All Questions
3
questions
5
votes
3
answers
258
views
How to find the exact value of $\sum_{n=1}^{\infty} \frac{\sin \left(\frac{n \pi}{4}\right)}{n^2 \cdot 2^{\frac{n}{2}}} $?
Once I met the identity
$$
\boxed{S_0=\sum_{n=1}^{\infty} \frac{\sin \left(\frac{n \pi}{4}\right)}{2^{\frac{n}{2}}}=1},
$$
I first tried to prove it by $e^{xi}=\cos x+i\sin x$.
$$
\begin{aligned}
\...
- 22.3k
3
votes
1
answer
215
views
Is there an analytic solution to $\int_a^b \frac{\arctan(A+Bt)}{C^2 + (t-Z)^2}dt$?
Is there a sensible analytic solution to the following integral: $$\int_a^b \frac{\arctan(A+Bt)}{C^2 + (t-Z)^2}dt$$ where all constants are real and $C>0$.
This integral is part of the third term ...
- 31
0
votes
2
answers
150
views
Complex logarithms when computing real-valued integral
My question arise when I try to calculate real-valued integral, specifically, I want to evaluate the integral
\begin{equation} \int_0^1 \frac{\ln \left(\frac{x^2}{2}-x+1\right)}{x} dx
\end{equation}
...
- 138