Tag Archives: residue

Prove that  $$\int^1_0 \frac{\cos(\log x)}{x^2+1}\,dx = \frac{\pi}{4}\mathrm{sech}\left( \frac{\pi}{2}\right)$$ First note that $$2 \int^1_0 \frac{\cos(\log x) }{x^2+1}\,dx = \int^\infty_0 \frac{\cos(\log x)}{x^2+1}\,dx$$ Integrate the following function $$f(z) = \frac{e^{i\log(z)}}{z^2+1}$$ Around a semi-circle in the upper half place. Where we avoid the branch … Continue reading

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Creating Difficult integrals by the residue theorem

Theorem Let $f$ be analytic function in the unit circle $|z|\leq 1$  such that $f\neq 0$ . Then $$\int^{2\pi}_0f(e^{it})\,dt =2\pi \, f(0)$$ $$\textit{proof}$$ Since the function $f$ is analytic in and on the … Continue reading

Integration related to gamma function using rectangle contour

[Ex 9] Watson’s complex integration $$\int^{\infty}_{-\infty}e^{-x^2}\,\cos(2ax)dx=e^{-a^2}\sqrt{\pi}$$ $$proof$$ Integrate the following function $$f(z) = e^{-z^2}$$ Use the following contour Note that the function is entire, hence $$\int^{L}_{-L} e^{-t^2}\,dt+\int^{L+ai}_{L} e^{-t^2}\,dt+\int^{-L}_{-L+ai} e^{-t^2}\,dt+\int^{-L+ai}_{L+ai}e^{-t^2}\,dt=0$$ For the forth integral use the substitution $x= t-ai$ $$\int^{-L+ai}_{L+ai}e^{-t^2}\,dt=\int^{-L}_{L}e^{-(x+ai)^2}\,dx=-e^{a^2}\int^{L}_{-L}e^{-x^2}\,e^{-2iax}dx$$ Take … Continue reading

$$\int^\infty_0\frac{\log\left(x^2+1 \right)\arctan^2\left(x\right)}{x^2}\,dx = \frac{\pi^3}{12}+\pi \log^2(2)$$ Lemma $$\int^\infty_0 \frac{\log^3(1 + x^2)}{x^2}\,dx = \pi^3+ 3 \pi \log^2(4)$$ Start by the following $$\int^{\infty}_0 x^{-p}(1+x)^{s-1} dx= \frac{\Gamma(1-p)\Gamma(p-s)}{\Gamma(1-s)}$$ Let $x\to x^2$ $$\int^{\infty}_0 x^{-2p+1}(1+x^2)^{s-1} dx= \frac{\Gamma(1-p)\Gamma(p-s)}{2\Gamma(1-s)}$$ Let $p = 3/2$ $$\int^{\infty}_0 \frac{1}{x^2(1+x^2)^{1-s}} … Continue reading Posted in Beta function, Contour Integration | | 4 Comments Integrating along the unit circle Prove that$$\int^{2\pi}_0e^{\cos \theta}\cos(n\theta -\sin \theta)\,d \theta=\frac{2\pi}{n!}\textit{proof}$$Consider the following function$$f(z)=e^{z^{-1}}z^{n-1}$$Now we integrate the function along a circle of radius 1 The contour encloses a pole at $z = 0$$$\oint_{|z|=1}e^{z^{-1}}z^{n-1} dz=2\pi i\mathrm{Res}(f(z),0) $$Now we … Continue reading Posted in Contour Integration | | Leave a comment Contour method for shifted logarithm branch Prove $a,b,c,d >0$$$\int^\infty_0 \frac{\log(a^2+b^2x^2)}{c^2+d^2x^2}\,dx = \frac{\pi}{cd} \log \frac{ad+bc}{d}$$Consider the function$$f(z) = \frac{\log(a-ibz)}{c^2+d^2z^2}$$We need the logarithm with the branch cut $y<-\frac{a}{b} , x =0$ . Note that this corresponds to$$\log(a+ibz) = \log\sqrt{(a+y)^2+b^2x^2}+i\theta … Continue reading
Prove that $$\sum_{n=1}^\infty \frac{H_n}{n^2} = 2\zeta(3)$$ $$\textit{proof}$$ Consider the function $$f(z) = \frac{(\psi(-z)+\gamma)^2}{z^2}$$ Note that $f$ has poles at non-negative integers By integration around a large circle $|z| = \rho$ Note that \oint f(z)\,dz = 2\pi … Continue reading