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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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 ...
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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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Simplify $\int_0^\infty \frac{\text{d}{x}}{e^x+x^n}$

I seem to have seen quite a lot of integrals in the form: $$\int_0^\infty \frac{\text{d}x}{e^x+(1+x^n)}$$ But none of those hold a closed forms (at least to my knowledge) ...
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Closed form of $\int_{x = 0}^{C} \exp\left(-\frac{x}{A}-\frac{B}{x}\right)\,dx$

Is there a closed-form expression for the following definite integral? $$\int_{x = 0}^{C} \exp\left(-\frac{x}{A}-\frac{B}{x}\right)\,dx,$$ where $A$, $B$, and $C$ are ...
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Integral $\int_0^\infty\Big[\log\left(1+x^2\right)-\psi\left(1+x^2\right)\Big]dx$

I found this intriguing integral: $$\int_0^\infty\Big[\log\left(1+x^2\right)-\psi\left(1+x^2\right)\Big]dx\approx0.84767315533332877726581...$$ where $\psi(z)=\partial_z\log\Gamma(z)$ is the digamma. ...
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Quick way to get closed form for this recurrence?

Is there supposed to be a fast way to compute recurrences like these? $T(1) = 1$ $T(n) = 2T(n - 1) + n$ The solution is $T(n) = 2^{n+1} - n - 2$. I can solve it with: Generating functions. ...
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Conjecture ${\large\int}_0^\infty\left[\frac1{x^4}-\frac1{2x^3}+\frac1{12\,x^2}-\frac1{\left(e^x-1\right)x^3}\right]dx=\frac{\zeta(3)}{8\pi^2}$

I encountered the following integral and numerical approximations tentatively suggest that it might have a simple closed form: ...
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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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Closed form of $\sum\frac{1}{k}$ where $k$ has only factors of $2,3$

Consider the set containing $A$ all positive integers with no prime factor larger than $3$, and define $B$ as $$B= \sum_{k\in A} \frac{1}{k}$$ Thus, the first few terms of the sum are: ...
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Need help with $\int_0^\infty\arctan\left(e^{-x}\right)\,\arctan\left(e^{-2x}\right)\,dx$

I was able to calculate: $$\int_0^\infty\arctan\left(e^{-x}\right)\,dx=G$$ $$\int_0^\infty\arctan^2\left(e^{-x}\right)\,dx=\frac\pi2\,G-\frac78\zeta(3)$$ $G$ is the Catalan constant. In both cases ...
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Rational series representation of $e^\pi$

This question is related to Why $e^{\pi}-\pi \approx 20$, and $e^{2\pi}-24 \approx 2^9$? by Tito Piezas III. Andrew Fraker (2014) found an almost-integer which is equivalent to the following ...
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How to solve even/odd divide-and-conquer problems?

I am looking into something called the Josephus problem, which seems to be popular, so I am sure there are lots of explanations online, but I want to do the work myself, but I do need a small push to ...
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OEIS A249665 generating function

I'm stuck at finding the general term of the sequence $$1, 1, 1, 2, 6, 14, 28, 56, 118, 254, 541, 1140, 2401, 5074, \ldots$$ According to OEIS, Colin Barker conjectured the recurrence relation to be ...
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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 ...
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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 ...
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Integrate the square root of the ratio of two quadratic polynomials

$$\int \sqrt{\frac{x^2+x-1}{x^2-1}} dx$$ I have been trying to find this integral for a while and I just can't. Does it even have a closed form?
How to solve $\int\ x^{\ln x} dx$? [closed]
How to solve this integral $$\int\ x^{\ln x} dx$$ step by step?
Conjecture $\sum_{m=1}^\infty\frac{y_{n+1,m}y_{n,k}}{[y_{n+1,m}-y_{n,k}]^3}\overset{?}=\frac{n+1}{8}$, where $y_{n,k}=(\text{BesselJZero[n,k]})^2$
While solving a quantum mechanics problem using perturbation theory I encountered the following sum $$S_{0,1}=\sum_{m=1}^\infty\frac{y_{1,m}y_{0,1}}{[y_{1,m}-y_{0,1}]^3},$$ where ...