# Tagged Questions

For questions about integration methods that use results from complex analysis and their applications

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### Choice of branch cut

I'm trying to rewrite an integral of the form $$\int_{-∞}^∞ (x^2+k^2)^{-s/2}e^{i(xc_1+kc_2)}dx$$ ($s>0,c_1,c_2,k\in\mathbb R$) in such a way that it is positive (for some $s$). To do so I'd like ...
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### Evaluating Improper Integrals with Residues - don't think I'm calculating the residues properly

I have to evaluate the integrals $\displaystyle \int_{-\infty}^{\infty}\frac{dx}{x^{2}+p^{2}}$, for $p > 0$, and $\displaystyle \int_{-\infty}^{\infty} \frac{dx}{(x^{2}+p^{2})^{2}}$, for $p > 0$ ...
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### Difficulty evaluating complex integral

The integral along the path $\gamma(t)=e^{2ti},\;t\in[0,2\pi]$ is $\begin{equation*} \int_{\gamma}\frac{1}{z^{2}-1}dz \end{equation*}$. I approached this like a real integral in the hopes things ...
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### Fourier transform of a Gaussian times a rational function

I am trying to compute the following Fourier transform $$\int_{-\infty}^\infty\text{d}x\,e^{i k x}e^{-x^2/a^2}\frac{P(x)}{Q(x)}$$ where $\text{deg}P(x)+1\leq\text{deg}Q(x)$, and the roots of $Q(x)$ ...
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### Cauchy-Riemann equation analogue but for the quaternions

given a function over the quaternions $$U(x,y,z,t)+iV(x,y,z,t)+jW(x,y,z,t)+kR(x,y,z,t)=f(x,y,z,t)$$ what are the analogues of the Cauchy Riemann equation for the quaternionic plane so the function ...
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### Determine the number of zeros using the Argument Principle

I'm tasked with finding the zeros of $f(z)=z^3+1$ that lie inside the first quadrant using the Argument Principle, which I have simplified below: $$N=\frac{1}{2\pi}[arg(f(z))]_C$$ where N represents ...
I have the integral $\int_{C(0;2)} \frac{e^z}{z^3+9}dz$ I was told I could use Cauchy's Integral formula but I'm still stuck, I'm not sure how to apply it. Any help would be great!