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This is not actually a Fourier series but a rather look like Fourier series. Anyway the solution will require us to take a look at the complex exponential series. Well mapping $x \mapsto xe^{ia}$ we have that: \begin{equation} \log \left ( 1-xe^{ia} \right ) = -\sum_{n=1}^{\infty} \frac{x^ne^{ina}}...

Let $\alpha, \beta$ be arbitrary positive integer numbers such that $\alpha>\beta$ and $\alpha^2 - \beta^2$ is prime. If a function $f:\mathbb{R} \rightarrow \mathbb{R}$ is continuous then evaluate the integral: $$\mathcal{J} = \int_{\alpha^2-\beta^2}^{\alpha + \beta} \frac{f^2(t) + f^4(t)}{1+f^{10}...

The $k=7$ case: \begin{align*} \int_{0}^{\pi/2} \frac{x^7}{\tan x} \, {\rm d}x &= \int_{0}^{\pi/2} x^7 \cot x \, {\rm d}x \\ &= -7 \int_{0}^{\pi/2} x^6 \log \sin x \, {\rm d}x\\ &= -7 \int_{0}^{\pi/2} x^6 \left ( -\log 2 - \sum_{n=1}^{\infty} \frac{\cos 2nx}{n} \right ) \, {\rm d}x\\ &a...

May we have a hint Vaggelis?

General notes on the polylogs. Let $n \in \mathbb{N}$. Then ${\rm Li}_n$ is analytic and single valued on $\mathbb{D}= \mathbb{C} \setminus [1,+\infty)$ and note that this domain is symmetric with respect to the real axis. $g(z)=\overline{{\rm Li}_n(\overline{z})}$ is also analytic on the same doma...

Now let us move on to the trilogarithm function. It is defined as $\displaystyle {\rm Li}_3 (z) = \sum_{n=1}^{\infty} \frac{z^n}{n^3}= \int_{0}^{z}\frac{{\rm Li}_2 (t)}{t} \, {\rm d}t$. 1. $\displaystyle {\rm Li}_3(i) = -\frac{3 \zeta(3)}{32} + i \; \frac{\pi^3}{32}$ since: 1st way: \begin{align*} {...

Based on the above fact here are some exercises that are left to the reader. 1. Prove that: $$\Im\operatorname{Li}_2 \left[\left(i\left(2\pm\sqrt3\right)\right) \right] =\frac{2 \mathcal{G}}{3}-\frac{\pi\,(2\pm3)}{12}\ln\left(2-\sqrt3\right)$$ ( Hint: Use another great equation the dilogarithm satis...

1. $\displaystyle \Im\left [ {\rm Li}_2 (i) \right ] =\mathcal{G}$ since in general \begin{align*} {\rm Li}_2 (iz) &= - \int_{0}^{z} \frac{\log (1-it)}{t} \, {\rm d}t \\ &= -\int_{0}^{z} \frac{\log \left [ \left ( 1+t^2 \right )^{1/2} e^{-i \arctan t} \right ]}{t} \, {\rm d}t\\ &= -\fra...

Maybe the following are worth deriving. 1. $\displaystyle \Im\left [ {\rm Li}_2 (i) \right ] =\mathcal{G}$ since in general \begin{align*} {\rm Li}_2 (iz) &= - \int_{0}^{z} \frac{\log (1-it)}{t} \, {\rm d}t \\ &= -\int_{0}^{z} \frac{\log \left [ \left ( 1+t^2 \right )^{1/2} e^{-i \arctan t}...

Let $p$ be a prime and let $\mathbb{F}_p=\mathbb{Z} / p \mathbb{Z}$. Find all $p \times p$ matrices $A$ and $B$ over $\mathbb{F}_p$ such that $AB - BA = \mathbb{I}$. Question: Can you do that in characheristic zero or for $n \times n$ matrices where $p$ does not divide $n$ ? Give a brief explanation.