Proving that $\pi(2x) < 2 \pi(x) - Mathematics Stack Exchange
Apr 12, 2016 · 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.
Fourier series for $f(x)=(\pi -x)/2$ - Mathematics Stack Exchange
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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 ...
Jan 17, 2020 · 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
integration - $\int_{-\infty}^{\infty}e^{-\pi x^2}\cdot e^{-2\pi ix\xi ...
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Show that $\\int\\limits_{-\\infty}^\\infty e^{-\\pi x^2}dx = 1$
Jun 1, 2020 · 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}
Nov 8, 2017 · 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 $\sqrt i$.
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 ...
How to calculate the Gibbs phenomenon for sawtooth function?
Oct 20, 2016 · 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.