Questions on the evaluation of integrals along a locus in the complex plane.

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$\int \frac{2+\sin(z)}{z} dz$

Please bear in mind that I am trying to teach myself complex integration having never taken a course in complex analysis, so assume I know very little. $\int \frac{2+\sin(z)}{z} dz$ traversing the ...
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1answer
31 views

$f_n$ converges uniformly on $\partial \Omega$ then $f_n$ converges uniformly on $\bar{\Omega}$

The problem states that $f_n$ is a sequence of functions which are continuous on the closure of $\Omega$ and holomorphic on $\Omega$ where $\Omega$ is a bounded region and were asked to show that if $...
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$\int_{-\infty}^\infty \frac{dz}{z - z_0}$ by contour integration

Consider the integral $\int_{-\infty}^\infty \frac{dz}{z - z_0}$. It has a simple pole at $z = z_0$. Assume $\Im (z_0) < 0$ so the pole is in lower half-plane. Divide $$ \oint_{C_0} = \int_{-R}^R +...
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1answer
29 views

Problem with integration limits using spherical substitution

Good night, i have a problem with this integral, please help me with the integration limits. \begin{align} \int_{-1}^{1}\int_{-\sqrt{1-x^{2}}}^{\sqrt{1-x^{2}}}\int_{-\sqrt{1-x^{2}-y^{2}}}^{\sqrt{1-x^{...
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An Alternate proof of Nyquist-Shannon

This problem is from Basic Complex Analysis, Part 2A, by Barry Simon. This problem will provide an alternate proof of the strong from of the Nyquist-Shannon sampling theorem (Theorem 6.6.16 of Part 1)...
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1answer
17 views

Problem with integration limits with cylindric cordinates.

Good night, i have a problem when i go to verify the integration limit $0\leq\theta\leq\varPi/2$ because i think the integration limit go to $0\leq\theta\leq\varPi$ because is an half a circle. $\...
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1answer
72 views

Integral of $p(x)\operatorname{csch}(x)$

I'd like to calculate the following integral $$\int_{-\infty}^{+\infty}\frac{x^4 \left(\frac 1 {a^2+x^2} +\frac 1 {b^2+x^2}\right)}{\sinh^2(x\pi /c)} \, dx$$ where $a$, $b$ and $c$ are positive ...
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1answer
24 views

Integral estimation

Is it true that $$\int_{-\pi}^{\pi}\left |\frac{iR^{x}e^{i \theta x}}{1-Re^{i\theta}}\right |\,\mathrm{d}\theta \leq 2\pi R\frac{R^{x}}{1-R}\xrightarrow{R\to\infty}0$$ $$\int_{-\pi}^{\pi}\left |\frac{...
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0answers
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Complex Contour Integral Involving Arg(z)

My question is regarding the following complex integral: $$\int_\gamma\frac{\operatorname{Arg}(z)}{z} dz$$ where $\gamma$ is the curve defined by:$\quad$ $\gamma(t) = e^{it}, 0\leq t\leq \frac{\pi}{...
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Splitting a contour integral into separate integrals?

Suppose you had a complex contour integral of the form $$ \int_{\alpha + \beta} f(z) \; dz $$ Is this equivalent to $$ \int_{\alpha} f(z) \; dz + \int_{\beta} f(z) \; dz $$ Thanks
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Integral problem with branch point from Physics

The question come from a Summation like this $${ \sum _{ { z=i\omega }_{ n } } { \frac { -\alpha E\pi }{ 4{ z }^{ 3 }\sqrt { -\alpha -z } } } }$$ I can use Cauchy theorem to transform it to a ...
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1answer
27 views

Proof of the Primitive Existence Theorem

In the first section of the proof of the primitive existence theorem we are trying to show that $$F\left(z+h\right)-F\left(z\right)=\int_{L_{z,z+h}}f$$ where $\alpha, z,z+h$ are collinear and $\alpha=...
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0answers
17 views

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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18 views

