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Questions tagged [complex-integration]

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

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23 views

How to solve this integration? Can Cauchy integration Theory be used under this circumstance?

About this integration, there is no z-ξ under the line, how to integrate it? $$\int_{|z|=4} \frac{z^{19}}{(z^2+1)^4(z^4+2)^3}$$ I am considering how to solve this integration with Cauchy theory, or ...
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62 views

Complex integral along the contour

I know answer for $$ \oint_{|z-\frac{\pi}{2}(1-i)|= \pi} \frac{z\,dz}{\cos z-\cosh z} = 2\pi i(1-e^{-1})$$ But I don't understand why it's true. I know that $$ \oint_{|z-\frac{\pi}{2}(1-i)|= \pi} \...
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44 views

If $f:\mathbb{C}\to\mathbb{R}$ is continuous and $f\ll1$ then $\int_{|z|=1}f\ll4$ [duplicate]

Show that if $f:\mathbb{C}\to\mathbb{R}$ is continuous and $f\ll1$ ($a\ll b$ means $|a|\leq |b|$), then $\int_{|z|=1}f\ll4$. Hint: Show first $\int f\ll\int_0^{2\pi} |\sin t|dt$. I'm not sure how to ...
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13 views

Phase marginal for multivariate complex Gaussian density

The following is a cross-post from stats.stackexchange, which I am including here since it is mostly about a hard integral. Suppose $z$ is a random variable taking values in $\mathbb{C}^n$ and ...
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1answer
21 views

Contour integration with the contour $\sigma=[0,1]+[1,i]$

$\sigma=[0,1]+[1,i]$ is a contour. I am asked to sketch the contour, and evaluate $\int_\sigma Re(z)$. Firstly, I am not sure how to visualise this contour, since there are two parts. What does it ...
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1answer
24 views

For a differentiable function at $0$, we have $\lim\limits_{r \rightarrow 0}\frac{1}{r^2}\int_{C(0,r)}f(z)dz=0$

Let $f$ be a continuous function on $D(0,1)$ and $\mathbb{C}$-differentiable at $0$. I want to prove that $$\lim_{r \rightarrow 0}\frac{1}{r^2}\int_{C(0,r)}f(z)dz=0$$ My idea is to use polar ...
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2answers
32 views

Evaluate $\int_{C}(z-i) \,dz$ where $C$ is the parabolic segment: $z(t) = t + it^2, −1 \le t \le 1$

Evaluate $\int_{C}(z-i) \,dz$ where $C$ is the parabolic segment: $$z(t) = t + it^2, −1 \le t \le 1$$ by integrating along the straight line from $−1+i$ to $1+i$ and applying the Closed Curve Theorem. ...
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2answers
58 views

Calculus of Residue

I finding difficulties calculating : $res(f,0)$. with $f(z)=\frac{1}{z^2sinz}$ I thought of the method : defining $g(z)=z^3f(z)$, since $0$ is a pole of order $3$. then : $res(f,0)=\frac{1}{2!}g^{(...
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Have I evaluated this complex integral correctly?

Let $C$ be the circle $\mid \:z \mid = 6$ traced one lap counterclockwise. Evaluate: $$\int_{C} \frac{\cos(z)}{(z+i)^3}\;dz$$ Since $\cos(z)$ is analytic, my solution was to use cauchys integral ...
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1answer
222 views

Integral involving complex exponential function

I am trying to solve the following integral:$$\int_0^∞\frac{e^{-α(u+iπ/2)}\exp(te^{u+iπ/2})}{(u+iπ/2)^{β+1}}\,\mathrm du-\int_0^∞\frac{e^{-α(u-iπ/2)}\exp(te^{u-iπ/2})}{(u-iπ/2)^{β+1}}\,\mathrm du$$...
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1answer
41 views

Compute $\oint_{\gamma}f(z)dz$, where $f(z)=\frac {ze^{\pi z}}{z^4-16}+ze^{\frac \pi z}$ where $\gamma:9x^2+y^2=9$

Compute $\oint_{\gamma}f(z)dz$, where $f(z)=\frac {ze^{\pi z}}{z^4-16}+ze^{\frac \pi z}$ where $\gamma:9x^2+y^2=9$ Using the residue theorem. I don't know how to start this, first I thought about ...
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2answers
72 views

Calculate $\oint_{\gamma}\frac {\sin z}{z(z-2i)}dz$ on $|z| =3$ in trigonometric sense and on the inverse trigonometric sense.

