# Tagged Questions

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### $\int_\gamma e^{\frac{1}{z^2-1}}\sin{(\pi z)}dz$ on a closed curve of index $N$ with respect to the point $1$.

Let $\gamma$ be a closed curve in the right half plane that has index $N$ with respect to the point $1$. Find $$\int_\gamma e^{\frac{1}{z^2-1}}\sin{(\pi z)}dz$$ This is a problem from an old ...
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### Writing the floor function as a contour integral

The function $f(z)=\frac{\pi}{\sin \pi z}$ has simple poles of residue 1 at the integers. Hence, by the residue theorem, I consider the interesting idea of drawing a (perhaps rectangular, for example) ...
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### Proper way to set up “Pac-Man” contour integral

I'm trying to evaluate $$\int_0^\infty \frac{x^a}{1+x} \: dx, \: -1<a<0$$ using contour integrals. Actually, I have found the correct answer using a "Pac-Man" contour and residues. My only ...
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### Contour Integral: $\int^{1}_{0}\frac{1}{\sqrt[n]{1-x^n}}dx$

I want to compute: $\int^{1}_{0}\frac{1}{\sqrt[n]{1-x^n}}dx$ for natural $n>1$ using Residue Calculus. I am thinking of using some kind of a keyhole or bone contour that could go around the ...
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### Calculating $\int_0^\pi \sin^2t\;dt$ using the residue theorem

I want to use the residue theorem to calculate $$I:=\int_0^\pi \sin^2t\;dt$$ Since $\sin^2$ is an even function, we've got $$I=\frac{1}{2}\int_0^{2\pi}\sin^2t\;dt$$ The solution of this exercise ...
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### Contour intergals of rational fuction

Consider $$F=\frac {x}{x^3+y^3}dx+\frac{y}{x^3+y^3}dy$$ 1) Show that $\int_GF=0$, where $G$ is the arc of a circle or radius $r$ in the first quadrant. 2) Compute the integral of $F$ along the ...
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An exercise given by my complex analysis assistant goes as follows: For $n \in \mathbb{N}$ and $x>0$ we define $$P_n(x) = \frac{1}{2\pi i} \oint_\Sigma ... 1answer 112 views ### Finding a generalization for \int_{0}^{\infty}e^{- 3\pi x^{2} }\frac{\sinh(\pi x)}{\sinh(3\pi x)}dx \;\;\;\;I was reading the introduction of Paul J. Nain's book "Dr. Euler's fabulous formula" where he talks about the sense of beauty in mathematics and quotes the G.N.Watson as saying that a ... 2answers 126 views ### Contour Integral of \int \frac{a^z}{z^2}\,dz. My task is to show$$\int_{c-i\infty}^{c+i\infty}\frac{a^z}{z^2}\,dz=\begin{cases}\log a &:a\geq1\\ 0 &: 0<a<1\end{cases},\qquad c>0.$$So, I formed the contour consisting of a ... 2answers 318 views ### Evaluation of the contour integral \int_\beta \frac{e^z}{e^z-\pi} dz Suppose \beta is a loop in the annulus \{z:10<\left|z\right|<12\} that winds N times about the origin in the counterclockwise direction, where N is an integer. Determine the value of ... 2answers 399 views ### Summation of series using residues Let P(n) and Q(n) be polynomials such that \displaystyle \sum_{n=-\infty}^{\infty} (-1)^{n} \frac{P(n)}{Q(n)} converges conditionally, that is, the degree of Q(n) is exactly 1 degree more than ... 2answers 505 views ### Contour integral \int_{|z|=1}\exp(1/z)\sin(1/z)dz Evaluate the contour integral$$\int_{|z|=1}\exp(1/z)\sin(1/z)\,dz along the circle $|z|=1$ counterclockwise once. The singularities are $\dfrac1{\pi k},k\in\mathbb{Z}$ plus the limit point $0$. So ...
I have been trying to evaluate the following sum using residues $\displaystyle \sum_{n=1}^{\infty}\frac{1}{\sinh^{2}(\pi n)}=\frac{1}{6}-\frac{1}{2\pi}$ I am mainly interested in using residues to ...
Let $g(z)$ be a branch of the square root on $\mathbb{C} \setminus \lbrace iy : y \leq 0 \rbrace$. For $0 < r < 1 <R$ and $0 \leq \theta \leq \pi$, let $\tau_r$ be the contour given by the ...