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

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32 views

How to calculate this residue which has a pole of order n-r?

So I have this complex integral: $$ \oint \frac{dz}{2\pi}\frac{e^{iz(br-(n-r)a)}}{\left(1-(1-q)e^{-ik_{1}-iza}\right)^{n-r}}$$ b,r,q,a,n are all constants in this context. However I'm not entirely ...
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1answer
32 views

Integration along a keyhole

(H. Priestley complex Analysis Chapter 7 Exercise 9) Suppose $f$ is holomorphic inside and on $\gamma(0,1)$. By integration around the usual keyhole like this one : Integration of $\ln $ around a ...
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1answer
83 views

Verifying the complex integral: $\int_0^{\infty}\frac{\cos{ax}}{1+x^4}dx$

Verifying the integral: $$\int_0^{\infty}\dfrac{\cos{ax}}{1+x^4}dx$$ I started considering: $$\cos{x}=\dfrac{e^{ix}+e^{-ix}}{2}\implies \cos{ax}=\dfrac{e^{iax}+e^{-iax}}{2}?$$ So: ...
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1answer
50 views

Complex integral with Fourier

So, I've been scratching my head over this for the whole day. I'm trying to solve the following integral $$\int^\infty_{-\infty} \frac{e^{i \alpha (X-\xi)}}{\sqrt{\alpha^2+ \beta^2 }} \, d\alpha$$ ...
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2answers
44 views

Solving $\frac{df}{dt}=\frac{i\cdot f}{|f|}$ where $f: \mathbb{R^+} \mapsto \mathbb{C}$

I've never seen a complex DE before, so this is uncharted territory for me. But it's separable so it's easy to turn it into an integral: $$f(t) = \int_0^t\frac{i \cdot f}{|f|} dt$$ Can this be solved? ...
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0answers
36 views

Show the integration with a complex variable

I want to show that there exists inverse Laplace transform, $f(t)$ of the function $F(\lambda)$. In other word, given $F(\lambda)$, existence of function $f(t)$ such that $$ ...
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1answer
59 views

$\int_0^{\infty} \exp(i(t-\alpha)^2) dt$

It's fairly straight forward to show that $$ \int_0^{\infty} \exp(it^2) dt = \frac{\sqrt{\pi}}{2}\exp\left(i\frac{\pi}{4}\right) $$ via complex contour integration over a contour shaped like a piece ...
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1answer
53 views

Integral with complex variable

I want to compute $$ \int_{-\infty}^{\infty} \frac{1}{\sqrt{x+yi +2}} dy $$ where $i$ is the imaginary number. How to compute this integral??
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0answers
31 views

Moving limit inside a contour integral

I'm trying to compute this integral as part of a larger problem I'm working on. I'm trying to solve the integral $\int_0^\infty \frac{\sin(x)}{x}dx$ and to do it I'm using the method where you ...
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1answer
32 views

Complex integral using Residue Theorem with a regularised pole

I need to prove the following integral computation by applying the residue theorem: $$\int_{-\infty}^{+\infty}ds \ e^{-i\Omega ...
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1answer
33 views

Complex integration: normally on a closed contour?

I have been studying complex integration for a few months now, and it seems my textbook mostly considers integration on closed contours. Is there no interest in integration on non-closed contours ?
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3answers
55 views

Complex integration with trigonometric and logarithm

Show that $\int_0^{2\pi}\log\sin^22\theta dx=4\int_0^\pi\log\sin \theta d\theta=-4\pi \log2$ I did $$\int_0^{2\pi}\log\sin^22\theta d\theta=4\int_0^{\frac{\pi}{4}}\log\sin^22\theta d\theta$$ ...
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2answers
163 views

Complex Integration with trignometric function

Verify that $\int_0^{\frac{\pi}{2}}\frac{d\theta}{a+\sin^2\theta}=\frac{\pi}{2[(a(a+1)]^\frac{1}{2}}$ I know that $\sin\theta=\frac{e^{i\theta}-e^{-i\theta}}{2}$ then I did ...
2
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2answers
40 views