$\oint_{\gamma}\frac {\sin z}{z(z-2i)}dz$ on $|z| =3$ in trigonometric sense and on the inverse trigonometric sense. On the trigonometric sense, we can apply the Residue theorem, we have : $z_1 = 0$ ...
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2answers
43 views

$\oint_{\gamma}\frac 1{z^2+1}dz$ on different curves

$\oint_{\gamma}\frac 1{z^2+1}dz$ on the curves: $\gamma_1: |z + i| = 1$, $\gamma_2:|z - i| = 1$, $\gamma_3: |z| = \frac 12$, $\gamma_4: |z - i| = \frac 32$. I would like to use the residue theorem and ...
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1answer
46 views

Integral $\oint_{\gamma}\frac {\cos z}{z}dz$ on 2 curves

$\oint_{\gamma}\frac {\cos z}{z}dz$ on the curve $\gamma = \gamma_1\colon|z| =1$ and $\gamma = \gamma_2\colon|z|=3$. I calculated on $|z| = 1$ but I don't see why it would be different if I would ...
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1answer
53 views

Entire function such that $\lim_{ z \to \infty} \frac{f(z)}{z^N} = 0$ then $f$ is a polynomial of degree at most $N$.

I am tried to prove this: Let a entire function $f$ such that $\lim_{ z \to \infty} \frac{f(z)}{z^N} = 0. $ Show that $f$ is a polynomial of degree at most $N$. I find this results Entire ...
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61 views

Evaluate this complex integral

I have the following complex integral that corresponds to a complex integral of a Wigner function of a 1-mode Gaussian state: $$I_n(\sigma) = \int^\infty_{-\infty} d^2\alpha \; L_n \left(\frac{4|\...
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1answer
51 views

Complex: If $|f|<\varepsilon$, then $\frac{1}{\pi}\int_E \frac{|f|}{|z-w|}<\varepsilon$?

Is this true? Let $f:E\xrightarrow{}\mathbb{C}$ be holomorphic on the interior of a compact set $E$, and let $\varepsilon>0.$ If $|f|<\varepsilon$ on $E$, then $$\frac{1}{\pi}\int_E \frac{|f(w)...
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25 views

Analytic continuation of Dirac delta distribution

My question is a simple one. Is it possible to analytically continue the Fourier transform of the Dirac delta distribution, i.e., can we analytically continue \begin{equation} \delta(x)=\frac{1}{2\pi}\...
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22 views

Cauchy's integral formula for modulus function on a set

I'm reading a paper by Rudin and I'm a little confused about how Cauchy's integral formula is used here on the modulus $|h_{i-1}-P|$. How do we go from contour integral to integrating on a set? I ...
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1answer
35 views

Doing contour integration of sin^z/(z-a)^4 in Maple

This is example 11.4.3 in Arfkin, Weber, and Harris: Calculate the following integral around a contour encircling $z=a$: $$I=\oint_C \frac{\sin ^2(z)dz}{(z-a)^4}$$ The answer is $-\frac{8\pi i}{3}\...
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1answer
25 views

evaluate $f(z) $ described in the anticlockwise ( i.e. positive direction)

Evaluate $$\int_{C} \frac{dz}{(z^2+ 4)^2}$$ where $C = \{ z \in \mathbb{C} \mid |z-i| = 2\}$ described in the anticlockwise ( i.e. positive direction) My attempt: I used Cauchy integral formula ...
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1answer
57 views

$\int_{[0, 1]} \frac{1}{w-z} dw$

I'm new to Complex Analysis so forgive me please. An exercise in Sarason's Complex Function Theory says to find, explicitely, the "Cauchy integral" of the constant function $1$ over $[0, 1]$ on the ...
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4answers
35 views

Evaluate $\int_{|z|=1}\frac{e^z-1}{z^2}dz$

How would you be able to evaluate $\int_{|z|=1}\frac{e^z-1}{z^2}dz$? Are you meant to perform some integration by parts to get it in a suitable form for Cauchy's Integral formula? The only problem ...
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0answers
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Solving $\sum\limits_{k=1}^n e(x-x_k) = h(x)$ for $e(x)$, where $x_k$ and $h(x)$ are given (updated)

I would like to find the function $e(x)$ which solves $\sum\limits_{k=1}^n e(x-x_k) = h(x)$, where $x_k$ and $h(x)$ are given. There are no restrictions on any of the $x_k$ or $h(x)$ except that $h(x)$...
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1answer
24 views

Mean value theorem for a disk (complex analysis) [closed]

