Category Archives: Beta function

Integrating a function around three branches using a semi-circle contour

[Ex] Watson’s complex integration $$\int^{\pi/2}_{0}\cos(nt)\cos^m(t)\,dt=\frac{\pi \Gamma(m+1)}{2^{m+1}\Gamma\left(\frac{n+m+2}{2}\right)\Gamma\left(\frac{2-n+m}{2}\right)}$$ $$\textit {solution}$$ Let us integrate the following function $$f(z) = z^{n-m-1}\left(1+z^2\right)^m$$ We choose the principle logarithm where $$\log(z) = \log|z|+\mathrm{Arg}(z)$$ Note that the function \(z^{n-m-1} = e^{(n-m-1)\log(z)}\) will have a branch cut on the … Continue reading

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Integral of arctan and log using contour integration

$$\int^\infty_0\frac{\log\left(x^2+1 \right)\arctan^2\left(x\right)}{x^2}\,dx = \frac{\pi^3}{12}+\pi \log^2(2)$$ 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

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Euler reflection formula proof using contour integration

$$\int_{0}^{\infty}\frac{x^{\alpha}}{x+1}\,dx=- \pi \csc(\pi \alpha)$$ $$\textit{proof}$$ Consider the following function $$f(z) = \frac{z^{\alpha}}{1+z} = \frac{e^{\alpha \log(z)}}{1+z}$$ As we know the function \( \log(z) \) is multi-valued defined as $$\log(z) = \ln|z|+i\theta +2k\pi i$$ This maps the complex plain more than once … Continue reading

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Proof of beta function using convolution

Prove the following $$\beta(x, y)=\int^{1}_{0}t^{x-1}\, (1-t)^{y-1}\,dt= \frac{\Gamma(x)\Gamma{(y)}}{\Gamma{(x+y)}}$$ $$\textit{proof}$$ Let us choose some functions $f$ and $g$ $$f(t) = t^{x} \,\, , \, g(t) = t^y$$ Hence we get $$(t^x*t^y)= \int^{t}_0 s^{x}(t-s)^{y}\,ds $$ So by the convolution rule we have the … Continue reading

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