The complex-integration tag has no wiki summary.
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Evaluating complex integrals involving log (finding bounds)
When evaluating real integrals involving log, I am having trouble with the step that involves finding a bound on circular segments. Let me explain what I mean:
If, for example, we have
$$
...
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Integrating $z^n$ and $(\overline{z})^n$ along a line segment in the complex plane
Let $z_1$ and $z_2$ be distinct points of $\mathbb{C}$. Let $[z_1,z_2]$ denote the oriented line segment starting at $z_1$ and ending at $z_2$. Evaluate the integral of $z^n$ and $(\overline{z})^n$ ...
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Problems in interpreting an integral that should be solved with residue method
Usually, when I solve an integral using residue method, I find real functions as integrands.
I am not able to provide an interpretation for the following complex integral
$$
\int_{-\infty}^{\infty} ...
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Is this OK: $\int_a^b \!\mathrm{d}x \,\,f(x) =^? \int_{\mathrm{i}\,a}^{{\mathrm{i}\,b}} \!\mathrm{d} (\mathrm{-i}y)\,\,f(\mathrm{-i}y).$ Any proof?
This is related to Wick rotation in QFT but it is not exactly it. I'll take a 2-dimensional spacetime to be brief but usually there are more.
I've checked with a few functions and with finite ...
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Complex form of gauss divergence theorem
Just as complex form of green's theorem $\int {f(z)}dz=i\int\int \frac{\partial f}{\partial x} + i\frac{\partial f}{\partial y}dxdy$ where $z=x+iy$ , do we have complex form of gauss divergence ...
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Show Smoothness by Morera
I'm trying to show smoothness on $(0,\infty)(\Re)$ of the following function:
$$ f(t,x)=\sum_{n=-\infty}^\infty e^{-\large \frac{(x-2\pi n)^2}{2t}}\frac{1}{\sqrt{2\pi t}} $$
The function is ...
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Laplace transform of a product of functions
While trying to compute the Laplace transform of a certain product, part of the calculation leaves me with a Bromwich integral which has the form:
...
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Residue of $\mathrm{exp} (z+1/z)/((z-z_1)(z-z_2))$ at z=0
The ultimate aim is to solve the following integral:
\begin{equation}
\label{eq:Icos1}
\begin{aligned}
I = \int\limits_{0}^{2\pi} \frac{\mathrm{exp}\left(c \cos(\varphi)\right)\mathrm{d}\varphi}{a - ...
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Solving an complex Integration with complex exp and other terms
I am trying to solve a partial differential equation and while solving I need to solve the following integral. If anyone could help me solve this integral that would be great.
$$y(x,t) = \int_{c-i ...
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Exercise of Complex Integration
Let $f(z)$ be such that along the path $C_N$ of the following figure
If $|f(z)|\leq \frac{M}{|z|^k}$ where $k>1$ and $M$ are constants independent of $N$.
How to prove that ...
