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 $\psi_1(x)$. This is clearly quite intuitive but I'm lost as to what I can use to prove the result.
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.
real analysis - Question about Fourier transform of $e^{-\pi x^2 ...
2020年1月17日 · Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
complex analysis - Fourier transform of $e^{-i\pi x^2}
2017年11月8日 · Completing squares as usual, it strongly suggests that its Fourier transform, in the sense of tempered distributions, is given by $$ \frac 1 {\sqrt i} e^{ i\pi \xi^2}. $$ The only problem is that I dont know how to determine the branch of $\sqrt i$.
calculus - Calculate $\pi$ precisely using integrals? - Mathematics ...
$$2\int_{-1}^1\sqrt{1-x^2}\,dx=\left.\left(x\sqrt{1-x^2}+\arcsin(x)\right)\right|_{-1}^1=\pi.$$ But this does not bring you any closer to the numerical value of $\pi$. There are many ways to obtain a desired number of decimals of $\pi$, using finite approximations of various sequences, series or integrals (this is a broad topic).
integration - $\int_{-\infty}^{\infty}e^{-\pi x^2}\cdot e^{-2\pi ix\xi ...
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For all $\\xi \\in \\mathbb{C}$ we have $e^{-\\pi\\xi^2}=\\int ...
Write $\xi=a+bi$. The exponent inside the integrand looks like $$\begin{array}{ll} -\pi x^2+2\pi ix\xi & =-\pi x^2+2\pi ix(a+bi) \\ & =-\pi x^2+2\pi iax-2\pi bx ...
Show that $\\int\\limits_{-\\infty}^\\infty e^{-\\pi x^2}dx = 1$
2020年6月1日 · Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Proof that $\\sin(x) > x/2$ - x/2$ - Mathematics Stack Exchange
I need to prove that $\sin(x) > \frac{x}{2}$ if $0<x<\pi/2$ I've started working with the derivative, but if it's possible, I'd rather something simpler than that. Skip to main content Stack Exchange Network