How would I start to prove $$ f(z_0) = \frac{1}{πr^2} \int\int_{|z-z_0|<r}f(x+iy)dxdy $$ for $0<r<R$, using the Mean Value Theorem $f(z_0) = \frac{1}{2π} \int_0^{2π} f(z_0 + re^{it})dt$?
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17 views

Modified Error Function Integral

I was given an identity recently without any stated derivation, so I am attempting a derivation on my own. The identity in question is $$\int_0^\infty\frac{e^{-s^2t}\sin(sy)}{s}\,ds=erf(\frac{y}{2\...
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44 views

Solutions or Hints to Solve a Complex Integral

I came across this problem in the practice of an old book. I looked into it, and find with surprise that it's really hard. This problem is in a Complex Analysis book, and it definitely uses some ...
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3answers
34 views

Show $\int_{0}^{2\pi}e^{int}\mathrm dt=0$ for all $n\neq0$

I am struggling to see how this result holds for non-integer $n$ because $$\int_{0}^{2\pi}e^{int}\mathrm dt=\int_{0}^{2\pi}[\cos(nt)+i\sin(nt)]\mathrm dt$$ and this works out to be $$\frac{\sin(2\pi n)...
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3answers
49 views

How to integrate e$^{int}$ with respect to $t$?

I am wanting to show that $\int_{0}^{2\pi}e^{nit}dt=0$ for $n\neq0$, but I am unsure if it is correct to write $$\int_{0}^{2\pi}e^{int}\mathrm dt=\left.\frac{e^{int}}{ni}\right|_{0}^{2\pi}$$ or $$\...
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0answers
18 views

$S$ is domain(open set and connected) and $\gamma$ is closed curve in $S$ . If $f(z)$ analytic on $S$ and $f'(z)$ continous on $S$

$S$ is domain(open set and connected) and $\gamma$ is closed curve in $S$ . If $f(z)$ analytic on $S$ and $f'(z)$ continous on $S$ . Show that $\int _\gamma \overline{f(z)}f'(z) dz$ is only imajiner ...
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0answers
40 views

Calculating infinite sum using Parseval's theorem

For$\alpha \in \mathbb{R} \backslash \mathbb{Z}$, consider the fnunction $[0,2\pi] \to \mathbb{C} : x \mapsto \frac{\pi}{\sin \pi \alpha} e^{i(\pi - x)\alpha}$, and prove that $\sum_{n=-\infty}^\infty ...
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2answers
67 views

Evaluate $\int_{-\infty}^{\infty}\frac{dx}{(1+x^2)^z}$ where $\Re(z)>\frac{1}{2}$

I am dealing with the following complex integration: $$\int_{-\infty}^{\infty}\frac{dx}{(1+x^2)^z} \quad\text{ where }\quad\Re(z)>\frac{1}{2}$$ But I do not know how exactly to deal with the $z$.
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60 views

Translation property of the Airy transform for complex values

Airy transform is defined as follows: $$ f_a(x)=\int_{-\infty}^{\infty}f(t)\mathrm{Ai}(t-x)dt $$ According to the book by Olivier Vallee and Manuel Soares it has the translation property: $f_a(x+c)$...
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28 views

Residue theorem and a two-dimensional integral: not working?

Consider the following integral: \begin{align} \iint_{\mathbb{R}^2}d t\,dT\, \frac{e^{-i(t-T)}e^{-t^2}e^{-{T}^2}}{(t-T-i\epsilon)^2}\,. \end{align} The $i\epsilon$ prescription simply tells me that if ...
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1answer
63 views

Integrating a complex $e$-function

Could someone explain in slow steps how to integrate the complex function: $$ \int_0^{2\pi} e^{i(\pi-x)2\alpha} dx $$ where $\alpha \in \mathbb{R} \backslash \mathbb{Z}$ is just some constant. I'm ...
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1answer
35 views

Evaluate $\frac{1}{2\pi i}\int_\gamma\frac{f(z)}{(z-z_1)(z-z_2)}-\frac{f(z)}{(z-z_0)^2}$

In Marsden's Complex Analysis, section 2.4, the main theorem is Cauchy's integral formula (C.I.F) and there appears this problem: Let $f$ be analytic inside and on $\gamma: |z-z_0|=R$. Prove that ...
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0answers
43 views

Change of variables integration for all real p>1?

