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Monthly Archives: February 2017
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 $$f(z) = \frac{\log(aibz)}{c^2+d^2z^2}$$ We need the logarithm with the branch cut \( y<\frac{a}{b} , x =0 \) . Note that this corresponds to $$\log(a+ibz) = \log\sqrt{(a+y)^2+b^2x^2}+i\theta … Continue reading
Posted in Contour Integration
Tagged analysis, branch, complex, logarithm, rational, residue, shifted, theorem
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Solving Euler sums using Contour integration
Prove that $$\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 nonnegative integers By integration around a large circle \( z = \rho \) Note that $$\oint f(z)\,dz = 2\pi … Continue reading
Posted in Contour Integration, Euler sum
Tagged analysis, complex, contour, Euler, integration, residue, sums
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Integral of rational function with cosine hyperbolic function using rectangle contour
$$ \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
Posted in Contour Integration
Tagged analysis, complex, cosine, hyperboic, rectangle
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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 multivalued defined as $$\log(z) = \lnz+i\theta +2k\pi i$$ This maps the complex plain more than once … Continue reading
Posted in Beta function, Contour Integration, Gamma function
Tagged analysis, complex, contour, Euler, formula, proof, reflection
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Solving an integral using Dogbone contour
Prove that $$\int^{1}_{0} \sqrt{x}\sqrt{1x}\,dx = \frac{\pi}{8}$$ $$\textit{proof}$$ Consider the function $$f(z) = \sqrt{zz^2} = e^{\frac{1}{2}\log(zz^2)}$$ Consider the branch cut on the xaxis $$x(1x)\geq 0\,\, \implies \, 0\leq x \leq 1 $$ Consider \( w= zz^2 \) then $$\log(w) = \logw+i\theta,\,\, … Continue reading
Contour integraion of a rational function of logarithm and exponential
$$\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) = \logr+i \theta \,\,\, , \theta \,\in (\pi , \pi]$$ Consider the following contour Then by … Continue reading
Posted in Contour Integration
Tagged contour, cosine, exponential, fraction, Integral, proof, residue, theorem
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Bromwich contour integration of the gamma function
$$\frac{1}{2\pi i}\int^{c+i\infty}_{ci\infty}\Gamma(a+t)\Gamma(bt) 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(bz) 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 nonpositive integer where … Continue reading
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 $$\small F'(a) = 3\int^\infty_0 \int^\infty_0 … Continue reading
Euler Hypergeometric transformation proof
$${}_2F_1 \left(a,b;c;z\right)=(1z)^{cab} {}_2F_1 \left(ca,cb;c;z\right)$$ $$\textit{proof}$$ In the Pfaff transformations let \( z \to \frac{z}{z1}\) , proved here $${}_2F_1 \left(a,b;c;\frac{z}{z1}\right)=(1z)^{a} {}_2F_1 \left(a,cb;c;z\right)$$ and $${}_2F_1 \left(a,b;c;\frac{z}{z1}\right)=(1z)^{b} {}_2F_1 \left(ca,b;c;z\right)$$ By equating the two transformations $$(1z)^{a} {}_2F_1 \left(a,cb;c;z\right)=(1z)^{b} {}_2F_1 \left(ca,b;c;z\right)$$ Now use the transformation … Continue reading
Posted in Hypergeoemtric function
Tagged Euler, hypergeoemtric, proof, transformation
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