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Tagged with polylogarithm fourier-series
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Complex polylogarithm/Clausen function/Fourier series
Sorry for the confusing title but I'm having a problem and I can phrase the question in multiple different ways.
I was calculating with WolframAlpha
$$\int \text{atanh}(\cos(x))\mathrm{d}x= i \text{Li}...
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How can you approach $\int_0^{\pi/2} x\frac{\ln(\cos x)}{\sin x}dx$
Here is a new challenging problem:
Show that
$$I=\int_0^{\pi/2} x\frac{\ln(\cos x)}{\sin x}dx=2\ln(2)G-\frac{\pi}{8}\ln^2(2)-\frac{5\pi^3}{32}+4\Im\left\{\text{Li}_3\left(\frac{1+i}{2}\right)\right\}$$...
- 25.8k
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Estimate of integral of Fourier series
Not really my area, so I appeal to you: any idea on how to estimate $$\zeta^{-k}(1+\alpha)\int_0^1 \left(\sum_{q=1}^\infty \cos(2\pi q x)q^{-1-\alpha}\right)^k dx$$ for $k\in\mathbb{N}$ and $0<\...
- 1,022
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Calculate $\sum_{n=-\infty}^{\infty}\frac{1-\cos(an)}{(an)^2}$
After playing with some series in a numerical math website, it seems to me like the following identity holds:
$$\sum_{n=-\infty}^{\infty}\frac{1-\cos(an)}{(an)^2}=\frac{\pi}{a}$$
It seems a little ...
- 1,192
2
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answer
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Real Part of the Dilogarithm
It is well known that
$$\frac{x-\pi}{2}=-\sum_{k\geq 1}\frac{\sin{kx}}{k}\forall x\in(0,\tau),$$
which gives
$$\frac{x^2}{4}-\frac{\pi x}{2}+\frac{\pi^2}{6}=\sum_{k\geq 1}\frac{\cos(kx)}{k^2}.$$
...
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Analytically solving an infinite series of periodic functions
I have the infinite series from a Fourier series problem:
$\sum_{n=1}^\infty \frac{1}{n^2}sin(n\pi x)sin(n\pi y)$
which is stated to be proportional to $x*(1-y)$ when $0 \leq x \leq 1$ and $x \leq y ...
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Closed form of a series (dilogarithm)
We are all aware of the dilogarithm function (Spence's function):
$$\sum_{n=1}^{\infty} \frac{x^n}{n^2}, \;\; x \in (-\infty, 1]$$
Also it is known that:
$$\sum_{n=1}^{\infty} \frac{\cos n x}{n^2}= ...
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