How to prove reflection positivity for $|x|^{-p}$ using Fourier transform (and contour integrals)

This question looks quite lengthy because I'm sketching the proof in the lecture - the two questions (look out for something bold) are actually relatively short. I need some help with a proof in our ...
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27 views

Evaluate the following contour integral

Evaluate the integral $$\int_{C}\left(\exp\left(z\right)-\dfrac{2}{z^3}\right)dz$$ where C is any contour from $i$ to $-i$ which does not pass through $0$ I understand that contour integration can ...
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1answer
16 views

Showing that a function is the restriction of another holomorphic function

Let $f$ be holomorphic on the annulus $\{z:\;1-\epsilon<|z|<1+\epsilon\}$. Define: $$\phi:\;D\to\mathbb{C};\quad \phi(w)=\frac{1}{2\pi i}\int_C \frac{f(z)}{z-w}\,\mathrm{d}z$$ where $C$ is the ...
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$ f(z) = \int _{0} ^\infty e^{-z t^2} dt $ is holomorphic in $Re(z) > 0$

I've been trying to find a reasonable way to approach this but nothing leads to a reasonable result. With $z = a + ib$ the last thing I tried looking at was $$| \frac{f(z) - f(z-h)}{h}| = |\frac{1}{...
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20 views

conversion from path integral to contour integral

I'm considering the following integral $$I=\int_{-i\infty-\epsilon}^{i\infty-\epsilon}\frac{\psi(v)}{v}\;dv$$ I want to use the following substitution to convert the path integral to a contour ...
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2answers
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Evaluate $\int_0^{2\pi}\frac{\sin^2(x)}{a + b\cos(x)}\ dx$ using a suitable contour

I need to find a good contour for $\int_0^{2\pi}\frac{\sin^2(x)}{a + b\cos(x)}\ dx$ but I don't know which one to choose. Both a semicircular, and rectangular contour look ugly for this. I've been ...
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1answer
24 views

Show that $\displaystyle \int_{\gamma_R}\exp(-w^2)\,\mathrm dw\to 0~(R\to\infty)$ along $\gamma_R(t)=R\exp(\mathrm it)$

Show that $\displaystyle \int_{\gamma_R}\exp(-w^2)\,\mathrm dw\to 0~(R\to\infty)$ along $\gamma_R(t)=R\exp(\mathrm it),t\in\left[0,\frac{\pi}{4}\right]$. Hint. Use that $\cos 2t\geq 1-\frac{4}{\...
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1answer
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Evaluate $I(x)=\int_{-\infty}^{\infty}\frac{e^{xv}}{1+e^v}dv$

This is the last part of a multistage evaluation of $I(x)=\Gamma(x)\Gamma(1-x)$. Through various substitutions we get the integral $$I(x)=\int_{-\infty}^{\infty}\frac{e^{xv}}{1+e^v}dv$$ We're also ...
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Compute an integral involving exterior Riemann mapping

Let $E$ be an infinite compact subset of the complex plane $\mathbb{C}$ such that $\overline{\mathbb{C}}\setminus E$ is simply connected. There exists a unique exterior conformal mapping $\Phi$ from $\...
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1answer
154 views

Show that $\int_0^{\pi}\frac {\cos {n \theta}}{1-2r\cos \theta+r^2}\, \mathrm d \theta = \frac {\pi r^n}{1-r^2}$

I am trying to calculate $$I=\int_0^{\pi}\frac {\cos {n \theta}}{1-2r\cos \theta+r^2}\, \mathrm d \theta$$ where $r\in(0,1)$ I tried substituting $u = e^{2 i \theta}$ and using the Cauchy integral ...
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21 views

Double Integral with Residues

I'm trying to solve the integral $$\int_a^b\int_a^b\frac{dxdy}{1+\left(x^2+y^2\right)^\alpha}$$ where the constant $\alpha$ is real-valued and in the range $\alpha\in[1/2,\infty)$. The bounds $a$ ...
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792 views

Integral of rational function over $\mathbb{H}^4$

Suppose I have a rational function of $8$ coordinates $a,b,c,d,e,f,g,h$ that I want to integrate over $\mathbb{H}^4$: ...
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2answers
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Complex integration problem via Cauchy's integral formula

I want to integrate the following :$$\int_{|z|=2} \frac{dz}{z^{2}-1}$$ in the positive direction. So my idea is two split the integral into a sum of two integral , something like $$\int_{|z|=2} \frac{...
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1answer
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How can I evaluate this contour integral?