Originally I have the following problem to show it holds for any real $p>1$, $$\int_0^\infty \frac{1}{x^p+1}\, \mathrm{d}x = \frac{\pi}{p}\sin\left(\frac{\pi}{p}\right).$$ However, since there are ...
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1answer
38 views

A puzzle with derivative of delta-functions

I will assume as a given the fact that in terms of complex variables $z,\bar{z}$ the following formula holds (normalization is not essential) $$\partial_{\bar{z}}\frac{1}{z}=\delta(z)$$ Then, by the ...
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23 views

Can we extend the idea of contour independence for complex contour integrals to several complex variables?

That is, given some function $f:\mathbb{C}^n \to \mathbb{C}$ entire/ sufficiently holomorphic, if we have two domains $D,D'$ in $\mathbb{C}^n$ with the same boundary i.e $\delta D = \delta D' $, will ...
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1answer
31 views

1st Extension of Cauchy Integral Formula

I was trying to obtain first extension of Cauchy Integral Formula which is $$ f'(z)= \frac{1}{2\pi i} \int_C \frac{f(s)}{(s-z)^2} ds$$ where $s$'s are points on $C$ contour and $z$ is any point in ...
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3answers
45 views

Computing $\int_{\gamma}\frac{\sin(\pi z)}{(z^2-1)^2} dt$

Compute the integral $I=\int_{\gamma}f(z)dz$, where $$f(z)=\frac{\sin(\pi z)}{(z^2-1)^2}$$ and $\gamma=\{z:|z-1|=1\}$ I thought of using the formula $$f^{(k)}(z_0)=\frac{k!}{2\pi i}\int\limits_{\...
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1answer
48 views

Residue theorem - simple integral around a singularity

Could anyone explain in detail how do you take the integration contours in order to calculate the following integral? $$\int_{y_c-\delta}^{y_c+\delta} \frac{1}{(y-y_c)^2} dy$$ where $\delta$ is very ...
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1answer
40 views

Using Cauchy theorem to compute $\int_{\gamma} z\overline{z}$

Compute $\int_{\gamma} z\overline{z}$ where $\gamma=\{z||z|=1\}$. I thought I could apply Cauchy Theorem and conclude the integral is zero since $\gamma$ is the unit ball hence connected and closed ...
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1answer
29 views

Compute the integral $I=\int_\gamma f(z)\, dz$, where $f(z)=\frac{e^{z^2}}{z^2-6z}$ and $\gamma=\{z:|z-2|=3\}$.

Compute the integral $I=\int_\gamma f(z)\,dz,\ $ where $f(z)=\frac{e^{z^2}}{z^2-6z}$ and $\gamma=\{z:|z-2|=3\}$. I thought of using $$f(z_0)=\frac{1}{2\pi i}\int\frac{f(z_0)}{z-z_0}\,dz$$ Rewriting $...
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1answer
33 views

$\int_{C}{\sin(\bar{z})}\,dz$ where C is the path connecting $-i$ to $i$ along the unit circle in the positive direction

I need help calculating $\int_{C}{\sin(\bar{z})}\,dz$ where C is the path connecting $-i$ to $i$ along the unit circle in the positive direction. I have already noticed that for $|z|=1$, $\bar{z} = \...
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2answers
34 views

Complex Integration - Cauchy's Formula

Wanted to check if I got the right answer/ idea for this question: $$\int_{|z|=1} \frac{\sin(z)}{z}\mathrm dz$$ Attempt: The region of the curve is the unit circle so there is a singularity at the ...
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0answers
17 views

Is there a way to categorise the valleys of a holomorphic function?

For an entire function $f$, the input space $\mathbb{C}$ is split into hills and valleys about the saddle points of $Re(f)$ by the maximum modulus theorem. Visually it is quite obvious whether or not ...
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1answer
20 views

Decomposition of the absolute value of a complex line integral

I just would like someone to help me understand how $$\Bigg\vert{\int^{b}_{a} g(t) dt} \Bigg\vert = e^{-i \theta} \int_{a}^{b} h(t) dt$$
0
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1answer
32 views

Integrating the following Complex Function

A problem I'm working on requires me to evaluate the following complex integral about some closed contour: $$\oint f(z)dz$$ where $$f(z) = \left(V-\frac{Va^2}{z^2}\right)^2$$ and $V$ and $a$ are real ...
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1answer
44 views

Integrals involving multi-valued functions [closed]

Show that $$\int_{0}^{\infty}{\frac{\cos{x}}{x^\alpha}dx}=\mathrm{\Gamma}\left(1-a\right)\sin{\left(\frac{\mathrm{\pi\alpha}}{2}\right)}\ ,\ \ \ \ 0<\alpha<1$$ what would be the contour of ...