# Questions tagged [closed-form]

A "closed form expression" is any representation of a mathematical expression in terms of "known" functions, "known" usually being replaced with "elementary".

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### $I(x) = -\int_0^1 \frac{1}{z}\ln\left(\frac{1-x z + \sqrt{1-2 x z+ z^2}}{2}\right)\,dz$

Is there a closed form integral for $$I(x) = -\int_0^1 \frac{1}{z}\ln\left(\frac{1-x z + \sqrt{1-2 x z+ z^2}}{2}\right)\,dz$$ for $-1 < x < 1$? This integral is related to Legendre polynomials ...
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### Determine closed forms for generating functions

I'm studying generating functions for my study of number theory and I'm trying to solve some exercises. I need to find closed forms for the generating functions of the following sequences: a) $a_n=n^3$...
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### formulas for binary expansion of irrational number between $0$ and $1$

One can write any irrational number between $0$ and $1$ composed as closed expression of popular known numbers, such as, for example, the expression $$\frac{1}{\sqrt{2}}$$ in binary by successively ...
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### Closed form for $\prod_{j=1}^n\prod_{k=1}^{m_{j}-1} (x_j-kN^{-1})$

Let $n,N\in\mathbb{N}$ and $m_1,x_1,\ldots,m_n,x_n\in\mathbb{N}$. I'm trying to rewrite following expression $$\prod_{j=1}^n\prod_{k=0}^{m_{j}-1} (x_j-kN^{-1})$$ I am only interested in the summands ...
Given $-1 \leq x \leq 1$ and $0 \leq \eta \leq 1$, I am interested in computing $$E(x,\eta) = \sum_{\ell = 0}^{+ \infty} |P_{\ell} (0)|^{2} \, P_{\ell} (x) \, \eta^{\ell} ,$$ with $P_{\ell}$ the ...