# Monthly Archives: February 2017

## Integral of rational function with cosine hyperbolic function using rectangle contour

$$\int^{\infty}_{-\infty} \frac{1}{(5 \pi^2 + 8 \pi y + 16y^2) }\frac{\cosh\left(y+\frac{\pi}{4} \right)}{\cosh^3(y)}dy=\frac{2}{\pi^3}\left(\pi \cosh\left(\frac{\pi}{4} \right)-4\sinh\left( \frac{\pi}{4}\right) \right)$$   $$\textit{proof}$$ Consider $$f(z) = \frac{\sinh(z)}{z \sinh^3(z-\pi/4)}$$ If we integrate around a contour of height $\pi$ and stretch it to infinity we … Continue reading

## 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

Prove that $$\int^{1}_{0} \sqrt{x}\sqrt{1-x}\,dx = \frac{\pi}{8}$$ $$\textit{proof}$$ Consider the function $$f(z) = \sqrt{z-z^2} = e^{\frac{1}{2}\log(z-z^2)}$$ Consider the branch cut on the x-axis $$x(1-x)\geq 0\,\, \implies \, 0\leq x \leq 1$$ Consider $w= z-z^2$ then $$\log(w) = \log|w|+i\theta,\,\, … Continue reading Posted in Contour Integration | Tagged , , | Leave a comment ## Contour integraion of a rational function of logarithm and exponential$$\int_{0}^\infty \frac{\log(x)\cos(x)}{(x^2+1)^2}\,dx = – \frac{\pi \mathrm{Ei}(1)}{4e}-\frac{\pi}{4e}\textit{proof}$$Consider the following function$$f(z) = \frac{\log(z) }{(z^2+1)^2}e^{iz}$$Now consider the principle logarithm where$$\log(z) = \log|r|+i \theta \,\,\, , \theta \,\in (-\pi , \pi]$$Consider the following contour Then by … Continue reading Posted in Contour Integration | | Leave a comment ## Bromwich contour integration of the gamma function$$\frac{1}{2\pi i}\int^{c+i\infty}_{c-i\infty}\Gamma(a+t)\Gamma(b-t) s^{-t}\,dt= \frac{\Gamma(a+b)}{(1+s)^{a+b}}s^a\textit{proof}$$Consider the following function$$f(z) = \Gamma(z+a)\Gamma(b-z) s^{-z}$$Suppose that $a,b \in \mathbb{R}$ and $a < b$. Note that the Gamma function has a pole of order 1 at each non-positive integer where … Continue reading Posted in Gamma function | Tagged , , , , | Leave a comment ## Triple integral with sines and cosines Find the integral$$\begin{align}\int^\infty_0 \int^\infty_0 \int^\infty_0 \frac{\sin(x)\sin(y)\sin(z)}{xyz(x+y+z)}(\sin(x)\cos(y)\cos(z)\\ + \sin(y)\cos(z)\cos(x) + \sin(z)\cos(x)\cos(y))\,dx\,dy\,dz \end{align}\textit{solution}$$This can be rewritten as$$3\small\int^\infty_0 \int^\infty_0 \int^\infty_0 \frac{\sin^2(x)\sin(y)\cos(y)\sin(z)\cos(z)}{xyz(x+y+z)}\,dx\,dy\,dz$$Now consider$$\small F(a) = 3\int^\infty_0 \int^\infty_0 \int^\infty_0\frac{\sin^2(x)\sin(y)\cos(y)\sin(z)\cos(z) e^{-a(x+y+z)}}{xyz(x+y+z)}\,dx\,dy\,dz$$Taking the derivative$$\small F'(a) = -3\int^\infty_0 \int^\infty_0 … Continue reading
$${}_2F_1 \left(a,b;c;z\right)=(1-z)^{c-a-b} {}_2F_1 \left(c-a,c-b;c;z\right)$$ $$\textit{proof}$$ In the Pfaff transformations let $z \to \frac{z}{z-1}$ , proved here  $${}_2F_1 \left(a,b;c;\frac{z}{z-1}\right)=(1-z)^{-a} {}_2F_1 \left(a,c-b;c;z\right)$$ and $${}_2F_1 \left(a,b;c;\frac{z}{z-1}\right)=(1-z)^{-b} {}_2F_1 \left(c-a,b;c;z\right)$$ By equating the two transformations $$(1-z)^{-a} {}_2F_1 \left(a,c-b;c;z\right)=(1-z)^{-b} {}_2F_1 \left(c-a,b;c;z\right)$$ Now use the transformation … Continue reading