any elementary functions whose derivatives cannot be expressed in a closed-form manner?
- 293
37
votes
5
answers
3k
views
Evaluating $\int_0^{\frac\pi2}\frac{\ln{(\sin x)}\ \ln{(\cos x})}{\tan x}\ dx$
I need to solve
$$
\int_0^{\Large\frac\pi2}\frac{\ln{(\sin x)}\ \ln{(\cos x})}{\tan x}\ dx
$$
I tried to use symmetric properties of the trigonometric functions as is commonly used to compute
$$
\...
- 945
3
votes
1
answer
278
views
Closed forms of sums $f(a)+f(a+d)+\cdots+f(a+nd)$ with $f$ sine, cosine or tangent
is/are there a closed form for
$\sin{(a)}+\sin{(a+d)}+\cdots+\sin{(a+n\,d)}$
$\cos{(a)}+\cos{(a+d)}+\cdots+\cos{(a+n\,d)}$
$\tan{(a)}+\tan{(a+d)}+\cdots+\tan{(a+n\,d)}$
$\sin{(a)}+\sin{(a^2)}+\...
- 691