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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Does this equation has a closed form solution?

We have $K$ non-negative coefficients: $a_1,a_2,\dots,a_K,A_1,A_2,\dots,A_K$, where $A_i\geq0,\;a_i\in(0,1),\;\sum A_i<T$. The equation is: $$\sum_{i=1}^K\frac{A_i}{1-a_ix}=T,\quad x\in(0,1)$$
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General formula for the higher order derivatives of composition with exponential function

Suppose I have a function $x:\mathbb{R} \to \mathbb{R}$ and consider: $$g(t) = e^{x(t)}$$ When I start differentiating with respect to $t$ I obtain: \begin{align} g'&=e^xx'\\ g''&=e^x((x')^2+x'...
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Integral of the square root of a trigonmetric function

Despite my best attempts, I have been unable to evaluate the following integral: $$\int_s^t\sqrt{9+(2+\cos3u)^2}\,du.$$ This integral showed up during an investigation of torus knots. It represents ...
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Prove $\int_0^\infty \frac{x^{k-1} + x^{-k-1}}{x^a + x^{-a}}dx = \frac{\pi}{a \cos(\frac{\pi k}{2a})}$.

I need help in proving this identity $$\int_0^\infty \frac{x^{k-1} + x^{-k-1}}{x^a + x^{-a}}dx = \frac{\pi}{a \cos(\frac{\pi k}{2a})}$$ for $0<k<a.$ It might be done using residues, but I don'...
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How to find Real Part of PolyLog[3,(1-i)] in closed form

$\Re \bigg(\text{Li}_3(1-i)\bigg)=\frac{\pi^a}{b}\ln(2)+\frac{c}{d}\zeta(e)$ has an approximate value of .8711588834109380 if $a=1 , b=-3415 , c=34 , d=39 , e=19$ are substituted into the closed ...
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Closed-form Solution to a Sum

I have some math questions for a programming course where it says to provide closed-form solutions for a list of sums. I've never taken an algorithms course, so I'm not quite sure what I'm doing. I ...
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Closed form for this integral $I=\int_0^{1}\frac{{\arcsin}({x^2})}{\sqrt{1-x^2}}dx$

I’m trying to find a closed form for this integral.Any help is appreciated.Thanks $$I=\int_0^{1}\frac{{ \arcsin}({x^2})}{\sqrt{1-x^2}}dx$$
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Geodesic distance between equidistant points on a sphere [closed]

On the unit sphere equidistant points can be found for $1, 2, 3, 4, 6, 8, 12, 20$. The geodesic distance between the points are $\pi$ for $2$, $2\pi\over 3$ for $3$, $\pi\over 2$ for $6$, etc... Is ...
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When is a multiple sum given in closed form?

Let $d$ be a positive integer and $a>0$. Consider a following multiple sum: {\mathcal S}^{(d)}_a(j) := \sum\limits_{0 \le j_1 \le j_2 \le \dots \le j_d \le j} \prod\limits_{l=1}^d ...
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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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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 ...
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 ...