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".

2,397 questions
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Integral $\int_{-\infty}^{\infty}\ln(2-2\cos(x^2))dx=-\sqrt{2\pi}\zeta(3/2)$

Prove that $$\int_{-\infty}^{\infty}\ln(2-2\cos(x^2))dx=-\sqrt{2\pi}\zeta(3/2)$$ I was given this integral in my post Request for crazy integrals. I have never seen an integral like this before ...
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Generalized central binomial coefficients convolution

It is well-known that \begin{align*} \sum_{i=0}^n \binom{2i}{i}\binom{2n-2i}{n-i} = 4^n, \end{align*} where one might use combinatorial arguments or generating function technique to prove this. Now I ...
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Methods to derive a closed form for $I_n=\int_0^\infty \frac{\ln^n(x+1)-\ln^n(x)}{x+1}dx$

I've stumbled onto this general integral that has closed form values for the $n\in \Bbb{Z^+}$ $$I_n=\int_0^\infty \frac{\ln^n(x+1)-\ln^n(x)}{x+1}dx$$ Obviously $I_0=0$ but higher values of $n$ yield ...
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Compute this following integral without Fourier series : $\int_0^{\pi/4}x\ln(\tan x)dx$

Compute the following integration without harmonic series or Fourier series : $I=\displaystyle\int_0^{\frac{π}{4}}x\ln(\tan x)dx$ Wolfram alpha give $I=\frac{7\zeta(3)-4πC}{16}$ Where $C$ : Catalan'...
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Integral $\int_0^\infty \frac{\ln(1+x+x^2)}{1+x^2}dx$

Prove that$$I=\int_0^\infty \frac{\ln(1+x+x^2)}{1+x^2}dx=\frac{\pi}{3}\ln(2+\sqrt 3)+\frac43G$$ I've found this integral in my notebook and perhaps I encountered it before since it looks quite ...
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