Question

Explicit formulas for $\operatorname{Re}(z^n)$ and $\operatorname{Im}(z^n)$

score 176 · Answer 1 · 2018-10-19 09:30:01

176

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Explicit formulas for $\operatorname{Re}(z^n)$ and $\operatorname{Im}(z^n)$

Published on 19 Oct 2018 - 9:30

score 501 · Answer 2 · 2018-10-19 14:14:28

501

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Show that $\int\limits_0^{\frac{\pi}2}4\cos^2(x)\log^2(\cos x)~\mathrm dx=-\pi\log 2+\pi\log^2 2-\frac{\pi}2+\frac{\pi^3}{12}$

Published on 19 Oct 2018 - 14:14

score 278 · Answer 3 · 2026-03-25 10:24:53

278

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Evaluate $\int_0^1 \log^n(x^a)\log^m(1-x^{\color{red}{\alpha}})x^b(1-x^{\color{red}{\beta}})^t\mathrm dx$ with $\alpha\ne\beta$

Published on 25 Mar 2026 - 10:24

score 829 · Answer 4 · 2018-10-20 13:50:02

829

Views

Proving that $\sum_{n=1}^\infty \frac{\sin\left(n\frac{\pi}{3}\right)}{(2n+1)^2}=\frac{G}{\sqrt 3} -\frac{\pi^2}{24}$

Published on 20 Oct 2018 - 13:50

score 330 · Answer 5 · 2026-03-25 16:00:08

330

Views

Is there a closed form for $\zeta(\pi)$?

Published on 25 Mar 2026 - 16:00

score 149 · Answer 6 · 2018-10-22 14:19:29

149

Views

Integration problem related to Gamma function: $ \int_{0}^{\infty} u^{\alpha + b - 1} \exp\left(-ub + u^{\alpha}c\right)du $

Published on 22 Oct 2018 - 14:19

score 5.7k · Answer 7 · 2018-10-22 16:22:07

5.7k

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Summation of $\sum_{i=1}^n i2^i.$

Published on 22 Oct 2018 - 16:22

score 135 · Answer 8 · 2026-03-25 01:40:03

135

Views

Is there a closed form for $\sum_{k=0}^n2^k\binom{2n+1}{2k}$, the sum of binomial coefficients times powers of two, for even indices?

Published on 25 Mar 2026 - 1:40

score 181 · Answer 9 · 2026-03-25 18:47:35

181

Views

Closed form for a product of sines

Published on 25 Mar 2026 - 18:47

score 47 · Answer 10 · 2018-10-25 17:18:05

47

Views

Express series by a closed-form expression

Published on 25 Oct 2018 - 17:18

score 426 · Answer 11 · 2018-10-26 21:27:53

426

Views

Can the binomial distribution be solved for k?

Published on 26 Oct 2018 - 21:27

score 185 · Answer 12 · 2018-10-28 09:05:48

185

Views

The $n$ th derivative of $\ln(xy)/(1-xy)$ with respect to $x$.

Published on 28 Oct 2018 - 9:05

score 81 · Answer 13 · 2018-10-29 08:54:17

81

Views

A closed form for the sum $\sum_{i=0}^j (-1)^i \binom{j}{i} (j+c-i)^k$

Published on 29 Oct 2018 - 8:54

score 64 · Answer 14 · 2018-10-29 11:28:42

64

Views

Antiderivative of : $t \mapsto (1-t^2)^\lambda$

Published on 29 Oct 2018 - 11:28

score 76 · Answer 15 · 2018-10-31 04:07:59

76

Views

Closed form of $\sum \frac {1}{p_1.p_2.….p_r}$ where $p_r$ is the $r^{th}$ prime

Published on 31 Oct 2018 - 4:07

List Question

Explicit formulas for $\operatorname{Re}(z^n)$ and $\operatorname{Im}(z^n)$

Show that $\int\limits_0^{\frac{\pi}2}4\cos^2(x)\log^2(\cos x)~\mathrm dx=-\pi\log 2+\pi\log^2 2-\frac{\pi}2+\frac{\pi^3}{12}$

Evaluate $\int_0^1 \log^n(x^a)\log^m(1-x^{\color{red}{\alpha}})x^b(1-x^{\color{red}{\beta}})^t\mathrm dx$ with $\alpha\ne\beta$

Proving that $\sum_{n=1}^\infty \frac{\sin\left(n\frac{\pi}{3}\right)}{(2n+1)^2}=\frac{G}{\sqrt 3} -\frac{\pi^2}{24}$

Is there a closed form for $\zeta(\pi)$?

Integration problem related to Gamma function: $ \int_{0}^{\infty} u^{\alpha + b - 1} \exp\left(-ub + u^{\alpha}c\right)du $

Summation of $\sum_{i=1}^n i2^i.$

Is there a closed form for $\sum_{k=0}^n2^k\binom{2n+1}{2k}$, the sum of binomial coefficients times powers of two, for even indices?

Closed form for a product of sines

Express series by a closed-form expression

Can the binomial distribution be solved for k?

The $n$ th derivative of $\ln(xy)/(1-xy)$ with respect to $x$.

A closed form for the sum $\sum_{i=0}^j (-1)^i \binom{j}{i} (j+c-i)^k$

Antiderivative of : $t \mapsto (1-t^2)^\lambda$

Closed form of $\sum \frac {1}{p_1.p_2.….p_r}$ where $p_r$ is the $r^{th}$ prime

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