Improper integral and residues

Evaluate $\int_0^\infty \frac{dx}{x^4+1}$ By the residue theorem $$\int_{-R}^Rf(x)dx+\int_{C_R}dz=2\pi i\sum Res(f,z_i)$$ but I have problems to evaluate it because $$z^4+1=0\Rightarrow ...
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1answer
62 views

Poles on the curve

Say I have this integral: $$\oint_\gamma f(z)\,{\rm d}z,$$and $f$ has a pole on $\gamma$. I understand that we "cut around" the pole with an arc of radius $\epsilon$ and then make $\epsilon \to 0$. ...
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2answers
33 views

Bounding a complex integral over a square

I'm solving the following exercise: Use the estimate lemma to prove that $$\left|\oint_\gamma \frac{z-2}{z-3}\,{\rm d}z\right| \leq 4\sqrt{10},$$where $\gamma$ is the square with vertices $\pm 1 ...
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1answer
30 views

Evaluate the improper integral with residues

Evaluate $\displaystyle\int_0^\infty\frac{dx}{x^2+1}$ I have that $z_0=i$ and $z_1=-i$ are singularity points but just $z_0=i$ is in the upper plane then ...
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1answer
18 views

Improper integrals and residues

I'm already read Conway, Churchill and Marsden but I'm still with doubts when it comes to improper integrals. Where come from this relation ...
2
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1answer
59 views

Calculation of an improper integral in the context of complex functions [duplicate]

I am facing the following improper integral: $$\int_0^\infty \frac{x^5\sin x}{(1+x^2)^3}dx.$$ Clearly the expression under the integral is a meromorphic function analytic on the nonnegative part of ...
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0answers
61 views

Using residue theorem find $\int_{\theta-\pi}^{\theta-2\pi}1/(r^2+1-2r\cos u)$

Find the value of $I$, using residue theorem. $$ I= \int_{\theta-\pi}^{\theta-2\pi} \frac{du}{r^2+1-2r\cos(u)} - \int_{\theta}^{\theta-\pi} \frac{du}{r^2+1-2r\cos(u)} $$ with $r<1$ and ...
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0answers
46 views

Integration using residue theorem

Can you find, using residue theorem, ($\epsilon >0$), the value of this integral ($I$)? \begin{equation} I=\lim_{\epsilon->0^{+}} \int_{-\infty}^{\infty} \frac{dw}{w+i\epsilon} \end{equation} ...
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1answer
56 views

How to calculate $\int_{-\infty}^{\infty} x^2 \cos(ax)e^{-x^2}dx$ using Cauchy's theorem

I want to calculate the integral $$\int_{-\infty}^{\infty} x^2 \cos(ax)e^{-x^2}dx$$ using complex analysis. I have a hint to look at the rectangle $(-R,0), (R,0), (R,h), (-R,h)$ for a certain ...
2
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3answers
58 views

Why $\lim_{R\to\infty}\int_{0}^{\pi}\sin(R^{2}e^{2i\theta})iRe^{i\theta}\:\mathrm{d}\theta = -\sqrt{\frac{\pi}{2}}$

This is a short question, but I'm simply not sure where to start, I know by Jordan's Lemma that the integral is not $0$, but I only know the below result due to Mathematica. ...
2
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2answers
217 views

When is this integral convergent?

