Fourier series for $f(x)=(\pi -x)/2$ - Mathematics Stack Exchange
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Proving that $\pi(2x) < 2 \pi(x) - Mathematics Stack Exchange
2016年4月12日 · In class we proved the prime number theorem, and then proceeded to prove several results such as $\pi(x) = Li(x) +O(x^\theta \ln x)$ and the explicit formula for …
What is the Fourier transform of $x \\cot(\\pi x/2) 1_{[-1,1]}$?
2025年1月8日 · Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for …
real analysis - Showing $g(x) = -2\pi x e^{-\pi x^2} = O(1/x^2)
2021年12月15日 · Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for …
Calculate Fourier series of $x(\pi^2-x^2)$ - Mathematics Stack …
A: Fourier series of derivative $\ f'(x) = \pi^{2}-3x^{2}$ The derivative has even parity and will be a sum of cosines.
Ramanujan's sum related to $\\tan^{-1}(e^{-\\pi x/2})$
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calculus - Proof of $\int_0^\infty \left(\frac{\sin x}{x}\right)^2 ...
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How do I prove $\\int_0^\\pi\\frac{(\\sin nx)^2}{(\\sin x)^2}dx
Here's another way. Recall that $\sin(x) =\frac{e^{i x}-e^{-i x}}{2i}$. Therefore $$ I = \int_0^{\pi} \frac{\sin^2(nx)}{\sin^2(x)}\mathrm{d}x = \int_0^{\pi} \frac ...
conic sections - Least error at $x=2$ for $\pi$ approximation ...
2024年12月20日 · $\begingroup$ Well, if not that, we do get a nice approximation of arctan for small x. $\arctan(x) \approx \frac{x(4\sqrt{x^2 + 1} - 1)}{3(x^2 + 1)}$, which can be used in …
Ways to prove $ \\int_0^\\pi \\frac{\\sin^2 nx}{\\sin^2 x} dx= n\\pi$
Here is a cannon to shoot a fly.. Rewrite the integral as \begin{equation} I(n):= \int_0^\pi\frac{1-\cos2nx}{1-\cos2x}\,dx\stackrel{2x\,\mapsto\, x}\Longrightarrow ...