Questions on the evaluation of integrals along a locus in the complex plane.
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62 views
Calculation of the Inverse Laplace Transform of $\frac{1}{p}$ by contour integration.
I am always told in my lessons of control engineering that the inverse Laplace Transform of $\frac{1}{p}$ is the Heaviside step function $\theta(t)$. But I have a problem when I calculate the inverse ...
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27 views
Contour integral redefining variables
I have the integral
${\operatorname{Im}} \left (\int^\infty_0 e^{ix} x^{s-1} \, \mathrm{d} x \right)$
and I wish to redefine $x \to iy$ but I am unsure of how to justify this using contour ...
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83 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 ...
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101 views
contour integration on indented path involving branch of the square root
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 ...
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149 views
Contour integration
In Penrose's 'Road to Reality', he states that for any integer n, $n \ne 1$, $ \oint z^n dz=0$. Qualitatively, why is this so, given that for any negative n poles in the complex graph exist (namely at ...
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24 views
Definition: “A contour respects causality”
When doing a contour integral, what does "the contour respects causality" mean?
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52 views
Justification in change of variables
it would be fantastic if anyone could help me with the following problem:
I have the integral
$$\operatorname{Im} \left( \int^\infty_0 e^{it} t^{s-1} \mathrm{d} t\right)$$
and I wish to make the ...
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42 views
Contour Integrals and positively oriented circles
If $C_0$ denotes a positively oriented circle $|z-z_0|=R$, then $\int_{C_0}$ $(z-z_0)^{n-1} dz$ = $\left\{
\begin{array}{lr}
0 & n=\pm1, \pm2, ...\\
2\pi i & n=0\\
...
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71 views
About evaluating $\mathcal{L}^{-1}_{s\to x}\left\{\dfrac{F(s)}{s}\right\}$ by considering contour integration with different entire functions $F(s)$
Detailedly compare the difficulties of different entire functions $F(s)$ where $F(0)\neq0$ when evaluating $\mathcal{L}^{-1}_{s\to x}\left\{\dfrac{F(s)}{s}\right\}$ by considering contour integration, ...
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55 views
About the inverse laplace transform of sinc function
How to calculate $\mathcal{L}^{-1}_{s\to x}\{\text{sinc}(s)\}$ ?
Note: $\text{sinc}(s)=\dfrac{\sin s}{s}$ when $s\neq0$ .
Also note that $\lim\limits_{s\to\pm\infty}\dfrac{\sin s}{s}=0$ .
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61 views
Contour Integrals and counterclockwise
$\int_C (z-z_0)^{(n-1)}\ dz$ for any integer $n$, where $C$ is the contour once around the circle $|z-z_0|=1$ counterclockwise and $z_0$ is any point in the plane. Also give the values of the ...
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67 views
Contour Integrals
Evaluate:
$\int_C \hat{z} dz$ where $C$ is the straight line from $i$ to $2-i$.
$\int_C \frac{dz}{z}$ where $C$ is the straight line from $3$ to $4i$
$\int_C (z-z_0)^{n-1}dz $ for any integer $n$, ...
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60 views
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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157 views
Find the value of the contour integral $ \oint \frac {3z^3 + 2}{(z-1)(z^2+9)} dz$
$ \oint \frac {3z^3 + 2}{(z-1)(z^2+9)} dz$ taken counterclockwise around circles:
(a) |z-2| = 2; (b) |z| = 4
My Attempt:
The circle of radius 2 centered at z = 2 only encloses the singularity z = 1 ...
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85 views
finding residue with $\oint_C \dfrac{3z^3 + 2}{(z-1)(z^2 + 9)} dz$
I am doing the integral $\oint_C \dfrac{3z^3 + 2}{(z-1)(z^2 + 9)} dz$, and I am trying to find the residue at the pole $3i$;I am unsure how to do this. Could I factor $z^2 + 9$ further?
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71 views
Evaluate $\int_{\gamma_R}f(z) dz$ when $R>2$
Let $$f(z)=\frac{e^{\pi iz}}{z^2-2z+2}$$
and $\gamma _R$ is the closed contour made up by the semi-circular contour $\sigma_1$ given by, $\sigma_1(t)=Re^{it}$, and $0\leq t\leq\pi$ the straight line ...
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134 views
$\int e^{-x^2}dx$ with contour integration
Is it possible to prove that
$$\int_{-\infty}^{\infty}e^{-x^2}dx=\sqrt\pi$$
integrating $e^{-z^2}$ along a suitable contour?
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25 views
How do I show this integral surface area relationship?
Let alpha=xdy-ydx and let M be a compact domain in the plane R^2. Show that the integral along the boundary of M of alpha is twice the surface area of M.
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232 views
Contour Integral of $f(z) \; \cot(\pi z)$
In lecture, my professor stated that
$$ \lim_{N \to \infty} \int_{\gamma_N} f(z) \cot(\pi z) \; dz = 0 $$
where $\gamma_N$ is the square contour with vertices at $\pm (N + \frac12) \pm i(N + ...
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1answer
31 views
Sequential Limit of Line Integral Is The Same As The Usual Limit of Line Integral? (Gamma Function Related)
Let $\epsilon > 0$, and $ n \in \mathbb{Z}^{+} $. Let $C_{n}$ be a positively oriented polygonal line that is from $-n + 1/2 - i \epsilon$ to $ 1/2 - i \epsilon$ and from $ 1/2 - i \epsilon$ to $ ...
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84 views
How to integrate $\frac{\sin^2 x}{x^2}?$ [duplicate]
Possible Duplicate:
Proof for an integral involving sinc function
How can one calculate $\displaystyle\int_0^{\infty} \frac{\sin^2 x}{x^2} dx$? I know that the answer is $\pi/2$ and I ...
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1answer
89 views
Finding an upper bound on an integrand.
I'm reading a passage on how to integrate $f(z)=z^\beta(1+z^2)^{-1}$ for $\beta\in(-1,1)$.
Take $\epsilon>0$, and let $\gamma_\epsilon$ be the arc $\{\epsilon ...
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96 views
Concept of Residue Cancellation
I am trying to understand how to apply the residue theorem to solve
$\frac{1}{2\pi j}\int^{\gamma+j\infty}_{\gamma-j\infty}\Gamma(n-s)\Gamma(s)\Gamma(1-s) {}_1F_1(s;b;c) ...
