# 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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### 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 ...
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### Integral with Legendre polynomial: $\int_{-1}^{1}x^{n+2k}P_{n}(x)dx$

How to compute the following integral? $$\int_{-1}^{1} x^{n+2k}P_{n}(x) dx$$ where $P_n(x)$ is the Legendre function, and $n, k = 1,2, \cdots.$
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### How to integrate $\frac{dx}{(x-p)\sqrt {(x-p)(x-q)}}$?

How to integrate $\frac{dx}{(x-p)\sqrt {(x-p)(x-q)}}$ ? I tried substituting $x=1/t$ but that's making it more complicated. Any suggestions?
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### Closed form of infinite product $\prod\limits_{k=0}^\infty 2 \left(1-\frac{x^{1/2^{k+1}}}{1+x^{1/2^{k}}} \right)$

I encountered this infinite product while solving another problem: $$P(x)=\prod_{k=0}^\infty 2 \left(1-\frac{x^{1/2^{k+1}}}{1+x^{1/2^{k}}} \right)$$ $$P(x)=P \left( \frac{1}{x} \right)$$ I strongly ...
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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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### Summation Closed form for floor$\left(\log_n\right)$

The closed sum for the floors of logs of consecutive integers is: $$\sum_{i=0}^n \lfloor \log_2i\rfloor = n\lfloor \log_2n\rfloor-2^{\lfloor \log_2n\rfloor+1}+\lfloor \log_2n\rfloor+2$$ This formula ...
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### Prove $\int_{0}^{2\pi}{x\sin^3(x)\over 1+\cos^2(x)}dx=2\pi-\pi^2$

Integrate $$I=\int_{0}^{2\pi}{x\sin^3(x)\over 1+\cos^2(x)}dx=2\pi-\pi^2$$ $${1\over 1+y}=\sum_{n=0}^{\infty}(-1)^ny^n$$ Setting $y=\cos(x)$ $\sin^3(x)={1\over 4}{(3\sin(x)-\sin(3x))}$ ...
### Family of definite integrals involving Dedekind eta function of a complex argument, $\int_0^{\infty} \eta^k(ix)dx$
The Dedekind eta function is denoted by $\eta(\tau)$, and is defined on the upper half-plane ($\Im \tau >0$). Put $\tau = i x$ where $x$ is a positive real number. The function has the following ...