query: Prove that $ {|\int_{\gamma} e^{iz^2} dz | \leq \frac{\pi(e-1)}{4e}}$
positive_passages.docid: 27347
positive_passages.text: Verify that $\left|\int_{\gamma} \exp(iz^2)dz\right| \leq \frac{\pi\big(1-\exp(-r^2)\big)}{4r}$ where $\gamma(t)=re^{it}$, for $0\leq t \leq \pi/4$.
negative_passages.docid: 126736
negative_passages.text: Show that $\lim_{\epsilon\to0^{+}}\frac{1}{2\pi i}\int_{\gamma_\epsilon}\frac{f(z)}{z-a} \, dz=f(a)$
negative_passages.docid: 124026
negative_passages.text: Prove $\int^\infty_0\frac x{e^x-1}dx=\frac{\pi^2}{6}$
negative_passages.docid: 105707
negative_passages.text: A proof of $\int_{0}^{1}\left( \frac{\ln t}{1-t}\right)^2\,\mathrm{d}t=\frac{\pi^2}{3}$
negative_passages.docid: 27564
negative_passages.text: Prove that $\|UVU^{-1}V^{-1}-I\|\leq 2\|U-I\|\|V-I\|$
negative_passages.docid: 80553
negative_passages.text: Show that,$\int_0^\pi \left|\frac{\sin nx}{x}\right|\mathrm{d}x \ge \frac{2}{\pi}\left(1+\frac12+\cdots+\frac{1}{n}\right)$
negative_passages.docid: 128790
negative_passages.text: Prove an integral inequality: $ \left(\int|f|^2dx\right)^2\le 4\left(\int|xf(x)|^2dx\right)\left(\int|f'|^2dx\right) $
negative_passages.docid: 90896
negative_passages.text: proof of inequality $e^x\le x+e^{x^2}$
negative_passages.docid: 58832
negative_passages.text: How prove this $\int_{a}^{b}[f''(x)]^2dx\ge\dfrac{4}{b-a}$
negative_passages.docid: 34081
negative_passages.text: Prove that $|f''(\xi)|\geqslant\frac{4|f(a)-f(b)|}{(b-a)^2}$
negative_passages.docid: 68821
negative_passages.text: Prove $\int_0^{\infty} \left(\sqrt{1+x^{4}}-x^{2}\right)\ dx=\frac{\Gamma^{2}\left(\frac{1}{4}\right)}{6\sqrt{\pi}}$
negative_passages.docid: 68097
negative_passages.text: Prove $\int\limits_{0}^{\infty} \mathrm{exp}(-ax^{2}-\frac{b}{x^{2}}) \mathrm{d} x = \frac{1}{2}\sqrt{\frac{\pi}{a}}\mathrm{e}^{-2\sqrt{ab}}$
negative_passages.docid: 10675
negative_passages.text: Proving that $\frac{1}{2\pi}\int_{0}^{2\pi}|f(re^{it})|^2dt=\sum_{n=0}^{\infty}|a_n|^2r^{2n}$
negative_passages.docid: 145152
negative_passages.text: How to prove that $\frac{x^2}{yz+2}+\frac{y^2}{zx+2}+\frac{z^2}{xy+2}\geq \frac{x+y+z}{3}$ holds for any $(x,y,z)\in[1,2]^3$
negative_passages.docid: 65562
negative_passages.text: Proving the inequality $\arctan\frac{\pi}{2}\ge1$
negative_passages.docid: 141194
negative_passages.text: Prove that $\frac{(n-2)s^2}{\sigma^2}\sim \chi^{2}_{n-2}$
negative_passages.docid: 148746
negative_passages.text: Calculating $\int\frac{e^{iz}}{z}\, dz$ on the semi-circle given by $re^{i\theta}$ where $\theta:\,0\to\pi$