# Tagged Questions

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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### Why is this the closed-form solution for this series? [duplicate]

I know this is simple, but I don't know very much at all about series, and I'm wondering how it's shown that: $$1 + 2 + 3 + \cdots + (n - 1) = \frac{n(n - 1)}{2}$$
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### Closed-form of spherical expansion of Legendre polynomial $P_k(\sin{\theta}\cos{\varphi})$

During the times of working on some problem in astro/geophysics I have come across a problem involving an expansion into spherical harmonic functions (this is the remnance of nomenclature there used ...
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### Pseudo-inverse of the Cumulative Distribution Function of X

The goal of these calculations is to write a Python function that generates pseudo-random values with the distribution described below. This isn't relevant to the question (or even to this community);...
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### Improper Integral $\int_0^1\frac{\arcsin^2(x^2)}{\sqrt{1-x^2}}dx$

$$I=\int_0^1\frac{\arcsin^2(x^2)}{\sqrt{1-x^2}}dx\stackrel?=\frac{5}{24}\pi^3-\frac{\pi}2\log^2 2-2\pi\chi_2\left(\frac1{\sqrt 2}\right)$$ This result seems to me digitally correct? Can we prove ...
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### Non-existence of closed-form solutions

An equation like $$a^x+b^x=1$$ can be turned to the form $$t^\alpha+t=1$$ by a suitable change of variable. When $\alpha$ is a rational we can put that in a polynomial form $$u^p+u^q=1$$ and ...
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### Closed form of this binomial expression?

Does a closed form for this binomial expression exists? $\sum_{K=2}^{N}\binom{N}{K}P^{K}(1-P)^{N-K}$ Thank you.
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### integral involving error function (erf)

Does anybody know if a closed form of this integral exist? $\int \mbox{erf}(x) \ln(\mbox{erf}(x)) \Bbb dx$ where erf is so called error function. In case there is no closed form solution. Is it ...
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### Extract imaginary part of $\text{Li}_3\left(\frac{2}{3}-i \frac{2\sqrt{2}}{3}\right)$ in closed form

We know that polylogarithms of complex argument sometimes have simple real and imaginary parts, e.g. $\mathrm{Re}[\text{Li}_2(i)]=-\frac{\pi^2}{48}$ Is there a closed form (free of polylogs and ...
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### What is asymptotics of this oscillatory double sum? (Fractal Dimension problem)

The term Gibbs Phenomenon refers to the peculiar way Fourier Series behave at sharp changes in a function's value. However, this problem becomes particularly annoying to deal with when trying to ...
### How to solve this integral $\int _0^{\infty} e^{-x^3+2x^2+1}\,\mathrm{d}x$
My classmate asked me about this integral:$$\int _0^{\infty} e^{-x^3+2x^2+1}\,\mathrm{d}x$$ but I have no idea how to do it. What's the closed form of it? I guess it may be related to the Airy ...
$$\int \sqrt{\frac{x^2+x-1}{x^2-1}} dx$$ I have been trying to find this integral for a while and I just can't. Does it even have a closed form?