# Questions tagged [contour-integration]

Questions on the evaluation of integrals along a locus in the complex plane.

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### Using a contour integral of $f(z)=x^{4n+3}e^{-z}$ to show $\int_0^\infty x^{4n+3}e^{-x}\cos x\;dx=(-1)^{n+1}(4n+3)!/2^{2n+2}$ [closed]

I am pretty close to finding the right-hand side of the desired proof (see below) but not able to finish it. I have done this far but, it seems too long to continue.
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### How to evaluate the following integral? [Integrand is a fraction under square root with each numerator and denominator being a 3rd degree polynomial]

I am stuck with a integration that is related to a pendulum problem. $$I = \frac{2\sqrt{2}}{\pi} \int_{x_1}^{x_2} \frac{\sqrt{-x^3+(1+E)x-\bar{\beta}}}{\sqrt{x} \sqrt{2-x^2}}dx\,$$ Note that, we ...
I was interested in solving for $\eta'(1)$ where $\eta(s)$ is Dirichlet eta function, and I am now able to expand it like this based on the fact that $\displaystyle\Gamma(s)\eta(s)=\int_0^\infty{t^{s-... 1answer 48 views ### Importance of a region being convex So I have been reading and solving problems from complex analysis ( A.R Shastri's book), many times we talk of certain results in the case of convex regions. Is there something special about convex ... 2answers 24 views ### Contour Integral and Removable Singularity Could someone check if what I have done is correct? To evaluate$\int_{C}\dfrac{z^3}{2z-i}$where$C$is the unit circle. My solution is as follows: Let$f(z):=\dfrac{z^3}{2z-i}$. There seems to be ... 2answers 45 views ### Evaluate$\int\frac{e^z-1}{z}\mathrm dz$along the unit circle How do I evaluate the following integral? $$\oint_{C}\frac{e^z-1}{z}\mathrm dz$$ where$C$is the unit circle (counter-clockwise). I have just learned Cauchy-Goursat's Theorem, but I cannot apply it ... 1answer 98 views ### Fourier transform of$H(x)\tanh(x)$I would like to compute the Fourier transform of the product of$\tanh(x)$and the Heaviside step function$H(x)$, i.e. $$\int_{-\infty}^{\infty} H(x)\tanh(x)e^{-ikx}dx = \int_{0}^{\infty} \tanh(x)e^{... 2answers 27 views ### Contour Integral from Laurent Series Disclaimer: I'm a Physics student, not a Maths student. I'm working on some problems involving Laurent series and the Residue theorem, and I've come across something I can't quite get my head around. ... 1answer 26 views ### Complex Integral \int_{C(0,2)} \frac{e^z}{i\pi -2z} Evaluate: \int_{C(0,2)} \frac{e^z}{i \pi -2z}dz So using the Cauchy Integral Formula: \int_{C} \frac{f(z)}{z-z_0} = 2\pi i f(z_0) If I define f(z)= \frac{e^z}{-2} then the z-z_0=\frac{-i\pi}{... 1answer 35 views ### Computing integral with contour I am trying to evaluate the integral \int_0^{\pi/2}\frac{1}{\sin^2 t+(\sin t)^{-2}}dt. The first step I took was using symmetry to get$$\int_{0}^{\pi/2}\frac{1}{\sin^2 t+(\sin t)^{-2}}dt=\frac{1}{4}... 0answers 14 views ### What is the difference between the trace operator and the contour integral operator? I noticed that the trace operator can be used to trace the boundary of a subspace of a spectrum. Some methods also use the contour integral in spectral analysis, however I am not sure if this is ... 1answer 68 views ### Show that$\int_{0}^{\infty}\frac{t^{\tau - 1}}{1+t}= \frac{\pi}{\sin(\pi \tau)}$, where$0<Re(\tau)<1$I need to show that Mellin Transform of the function$\frac{1}{1+t}$is$\frac{\pi}{\sin(\pi \tau)}$. So, by definition$(Mf)(\tau)=\int_{0}^{\infty}f(t)t^{\tau -1}dt=\int_{0}^{\infty}\frac{t^{\tau - ...
I'm trying to understand the complex line integral equation as given in these notes. It's given as $$\int_{\gamma} f(z) dz = \int_a^b f(\gamma(t)) \gamma'(t) dt$$ where $\gamma(t)$ is a ...