Let $a \in \mathbb{C}$. Consider the integral $$\int_{-\infty}^{+\infty} \frac{e^{-ax}}{1 + e^x} dx,$$ for which values of $a$ is this convergent? Is it right to say that $a$ has to be purely ...
2
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1answer
25 views

Complex contour integral of fraction of polynomials

Let $n \in \mathbb{N}_0$ and set $p(z) = z^n + a_1 z^{n-1} + \cdots$ and $q(z) = z^{n+1} + b_1 z^{n} + \cdots$ to be two monic complex polynomials with no common zeros. I want to prove that ...
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1answer
26 views

Polar form of Taylor's theorem for complex analysis

(H.Priestley Exercise 5.7) Let $f \in H(D(0,R))$ and $f=\sum_{n=0}^{\infty} c_{n} z^n$ Using the integral formula for $c_n$ and the fact that $\int_\gamma f(z)z^{n-1}dz=0 \quad\forall n\ge1$ Show ...
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0answers
42 views

Integrate complex function over $\mathbb{C}^2$

I have a question in mind and I would appreciate your help. Usually in complex analysis we consider integrals of the form $\int_\gamma f(z) dz$ where $\gamma $ is a contour and ...
2
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2answers
43 views

Results for values of the residues in the Residue Theorem

If the sum of the residue is $0$, what can I conclude: that the value of the integral is $0$ or that the integral diverges? If the residue tends to $\infty$, should I conclude that the integral ...
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5answers
269 views

Evaluate the complex integral of function

Use the residue theorem to evaluate $\int_\gamma \frac{z^5}{1-z^3}dz$ where $\gamma$ is the circle $|z|=2$. I have that $z_0=1$ is a singularity point and taking $g(z)=z^5$ and $h(z)=1-z^3$ and ...
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1answer
56 views

Integrate $\int \log(1+2m\cos x+m^2) dx $

How do I integrate $\int \log(1+2m\cos x+m^2) dx $ ? I tried 2 things. First, I tried complex numbers. Putting $\cos x = \frac{e^{ix}+e^{-ix}}{2} $ which led me to $\int \log((me^{ix} ...
3
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2answers
73 views

How to integrate $e^{-\cos(\theta)}\cos(\theta + \sin(\theta))$

I am struggling to find a way to evaluate the following real integral: $$\int_{0}^{2\pi}e^{-\cos(\theta)}\cos(\theta + \sin(\theta))\:\mathrm{d}\theta$$ The exercise started by asking me to ...
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2answers
55 views

Find the poles of $f(z)=\frac 1{1+z^w}$ for $w \gt 1$

I am trying to use contour integration on the following integrand between $0$ and $\infty$, however I am not sure how to go about finding the poles for it: $$f(z)=\frac 1{1+z^w},w \in \mathbb Z:w \gt ...
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1answer
38 views

Solve this double integration

$$\int_0^{\pi/2}\int_x^{\pi/2}\frac{\cos y}{y} \, dy \, dx=\text{ ?}$$ I have tried this question but don't have any idea how to integrate $\dfrac{\cos y}{y}$. I have studied math up to 12th. Many ...
3
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2answers
35 views

Contour Integral of $\frac{1}{z} dz$ for any contour in the right half-plane from $z=-3i$ to $z=3i$

Contour Integral of $\frac{1}{z} dz$ for any contour in the right half-plane from $z=-3i$ to $z=3i$. I've seen some examples where I can just take the definite integral of $\frac{1}{z} dz$ from $-3i$ ...
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1answer
41 views

Contour Integral of $e^z dz$ from $z=1$ to $z=-1$

Evaluate the contour integral of $e^z dz$ along the upper half of the circle absolute value of $z=1$, from $z=1$ to $z=-1$. I did integral of $e^z dz$ from $z=1$ to $z=-1$ and got $e-e^{-1}$. But the ...
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2answers
163 views

A (basic?) contour integration problem

I am trying to prove the following using complex analysis: $$\sum_{n=-\infty}^{\infty}\frac{(-1)^{n}}{a^{2}+n^{2}}=\frac{\pi}{a\sinh(a\pi)}$$ I am told to use the following function: ...
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0answers
33 views

About the support of an equivalence class of chains

Let $\Omega$ be a non empty open set of $\mathbb{C}$. Let $\mathscr{C}(\Omega)$ be the set of continuous curves in $\Omega$. Let $H(\Omega)$ be the set of holomorphic functions defined on $\Omega$. ...
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3answers
49 views

