
Recent Posts
 Integrating a cosine log integral around a semicircle contour
 Creating Difficult integrals by the residue theorem
 Proving a trigonometric integral by integrating around an ellipse in the complex plain
 Integrating a fraction of exponential and trignometric using rectangular contour
 Integrating around a triangular contour for Fresnel integral
Recent Comments
 Ricardo on Integral representation of the digamma function using Abel–Plana formula
 Zaidalyafeai on Integral representation of the digamma function using Abel–Plana formula
 Ricardo on Integral representation of the digamma function using Abel–Plana formula
 Zaidalyafeai on Integral of arctan and log using contour integration
 tired on Integral of arctan and log using contour integration
Archives
Categories
Meta
Tag Archives: analysis
Integrating a fraction of exponential and trignometric using rectangular contour
[Ex 41 ] Watson’s complex integration $$\int^\infty_0 \frac{\sin(ax)}{e^{2\pi x}1}\,dx = \frac{1}{4}\coth\left(\frac{a}{2} \right)\frac{1}{2a}$$ $$\textit{solution}$$ By integrating the following function $$f(z) = \frac{e^{iaz}}{e^{2\pi z}1}$$ The function is analytic in and on the contour, indented at the poles of the function Hence by … Continue reading
Posted in Contour Integration
Tagged analysis, complex, contour, exponent, fraction, rectangle, residue, theorem, trignometric
Leave a comment
Integrating around a triangular contour for Fresnel integral
$$\int^\infty_0\frac{\cos(x)}{\sqrt{x}}\,dx =\int^\infty_0\frac{\sin(x)}{\sqrt{x}}\,dx =\sqrt{\frac{\pi}{2}}$$ $$\textit{solution}$$ Consider the following function $$f(z)=z^{1/2}\,e^{iz}$$ Where we choose the principle root for \( z^{1/2}=e^{1/2\log(z)}\). By integrating around the following contour $$\int_{C_r}f(z)\,dz+\int_{r}^R f(x)\,dx+\int_{\gamma}f(z)\,dz+\int^{iR}_{ir}f(x)\,dx = 0$$ Taking the integral around the small quarter circle with $r\to 0$ $$\left … Continue reading
Posted in Contour Integration
Tagged analysis, complex, Fresnel, Integral, residue, theorem, triangle
Leave a comment
Integrating a function around three branches using a semicircle contour
[Ex] Watson’s complex integration $$\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{2n+m}{2}\right)}$$ $$\textit {solution}$$ Let us integrate the following function $$f(z) = z^{nm1}\left(1+z^2\right)^m$$ We choose the principle logarithm where $$\log(z) = \logz+\mathrm{Arg}(z)$$ Note that the function \(z^{nm1} = e^{(nm1)\log(z)}\) will have a branch cut on the … Continue reading
Posted in Beta function, Contour Integration
Tagged analysis, circle, complex, contour, halfcircle, integration, semi
Leave a comment
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 \(x= tai\) $$\int^{L+ai}_{L+ai}e^{t^2}\,dt=\int^{L}_{L}e^{(x+ai)^2}\,dx=e^{a^2}\int^{L}_{L}e^{x^2}\,e^{2iax}dx$$ Take … Continue reading
Posted in Contour Integration, Gamma function
Tagged analysis, complex, contour, Dawson, Gamma, Integral, rectangle, residue
Leave a comment
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 $$f(z)=e^{z^{1}}z^{n1}$$ Now we integrate the function along a circle of radius 1 The contour encloses a pole at \(z = 0\) $$\oint_{z=1}e^{z^{1}}z^{n1} dz=2\pi i\mathrm{Res}(f(z),0) $$ Now we … Continue reading
Posted in Contour Integration
Tagged analysis, circle, complex, contour, integration, residue, theorem, unit
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 $$f(z) = \frac{\log(aibz)}{c^2+d^2z^2}$$ We need the logarithm with the branch cut \( y<\frac{a}{b} , x =0 \) . Note that this corresponds to $$\log(a+ibz) = \log\sqrt{(a+y)^2+b^2x^2}+i\theta … Continue reading
Posted in Contour Integration
Tagged analysis, branch, complex, logarithm, rational, residue, shifted, theorem
Leave a comment
Solving Euler sums using Contour integration
Prove that $$\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 $$\oint f(z)\,dz = 2\pi … Continue reading
Posted in Contour Integration, Euler sum
Tagged analysis, complex, contour, Euler, integration, residue, sums
Leave a comment
Integral of rational function with cosine hyperbolic function using rectangle contour
$$ \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
Leave a comment
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
Leave a comment