Suppose we have the following contour integral, in the complex plane: $$ \int_{\gamma} \frac{e^{\frac{1}{z}}}{z^{2}} \; dz $$ where $\gamma (t) = e^{it}$ for $0 \leq t \leq 2 \pi$. To solve this, I ...
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Find all possible values of the integral

Find all possible values of $\displaystyle I= \int_C \frac{dz}{1+z^2}$, where $C$ is a curve with initial point $0$ and final point $1$ that does not meet the poles of $\dfrac{1} {1+z^2}$. It looks ...
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1answer
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Is integrating $e^{iz^{2}}$, along the real axis in the complex plane the same as integrating the riemann integral of $e^{x^2}$?

In the title, $z\in \mathbb{C}$ and $x\in\mathbb{R}$. More specific to my problem, I am hoping that $\int_{0}^{R}e^{iz^{2}}dz=\int_{0}^{R}e^{x^{2}}dx$. Maybe this is obvious but I want to make sure.
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1answer
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Gaussian integral $\int_{-\infty}^\infty \exp(-(x+\mathrm iY)^2)\,\mathrm d x$ along $[-R,R]+\mathrm i[0,Y]$

Use integration along $\partial Q$ of $Q=[-R,R]+\mathrm i[0,Y]$ to show that for all $Y\geq 0$ it holds that $$\int_{-\infty}^\infty \exp(-(x+\mathrm iY)^2)~\mathrm dx = \int_{-\infty}^\infty \...
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0answers
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Contour integral along a parabola

The question reads: Evaluate $$ \int_\gamma f(z)dz$$ where $$f(z)=x^2: x,y \in \mathbb{R} $$ and $\gamma$ is the parabola $y=2x^2$ from $x=0$ to $x=2$. This is the first question I've encountered ...
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1answer
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How would you plot this equation (in the complex plane)?

So I am currently looking at how to calculate the integral of a complex function $f(z)$ within a contour $\gamma$. That is, an integral of the form $$ \int_{\gamma} f(z) \; dz $$ Where the contour is ...
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2answers
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Mistake while evaluating the gaussian integral with imaginary term in exponent

I am trying to evaluate the integral $I=\int_0^\infty e^{-ix^2}\,dx$ as one component of evaluating a contour integral but I am dropping a factor of $1/2$ and after checking my work many times, I ...
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1answer
69 views

Integrating $\int_{-\infty}^{\infty}\frac{e^{ax}}{1+e^x}dx$ for $0<a<1$.

I'm supposed to find the value of $\int_{-\infty}^{\infty}\frac{e^{ax}}{1+e^x}dx$ for $0<a<1$. I wanted to integrate over the upper semicircle of radius $R$, and take the limit as $R\to\infty$. ...
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Deformation of Gamma function integral contour

Terence Tao has described the gamma function as the inner product of a multiplicative and an additive character with respect to the Haar measure on $\Bbb R^+$. The gamma function is defined as follows:...
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41 views

Contour integral of continuous but not holomorphic functions

This question was transferred here following Mathoverflow suggestions. Let us consider two functions $f(z)$ and $g(z)$, both holomorphic on a domain $U$ (a simply connected subset of $\mathbb{C}$). ...
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3answers
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Calculate $\int_0^{\infty}\frac{1}{(x+1)(x-2)}dx$ using residues

I'm supposed to calculate $$\int_0^{\infty}\frac{1}{(x+1)(x-2)}dx$$ using residues. The typical procedure on a problem like this would be to integrate a contour going around an upper-half ...
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Contour Integral of $\int\limits_0^{2\pi}\frac{d\theta}{1+a\cos\theta}$ for $a^2<1$ (textbook wrong?)

