# 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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### Closed form for $\sum_{n=1}^{\infty}\frac{x^n}{n!\sqrt{n}}$, or an asymptotic for it [duplicate]

In my study I came about the function defined by $$f(x)=\sum_{n=1}^{\infty}\frac{x^n}{n!\sqrt{n}}$$ and I have not been able to find any sort of closed form for it, nor have I been able to find any ...
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### Closed form of an integral that seems “alike” the Laplace integral

It is well-known that the Laplace integrals $$\int_0^{+\infty} \frac {\cos (ax)}{b^2 + x^2} \mathrm dx, \quad \int_0^{+\infty} \frac {x\sin (ax)}{b^2 + x^2} \mathrm dx$$ are computable to get a ...
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### Can we find a closed form for $\sum _{i=0}^{\infty } ((-1)^{i}x^i\prod_{j=1}^{i}\frac{e}{e^j-1})$?

Can we find a closed form for this infinite sum / product? \begin{align*} f(x) &= \sum _{i=0}^{\infty } \biggl((-1)^{i}x^i\prod_{j=1}^{i}\frac{e}{-1 + e^j} \biggr) \\ &= 1 - \biggl(\frac{e}{-1 ...
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### Closed form of $a_{k+2} = 4a_k + c^2$ when $c$ is some constant.

What is the closed form of $a_{k+2} = 4a_k + c^2$ when $c$ is some constant. How can we find the closed form of this recurrence with constant? Usually , I’ll use the characteristic root technique, but ...
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### Find the exact decay from a discrete sum of powers

In a discrete function of the form $h_n=\sum_k{A_k\,p_k^n}$, with $h_n$ being an unknown vector of real values, $A_k$, $p_k$ vectors of complex numbers, and $n$ integer, is it possible to find out the ...
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### Is there a closed form to this summation? [closed]

I cannot get a closed form for $\sum\limits_{r=0}^{m} \frac{(m+r) !}{(m-r)! (2 r)!}$ ‘ Does anyone have any idea on what it is?
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### Is the generalization $\sum_{n=1}^\infty\frac{H_{\frac np}}{n^q}$ known in the literature?

I managed to derive the following generalization $$\sum_{n=1}^\infty\frac{H_{\frac np}}{n^q}=(-1)^qp \sum_{n=1}^\infty\frac{H_{pn}}{(pn)^q}-\sum_{j=1}^{q-2}(-p)^{-j}\zeta(q-j)\zeta(j+1)\tag1$$ and ...
It is known that $$\frac{d^n\Bigl(f(x)+g(x)\Bigr)}{dx^n}=\frac{d^nf(x)}{dx^n}+\frac{d^ng(x)}{dx^n}$$ It can also be shown that $$\frac{d^n\Bigl(f(x)g(x)\Bigr)}{dx^n}=\sum_{k=0}^n\binom{n}{k}\frac{d^kf(... 3answers 271 views ### Closed form of \int_0^\infty \arctan^2 \left (\frac{2x}{1 + x^2} \right ) \, dx Can a closed form solution for the following integral be found:$$\int_0^\infty \arctan^2 \left (\frac{2x}{1 + x^2} \right ) \, dx.$$I have tried all the standard tricks such as integration by ... 3answers 79 views ### A closed form for the dilogarithm integral \int _{ 0 }^{ 1 }{ \frac { \operatorname{Li}_2\left( 2x\left( 1-x \right) \right) }{ x } dx }$$\int _{ 0 }^{ 1 }{ \frac { \operatorname{Li}_2\left( 2x\left( 1-x \right) \right) }{ x } dx } $$when I was solving an infinite series by using the beta function I encountered the above ... 1answer 78 views ### A parameterized log-sine integral equating to square of arctan I have a really convoluted proof of the following:$$ (1) \quad \quad - \int_0^\pi \frac{\sin(2 y) \log(\sin (y/2)) } {r + 1/r + 2 \cos{(2 y)} } dy = \big(\arctan(\sqrt{r})\big)^2 $$I proved it for ... 0answers 26 views ### How many degrees of freedom does a symplectic form have? A symplectic 2-form on a 2n dimensional manifold has to be closed, nondegenerate and antisymmetric. My question is: Do these conditions imply how many degrees of freedom the symplectic form has? I ... 0answers 47 views ### Is there a known closed form for these polynomials? The polynomials are like: P_0 = 1\\ P_1 = -2+4x\\P_2 = 8-32x+16x^2\\P_3=-48 + 288 x - 288 x^2 + 64 x^3 ? I have checked Legendre, Chebyshev, Hermite... Any help will be appreciated! This comes ... 1answer 41 views ### Request for reference links related to \sum_{k=1}^{\infty}(-1)^k(\zeta(ak) - 1) I was curious if there were any papers, Wikipedia articles, or YouTube videos that cover videos similar to the sum in the title. I’ve been looking for a while now, and the closest thing I got to was ... 2answers 96 views ### Closed form solution of Integral with many “free variables” I have a pretty complicated expression that I'm interested in integrating. There's a lot of parameters, so it looks pretty involved:$$ \int_{-\infty}^{\infty}d\Delta\frac{W \sqrt{\frac{\log (2)}{\pi ...
A very famous log-gamma integral due to Raabe is $$\int_0^1 \log \Gamma (x) \, dx = \frac{1}{2} \log (2\pi).$$ Several proofs of this result can be found here. I would like to known about the ...