# Tagged Questions

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### Is there an analytic function defined on $\Bbb C$ except for Gaussian integers where it has poles of order 1 and residue 1?

I need a function defined for all complex variables $z$, except for at all the Gaussian integers, where it has poles of order 1 and residue 1. The function has to be complex-analytic. Can anyone ...
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### holomorphic function over the disk that is real on a closed curve must be constant

Let $f$ be holomorphic on $\{z\in \mathbb{C}\mid |z|\leq 3\}$ and real on the boundary of the square $\{z\in\mathbb{C}\mid Re(z)\leq1 \text{ and } Im(z)\leq 1 \}$. Prove $f$ is constant. How to ...
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### A difficult integral evaluation problem

How do I compute the integration for $a>0$, $$\int_0^\pi \frac{x\sin x}{1-2a\cos x+a^2}dx?$$ I want to find a complex function and integrate by the residue theorem.
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### Computing the contour integral of $\frac{\log(z)}{z^2 +a^2}$.

I'm still a bit insecure when it comes to complex analysis and I wondered if you guys could take a look at my solution to this problem. Let $a > 0$ and define $$f(z) = \frac{\log(z)}{z^2 +a^2}$$ ...
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### calculation of $\int^{\frac{\pi}{2}}_{0}\cos^{n}x\cos (nx)\ dx$, where $n\in \mathbb{N}$

Calculation of $\displaystyle \int^{\frac{\pi}{2}}_{0}\cos^{n}x\cos (nx)\ dx$, where $n\in \mathbb{N}$ $\bf{My\; Try}::$ Using $\displaystyle \cos (x) = \frac{e^{ix}+e^{-ix}}{2}$, we get ...
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### question on integrals

Let $\displaystyle A=\int_0^1 \frac{dx}{1+x^8}$. Then which of the following are true: 1) $A\lt 1$, 2) $A\gt 1$, ...
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### real integrals using residues

How to evaluating this integral using residues where $a>0$: $$\int _0^{\infty }\frac{x^3dx}{x^5-a^5}$$ Any help is appreciated
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### Summing a series by using residues

For the same series $$\sum_{n=0}^{\infty}\binom{3n}{2n} x^n$$ I am trying to calculate te sum by using residue theory. At the last line, I need to find the roots of $z^2-(z+1)^3x=0$ and one of ...
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### Applications of Residue Theorem in complex analysis?

Does anyone know the applications of Residue Theorem in complex analysis? I would like to do a quick paper on the matter, but am not sure where to start. The residue theorem The residue ...
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### Graphing $r =2\sin(2\theta)$

I am calculating some residue calculus stuff, where I need to know if the prescribed poles are inside the curve given above, namely $2\sin(2\theta)$ for $0\leq \theta<2\pi$. I actually need to know ...
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### Use Residue Theorem to evaluate $\ \oint_{C_3 (0)} \frac{z+7}{z^4 + z^3 - 2 z^2}\,dz \$?

How do I use Residue Theorem to evaluate $\ \oint_{C_3 (0)} \frac{z+7}{z^4 + z^3 - 2 z^2}\,dz \$ where $C_3(0)$ is the circle of radius 3 centered at the origin, oriented in the counter- clockwise ...
### Singularities of $\ \frac{z-1}{z^2 \sin z} \$
Find all singularities of $\ \frac{z-1}{z^2 \sin z} \$ Determine if they are isolated or nonisolated. This is not hard, it is z = 0 and z = k*pi. But how do I: For isolated singularities, ...
The integral in question is $$\int_{_C} \frac{z}{z^2+1}\,dz,$$ where $C$ is the path $|z-1| = 3.$ The two pole of $f(x)$ where $f(x)=\frac{z}{z^2+1}$ is $-j$ and $j$ {\rm ...