My book is telling me that the answer is $\frac{2\pi}{\sqrt{1-a^2}}$. I'm getting an extra a on the numerator. Could somebody verify if I'm wrong, or if it's my book (it has been wrong numerous times)....
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1answer
33 views

Prove the integral is always imaginary

Show that if f is analytic on D and γ is a closed curve in the region then the integral $$\int \overline{f(z)}f'(z)$$ is purely imaginary. I think this problem would use some extension of cauchy ...
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$\frac{\omega}{2\pi}\int_0^{\frac{2\pi}{\omega}}\frac{\sin^2\theta \cos^2\omega t}{(1+\beta\cos\theta \sin\omega t)^5} dt$

I'm going to write out the whole problem as it is given to me (bad grammar and all) even though some of the info may be irrelevant to finding a solution. A charge $e$ moving along a straight line ...
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1answer
39 views

Cauchy integral formula with singularities

I am stuck on this question. $$\int \frac{e^{sin z^{2}}}{(z^{2}+1)(z-2i)^{3}}dz $$ along the path γ where γ is a circle centered at the origin of radius different from 1 or 2. I initially thought of ...
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1answer
22 views

Big-O Notation?

The problem is to to evaluate the following contour integral along a path $C$ defined/parameterized as $z(t)=εe^{it}$: $ \int_C \frac{e^{iz}}{z} dz$ The solution for the problem proceeds to say ...
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Evaluation of complex integral?

I'd like to verify the result of this integral, or find if I've made a mistake. In the following, $\mathbf x, \mathbf a, \mathbf b$ are all real vectors in $\mathrm R^3$. I do the following: group ...
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1answer
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$x^2+y^2\le 1$; $z=\sqrt{x^2+y^2}$; and $x^2+y^2=4-z$

I need to find a value and "surface" of a body which is contained in the following contours: $x^2+y^2\le 1$; $z=\sqrt{x^2+y^2}$; and $x^2+y^2=4-z$. Some hints and directions will be helpful. Sorry for ...
2
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1answer
48 views

Modulus of roots of polynomial tend to infinity

Define $f_n:\mathbb{C}\to\mathbb{C}$ and $(\alpha_n)$ such that:$$f_n(z)=\sum_{k=0}^n \frac{z^k}{k!}$$ and $f_n(\alpha_n)=0$. Prove $|\alpha_n|\to\infty$ as $n\to\infty$. I guess this makes sense ...
3
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2answers
51 views

Contour Integration of $\sin^2(x)/(1+x^2)$

How should I calculate this integral $$\int\limits_{-\infty}^\infty\frac{\sin^2x}{(1+x^2)}\,dx\quad?$$ I have tried forming an indented semicircle in the upper half complex plane using the residue ...
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2answers
46 views

Basic contour integration

I am trying to evaluate $$\int_{\gamma(0;2)}\frac {e^{i\pi z/2}}{z^2-1}\, \mathrm d z$$ using the Cauchy integral formula The problem is it is not holomorphic at $1$ and $-1$. My textbook suggests ...
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1answer
40 views

Contour Integral over a Closed Circle (Complex Analysis)

I'm having trouble understanding the difference, other than notation, between a contour integral over an open curve and a contour integral over a closed curve. So far, it seems to me that the ...
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0answers
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To get boundary of xy coordinates around object inside an image using active contour method

Trying to extract the XY coordinates around the boundary of an lesion object inside an image of plain background. I am using the active contour method as suggested to find x and y coordinates along ...
3
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3answers
43 views

Improper integral with complex limits

I would like to compute an integral of the form ($a,b \neq 0$) $$\int_{-\infty}^{\infty} e^{-(ax+ib)^2} dx = \frac{1}{a} \int_{-\infty+ib}^{\infty+ib} e^{-z^2} dz$$ where we made the substitution $z ...