# 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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### Polynomials with degree $5$ solvable in elementary functions?

Quadratic, cubic and quartic polynomials are solvable in radicals, so there is no question here. What about the polynomials of degree $5$ (quintic)? Do we know all such polynomials (classes of ...
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### Challenging integral: $\int_0^Z\frac{\alpha^{(1-x^2)}}{1-x^2} dx$

I'd like to find a symbolic form for the following integral: $$f(\alpha, Z) = \int_0^Z\frac{\alpha^{(1-x^2)}}{1-x^2} dx$$ It is given that $0 \le \alpha \le 1$ and $0 \le Z < 1$. The following ...
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### Can anyone identify the function that represents this infinite product?

$$\lim_{\omega \to \infty} \prod_{N=1}^{\omega} {{1+e^{b \cdot c^{-N}}} \over 2}$$ For instance, the Lerch Transcendent is a analogous example of a special function that defines the sum of a useful ...
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### Is there a closed-form expression for Shapley value of glove game?

Suppose we have a coalition game with transferable utilities, with $m$ players having a right-handed glove and $n$ players having a left-handed glove. The value of a coalition is equal to the number ...
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### How do I find the finite limits of this infinite product?

What is... $$\lim_{\omega \to \infty} \left( {1 \over {a^{\omega}}} \cdot \prod_{N=1}^{\omega} (1+e^{b \cdot c^{-N}}) \right)$$ I'd like closed form solutions, and in this case that means any ...
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### Closed-form expression for a hypergeometric series

What is the closed-form expression for $${}_2 F_1 \left(1+2\lceil n/2\rceil,-n;1/2;-z/4\right)$$ According to the book Concrete Mathematics (R.Graham, D.Knuth, O.Patashnik 2nd), the authors say the ...
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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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Recently, when facing a baby Rudin's exercise, I proved that: $$\int_{0}^{y}\frac{|\cos x\,|}{1+x}\,dx = \frac{2}{\pi}\log(1+y)+O(1)$$ holds by integration by parts. Now I wonder if $$\color{red}... 0answers 86 views ### Closed form expression for a sum I want to calculate a sum of the form$$\sum_{k=0}^m \frac{\Gamma[m+1+\alpha-k]^2}{\Gamma[m+1-k]^2}\frac{\Gamma[x+k]}{\Gamma[x]k!}$$where m>0 and belongs to integers and \alpha takes half ... 0answers 47 views ### generalization of geometric series  \sum_{k=0}^\infty x^{p(k)}  I have been playing around with infinite series and I wondered if it is possible to find an expression for the series:$$ \sum_{k=0}^\infty x^{p(k)} $$as a generalization of geometric series. p(k) ... 0answers 38 views ### How to work with a recursive function with 2 recursive instances? In class, we figured out how to find the closed form of a recursive definition through the "basic 5 steps method". Example function T(n): If n = 1, T(1) = 1 If n > 1, T(n) = T(n-1)+1 Step 1: ... 0answers 98 views ### Closed-form expression of a definite integral Does this definite integral admit a closed-form in terms of elementary functions?$$\int_0^{\infty } \frac{x}{\left(x^4+1\right) \left(2 x^2-2 \arctan\left(x^2\right)+\pi \right)} \, dx.$$0answers 52 views ### Solving equation with LambertW function? Does the equation$$ a = b x e^x + c x + d e^x $$have a solution form solution? I tried to look for it by using the LambertW funcion, but I did not succeed. Thanks in advance. 0answers 73 views ### Closed form of specific series I'm working on a problem that involves the integrals of various Bessel functions that Mathematica can't symbolically handle. I've managed to grind out the transformations and integrals by hand, and ... 0answers 118 views ### A photon in expanding Universe (a snail on a tree) I want to know how far a snail can reach in expanding universe. It has a constant speed c = 1 and tree is expanding at speed v= H_0 D, with Hubble constant H_0 = 1. Here D(T) is the distance of ... 0answers 108 views ### How to resolve this equation for f(n) without using f(n-1) I have an equation related to some work I'm doing on Poisson distribution where I'm calculating a sequence of 100 values between a minimum and maximum value which is set by another formula. ... 0answers 28 views ### Maxima of f(x)/e^x where f(x) is an approximation of e^x using Stirling's Let$$f(x)=1+\sum_{n=1}^\infty\frac{x^n}{\sqrt{2\pi n}(n/e)^n}\tag1$$and let$$g(x)=\frac{f(x)}{e^x}\tag2 If we plot $g(x)$ we get a graph that looks like this: Clearly there is a maximum at ...
Assume $I=\int_0^\infty f(x)\text{ d}x$ and $J=\sum_{n=0}^\infty f(n) ; I,J\in\Bbb{R}$ Conjecture: If $I$ has a closed form, then $J$ must carry a closed form. Can someone find a proof or ...