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by whitexlotus
Wed Sep 07, 2016 3:03 am
Forum: Calculus
Topic: Fresnel Cosine Integral
Replies: 2
Views: 1146

Re: Fresnel Cosine Integral

Prove that: $$\int_{0}^{\infty}\cos(x^a)\,dx=\frac{\pi\csc\left ( \frac{\pi}{2a} \right )}{2a\Gamma (1-a)},\,\,\,\,\,a\in \mathbb{N}$$ without contours or complex analysis. $\displaystyle \int\limits_{0}^{+\infty }{\cos \left( x^{a} \right)dx}\underbrace{=}_{\sqrt[a]{y}=x}\frac{1}{a}\int\limits_{0}...
by whitexlotus
Wed Sep 07, 2016 12:47 am
Forum: Real Analysis
Topic: Putnam Question 2014 (Infinite product)
Replies: 1
Views: 977

Re: Putnam Question 2014 (Infinite product)

Let \(a_k \) be a recursive sequence defined as \( \displaystyle a_0=\frac{5}{2} , \; a_k=a_{k-1}^2-2 ,\;\; k \in \mathbb{N} \). Evaluate the product: $$\prod_{k=0}^{\infty} \left( 1-\frac{1}{a_k} \right) $$ My solution :D Note that $a_{0}=\frac{5}{2}=\frac{2^{2}+1}{2^{1}}=\frac{2^{2^{1}}+1}{2^{2^{...
by whitexlotus
Mon Sep 05, 2016 12:39 am
Forum: Calculus
Topic: Integral
Replies: 3
Views: 1769

Integral

$$\int\limits_{0}^{\frac{\pi }{2}}{x\log \left( 1-\cos x \right)dx}=\frac{35}{16}\zeta \left( 3 \right)-\frac{\pi ^{2}}{8}\log 2-\pi G$$
:P
by whitexlotus
Sun Sep 04, 2016 10:05 pm
Forum: Calculus
Topic: Gamma, trigonometric and iterated integral
Replies: 1
Views: 1022

Re: Gamma, trigonometric and iterated integral

This is very beautiful, can you give me some hint ?
by whitexlotus
Sun Sep 04, 2016 9:59 pm
Forum: Calculus
Topic: A tough product
Replies: 3
Views: 1787

Re: A tough product

http://zerocollar.blogspot.cl/" onclick="window.open(this.href);return false;
by whitexlotus
Sun Sep 04, 2016 9:53 pm
Forum: Calculus
Topic: Another challenging integral
Replies: 1
Views: 1135

Re: Another challenging integral

Tolaso J Kos wrote:Let $\Omega$ denote the unique real root of the equation $xe^x=1$. Prove that:

$$\int_{-\infty}^{\infty} \frac{{\rm d}x}{(e^x-x)^2+\pi^2}= \frac{1}{1+\Omega}$$
here http://zerocollar.blogspot.cl/2014/08/a ... stant.html" onclick="window.open(this.href);return false;
by whitexlotus
Sun Sep 04, 2016 5:41 pm
Forum: Calculus
Topic: An integral!
Replies: 2
Views: 1177

Re: An integral!

Prove that: $$\int_{0}^{\pi/2} \theta^2 \cot \theta \, {\rm d}\theta= \frac{\pi^2}{4}\log 2 - \frac{7}{8}\zeta(3)$$ $$I=\int\limits_{0}^{\frac{\pi }{2}}{x^{2}\cot xdx}=\left( x^{2}\ln \sin x \right)\left| _{\left( \frac{\pi }{2},0 \right)} \right.-\int\limits_{0}^{\frac{\pi }{2}}{2x\ln \sin xdx}=\i...
by whitexlotus
Sun Sep 04, 2016 4:45 pm
Forum: Calculus
Topic: $\int_0^{\pi/2} \log^3 \cos x \, dx$
Replies: 5
Views: 1953

Re: $\int_0^{\pi/2} \log^3 \cos x \, dx$

In continuity of this post by Grigorios let us see the same integral with a higher power. Prove that: $$\int_0^{\pi/2} \log^3 \cos x\, {\rm d}x = - \frac{3\pi \zeta(3)}{4}- \frac{\pi^3 \log 2}{8} - \frac{\pi \log^3 2}{2}$$ $$I=\int\limits_{0}^{\frac{\pi }{2}}{\log ^{3}\cos xdx}=\int\limits_{0}^{\fr...
by whitexlotus
Sun Sep 04, 2016 4:10 pm
Forum: Calculus
Topic: \( \int_{0}^{\infty}{e^{-x}\ln\bigl({\ln\bigl({e^x+\sqrt{e^{2x}-1}\,}\bigr)}\bigr) \;dx}\)
Replies: 1
Views: 1076

Re: \( \int_{0}^{\infty}{e^{-x}\ln\bigl({\ln\bigl({e^x+\sqrt{e^{2x}-1}\,}\bigr)}\bigr) \;dx}\)

Tolaso J Kos wrote:Evaluate the following integral :

$$ \displaystyle\int_{0}^{\infty}{e^{-x}\ln\bigl({\ln\bigl({e^x+\sqrt{e^{2x}-1}\,}\bigr)}\bigr) \;{\rm d}x}$$
Here http://mathimatikoi.org/forum/viewtopic ... 2161#p2161" onclick="window.open(this.href);return false;
Regards
by whitexlotus
Sun Sep 04, 2016 4:08 pm
Forum: Calculus
Topic: Laplace transform of a function!
Replies: 2
Views: 1468

Re: Laplace transform of a function!

Prove that: $$\int_{0}^{\infty}e^{-x} \ln \left ( \ln \left ( e^x+ \sqrt{e^{2x}-1} \right ) \right )\, {\rm d}x = -\gamma+ 4 \log \Gamma \left ( \frac{1}{4} \right )-3\log 2 -2 \log \pi$$ where $\gamma$ stands for the Euler - Mascheroni constant and $\Gamma$ is the Gamma function. Hi tolaso, I'm pp...