61 Prove $\int\limits_{0}^{\pi/2}\frac{dx}{1+\sin^2{(\tan{x})}}=\frac{\pi}{2\sqrt{2}}\bigl(\frac{e^2+3-2\sqrt{2}}{e^2-3+2\sqrt{2}}\bigr)$ 45 Prove $\int_0^{\infty}\! \frac{\mathbb{d}x}{1+x^n}=\frac{\pi}{n \sin\frac{\pi}{n}}$ using real analysis techniques only 38 How to prove that $\frac{\zeta(2) }{2}+\frac{\zeta (4)}{2^3}+\frac{\zeta (6)}{2^5}+\frac{\zeta (8)}{2^7}+\cdots=1$? 36 A series involving the harmonic numbers : $\sum_{n=1}^{\infty}\frac{H_n}{n^3}$ 33 How to evaluate $\int_0^\infty\operatorname{erfc}^n x\ \mathrm dx$?

Reputation (26,084)

 +15 Show that the function $z\to \int_{\sqrt{5}}^{z} \frac{2w}{w^{2}-4}dw$ is well-defined in $\mathbb{C}\setminus [-2,2]$ +5 Choosing between a semicircular contour and a rectangular contour +10 Series expansion of $\coth x$ using the Fourier transform +10 Closed form for an infinite series of Bessel functions

Questions (86)

 66 Evaluating the log gamma integral $\int_{0}^{z} \log \Gamma (x) \, \mathrm dx$ in terms of the Hurwitz zeta function 38 Evaluating the sum of the series $\sum\limits_{n=1}^{\infty} \frac{1}{n^{3} \binom{2n}{n}}$ 31 Evaluating the series $\sum_{n=1}^{\infty} \frac{n}{e^{2 \pi n}-1}$ using the inverse Mellin transform 25 When can't a real definite integral be evaluated using contour integration? 24 Evaluating $\int_{0}^{\infty} \frac{x^{3}- \sin^{3}(x)}{x^{5}} \ dx$ using contour integration

Tags (92)

 1k integration × 208 351 sequences-and-series × 53 1k definite-integrals × 122 335 improper-integrals × 48 995 calculus × 114 318 special-functions × 49 443 real-analysis × 50 295 closed-form × 26 396 complex-analysis × 124 240 contour-integration × 61

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