Questions tagged [residue-calculus]
Questions on the evaluation of residues, on the evaluation of integrals using the method of residues or in the method's theory.
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Using Residue Theorem for functions with removable singularities
So for an Integral of the form $\int_{-\infty}^{\infty} \frac{e^{ixa} -1}{x(x^2 + 1)} dx$. My intuition is to use complex contour integration and use a contour that is a semi-circle on the upper-half ...
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Complex integrals that look like they agree, differ by sign (according to Mathematica)
Consider the integral
$$\int_0^\infty \frac{dz}{1-z^2 +i0^+},$$
I would assume it to agree with the integral
$$\int_0^\infty \frac{dz}{(1-z+i0^+)(1+z+i0^+)}. $$
However, according to Mathematica the ...
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Using the residue theorem to compute two integrals [closed]
Classify the singular points for the function under the integral and using the residue theorem, compute:
(a) $$ \int_{|z-i|=2} \frac{z^2}{z^4 + 8z^2 + 16} \, dz, $$
(b) $$ \int_{|z|=2} \sin\left(\frac{...
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Compute residue of pole of order $m$ of complicated function
Consider the following function:
$$f(z)=\frac{\left(z^{-m}-z^m\right)^2 \left(z^{-n}+z^n\right)}{\omega z-z^2-1}$$
with integers $m,n\geq 0$ and an arbitrary constant $\omega\in \mathbb{C}$ with $\rm{...
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Validity of Python-derived solution for contour integral $\oint f(z)f(z-\overline{z})~dz$
$\newcommand{\on}[1]{\operatorname{#1}}$
$$
\mbox{Consider the function:}\quad
\on{f}\left(z\right) =
\frac{{\rm e}^{tz}}{\left(1 + z^{2}\right)^{3}}\,
\left(\sqrt{t} - t\right)\ \ni\ t,z \in \mathbb{...
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Residue of a removable singularity at inifinity
Exercise:
Find all the singularities of $$\frac{z^3e^{\frac{1}{z^2}}}{(z^2+4)^2},$$ classify them, and find each residue.
I found that $+2i, \ -2i$ are poles of order two. I was able to calculate ...
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Classify the singular points for the function under the integral and using the residue theorem
Classify the singular points for the function under the integral and using the residue theorem, calculate:
(a)
$\displaystyle\int_{|z|=3} \frac{1 - \cosh z}{z^6 + 2z^5} \, dz $ and
(b)
$\displaystyle \...
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The function $ f $ is given by the formula $ f(z) = \frac{1 - \cos z}{z^5 + z^7}$
The function $ f $ is given by the formula
$$ f(z) = \frac{1 - \cos z}{z^5 + z^7}. $$
(a) Classify the singularity at the point $ z = 0 $ and write down the principal part of the Laurent series ...
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Given the function $f$ with the rule $f(z) = \frac{z \sinh\left(\frac{1}{z^2}\right)}{z^2 + 1}$.
Given the function $f$ with the rule $f(z) = \frac{z \sinh\left(\frac{1}{z^2}\right)}{z^2 + 1}$.
(a) Determine and classify the singular points of the function $f$ and calculate the residues at these ...
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Computing residue at infinity
Let $g$ be holomorphic function in $G = \{|z| > 100\}$ where
$$g (z) = \frac{z^{99}}{\prod_{k=1}^{100} (z-k)}$$
I would like to compute $Res(g,\infty)$. By definition
$$Res(g,\infty) = -Res(\frac{1}...
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What is the residue of $f$ at $\infty$?
How do I compute the residue at $a = \infty$ of the function $\operatorname{f}$ ?:
$$
\operatorname{f}: \mathbb{C} \setminus \left\{{\rm i},-{\rm i}\right\} \to \mathbb{C},\quad z \mapsto \frac{1}{z^{...
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Obtaining the residues of $\frac {(z-1)^2} {(e^z - 1)^3}$
I want to calculate the residues of $$f(z) = \frac {(z-1)^2} {(e^z - 1)^3}.$$ I already know that the isolated singularities are of the form $2\pi i \cdot \mathbb Z$, and that they are poles of order $...
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Use residues to verify the integral $\int_0^{\infty} \frac{x^{1/2}}{(x+1)^2} dx$ [duplicate]
I found this question on page $285$ of the book COMPLEX ANALYSIS FOR MATHEMATICS AND ENGINEERING by J, Mathews.
I tried to draw the contour and calculate the contour integral and each path integral,...
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Finding mistake in contour integral. $f(z)=\frac{\exp{(-1+i)z}}{z \cdot z^{1/2}}$
I'm trying to calculate the following real integral:
$$I=2\int_{0}^{\infty}\frac{e^{-x^2}\sin(x^2)}{x^2}\mathop{\mathrm{d}x} = \int_{0}^{\infty}\frac{e^{-t}\sin{t}}{t^{3/2}}\mathop{\mathrm{d}t}.$$
I ...
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Integration over Hankel contour
In a write-up by Paul Garrett, he claims that he can apply the Residue Theorem to the equality
$$
\zeta\left(s\right) = \frac{1}{\Gamma\left(s\right)\left(1-{\rm e}^{2\pi{\rm i}s}\right)}
\lim_{\...
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