Complex Integration, residues

Evaluate the following integrals by the method of residues i)$\int_0^{\infty}\frac{\cos x}{x^2+a^2}dx$, a real ii)$\int_0^\infty \frac{x^\frac{1}{3}}{1+x^2}dx$ I'm a little lost to ...
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1answer
34 views

Complex integral computation with $\sinh$

I need to prove the following integral computation by applying the residue theorem: $$\int_{-\infty}^{+\infty}ds\frac{e^{-i\Omega ...
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1answer
47 views

Question regarding Cauchy Integral formula

I have a question regarding Cauchy Integral formula, I was given an assignment questions, and my professor uploaded a solution and i do not understand how he reached to an answer with his method. Can ...
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1answer
43 views

Complex Number question [Cauchy Integral/Series]

I'm going through the practice finals that my professor uploaded on his site, and I came across this question, and I have absolutely no one clue how to approach it and never seen anything like this on ...
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2answers
28 views

Residues theorem, calculate the integrates

Find the residue of $f(z)=\frac{(z-1)^3}{z(z+2)^3}$ at $z=\infty$ In my refference they say that $Res(f;\infty)=-Res(\frac{1}{z^2}F(z);0)$ where $F(z)=f(\frac{1}{z})$ $$-lim_{z\rightarrow ...
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1answer
38 views

definite integral of form G(cos(x), sin(x)) by complex integration

Given: the following integral: $$ \int_0^{2\pi} \frac{\mathrm{sin}(3x)}{5-3\mathrm{cos}(x)}\,\mathrm{d}x = 0 $$ Prove it by using complex integration and the residue theorem. But I do something wrong ...
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1answer
50 views

Residues theorem, and integrate

Evaluate $\int_\gamma\frac{z}{z^2+2z+5}dz$ where $\gamma$ is the unit circle I did but I don't know if it's right $$z^2+2z+5=0\Leftrightarrow z=-1\pm 2i$$ this I have that $z_1=-1+2i$ and ...
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1answer
45 views

Proving Cauchy's integral formula

How do I prove Cauchy's integral formula? Namely: Let $D$ be a simple, connected domain in $\mathbb C$ and $C$ be a simple, closed, anti-clockwise oriented curve contained in $D$. Let $z_0$ ...
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2answers
59 views

Fundamental theorem of calculus and complex integration

I am teaching myself complex integration, and unfortunately my text book has left me confused as to when I can apply the Fundamental theorem of calculus for complex integration. Consider the ...
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3answers
57 views

Cauchy's Integral Question Complex Number

I have a question and I'm kind of stuck, I was wondering if you were able to help me move forward. The question is, Use Cauchy's integral formula to evaluate, $$ \int_{|z| = 1}\frac{e^{2z}}{z^2}dz ...
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1answer
45 views

Solving an Integral using Cauchy's integral theorem

I have got a question about the calculation of two specific integrals: \begin{align*} &\int_C z^n e^z \, dz \quad n\in\mathbb{N}_0 \\ &\int_{\vert z \vert = 1} z^{-n} e^z \, dz \quad ...
3
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2answers
142 views

Residues and poles proof

Let the degree of the polynomials $P(z)=a_0+a_1z+a_2z^2+\cdots+a_nz^n$ $a_n\neq0$ and $Q(z)=b_0+b_1z+b_2z^2+\cdots+b_mz^m$ $b_m\neq 0$ be such that $m\geq n+2$. Show that if all the zeros of ...
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2answers
38 views

Complex analysis, residues and integrate

Let $C$ denote the circle $|z|=1$ oriented counterclockwise. Show that i)$\int_Cz^ne^{\frac{1}{z}}dz=\frac{2\pi i}{(n+1)!}$ for $n=0,1,2$ ii)$\int_C e^{z+\frac{1}{z}}dz=2\pi ...