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.
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 developers to learn, share their knowledge, and build their careers.
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.
Prove that $ \\int_0^\\infty \\frac{\\cos(2\\pi x^2)}{\\cosh^2(\\pi x ...
2016年10月28日 · Inspired by Ramanujan's work I decided to try to compute numerically various similar looking integrals and by coincidence it turned out that $ \int_0^\infty \frac{\cos(2\pi x^2)}{\cosh^2(\pi x)}dx=0.250000000000000000... $ It looks like this integral is $\frac 14$, but I have no clue how to prove it.
calculus - Proof of $\int_0^\infty \left(\frac{\sin x}{x}\right)^2 ...
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Ramanujan's sum related to $\\tan^{-1}(e^{-\\pi x/2})$
2015年5月16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.
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 ...
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.
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 Machin like formulae for pi.