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

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2
votes
1answer
123 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 ...
1
vote
0answers
63 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 ...
0
votes
0answers
54 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} ...
2
votes
1answer
63 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
votes
3answers
61 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
votes
2answers
220 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 ...
3
votes
1answer
29 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 ...
1
vote
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 ...
4
votes
0answers
46 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
votes
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 ...
3
votes
5answers
289 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 ...
1
vote
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
votes
2answers
86 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 ...
2
votes
2answers
60 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 ...
1
vote
1answer
43 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
votes
2answers
37 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 $-3i$ ...
2
votes
1answer
51 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 ...
5
votes
2answers
180 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: ...
0
votes
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 $\Omega$. ...
1
vote
3answers
59 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 ...
1
vote
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 ...
1
vote
1answer
50 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 ...
0
votes
1answer
46 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 ...
0
votes
2answers
30 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 ...
1
vote
2answers
45 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 ...
2
votes
1answer
57 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 ...
1
vote
1answer
50 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 $z_0$ ...
4
votes
2answers
60 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 ...
3
votes
3answers
63 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 ...
1
vote
1answer
50 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
votes
2answers
146 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 ...
1
vote
2answers
41 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 ...
1
vote
3answers
30 views

Evaluate the integral along a contour containing 2 interior points by using Cauchy's Integral Formula

Evaluate the integral $$\int_{C}\frac{z^2}{z^2+9}dz$$ where C is the circle $|z|=4$ I know that if f is analytic in simply connected domain $D$, $C$ a simple closed positively oriented contour that ...
0
votes
0answers
29 views

Help with $\int _{R_0<|z|<R_1}\frac{1}{z} dz$.

Consider the integral in $\mathbb{C}\simeq \mathbb{R}^2$ $$ \int_{R_0<|z|<R_1} \frac{1}{z}\; dx_1 dx_2 $$ where $0<R_0<R_1$ and $z=x_1+i x_2$ and $|z|=(x_1^2+x_2^2)^{\frac{1}{2}}$. So ...
0
votes
2answers
29 views

Region of Convergence of power series

The power series $\sum_{n=0}^\infty 2^{-n} z^{2n} $ converges if a)$|z|\le 2$ b)$|z|\lt 2$ c)$|z|\le\sqrt2$ d)$|z|\lt\sqrt 2$ I tried this problem,my answer is d).I am not sure whether it is correct ...
1
vote
1answer
44 views

Best way to calculate residues

Basically, what is the best method to calculate residues, specifically, something like this: \begin{equation*} f(z)=\frac{1+z}{1-\cos(z)}. \end{equation*} For simple poles, I can just use L'Hopital ...
0
votes
0answers
23 views

Integrating $\operatorname{Log}(z+2)$ along the unit circle [duplicate]

For the function $f(z) = \operatorname{Log}(z + 2)$, where we choose the principal branch of logarithm (namely, $−\pi < \operatorname{Arg}(z) < \pi$), and the contour $C := \{z \in ...
2
votes
4answers
452 views

Cauchy integral formula

Can someone please help me answer this question as I cannot seem to get to the answer. Please note that the Cauchy integral formula must be used in order to solve it. Many thanks in advance! ...
1
vote
0answers
30 views

calculate complex integral $\int_{0}^ {\pi} \frac{(x \sin x)dx }{1-2a \cos x+a^2}, a>0$

I don't know how to calculate this complex integral: $$\int_{0}^ {\pi} \frac{(x \sin x)dx }{1-2a \cos x+a^2}, a>0$$
2
votes
2answers
85 views

Gaussian integral with a shift in the complex plane

$$ \int_{-\infty}^\infty e^{-(x+ia)^2} \text{d}x $$ where $a\in \mathbb{R}$. I don't know where to start but have reasons to believe the answer is $\sqrt{\pi}$. Namely $\int_{-\infty}^\infty ...
3
votes
2answers
71 views

$\int \frac{\exp (z)(\sin(3z)}{(z^2-2)(z^2)} dz$ on $|z|=1$

So I need to calculate \begin{equation*} \int \frac{\exp(z) \sin(3z)}{(z^2-2)z^2} \, dz~\text{on}~|z|=1. \end{equation*} So I have found the singularities and residues and observed that the ...
0
votes
0answers
10 views

Finding the limits when integrating a complex number

Evaluate $\int_c f(z) dz$ from $z(0,0)$ to $z=2+4i$ where $f(z)=x^2 -iy^2$ I know how to work this out and I know the answer is $24+\frac{8}{5}i$ However I do not understand why the limits for x are ...
3
votes
2answers
78 views

The value of $\sum_{n=-\infty}^{n=\infty}\frac{1}{n^2-z^2}$ on $\mathbb{C}\setminus\mathbb{Z}$

Show, for $z\not\in\mathbb{Z}$, that $$\sum_{n=-\infty}^{n=\infty}\frac{1}{n^2-z^2} = \frac{-\pi}{z\tan(\pi z)}$$ Hint: You may assume that there exists $C$ such that $|\pi\cot(\pi w)|\leq ...
2
votes
2answers
84 views

Apply the Cauchy-Goursat theorem to show that $\int_C \operatorname{Log}(z+2)\, dz=0$ on a unit circle.

Cauchy-Goursat theorem. If a function $f$ is analytic at all points interior to and on a simple closed contour $C$, then $$\int_C f(z) \,dz=0.$$ This is a problem from Churchill's Complex Variables. ...
0
votes
1answer
21 views

How is the integral of $\frac{f(\zeta)-f(z)}{\zeta - z}$ over $C_{\epsilon}$ $0$?

I am trying to understand a proof of this theorem: Suppose $f$ is holomorphic in open set that contains the closure of a disk D. If C denotes the boundary circle of this disk with positive ...
2
votes
1answer
90 views

Evaluate integral over complex path numerically to show that $C_\infty$ is equivalent to $-I$

I would like to evaluate $$C_\infty = \int_{R = -a}^{R = a} H_0^{(1)}(z) e^{-izt} dz $$ where $H_0^{(1)}(z)$ is the Hankel function of the first kind, $a \rightarrow \infty$, and $$ z = R - i ...
0
votes
1answer
35 views

Complex analysis integration method

How do you solve the integral $$\int^\infty_{-\infty}\frac{cos z}{z^2+9}dz$$ If I first find the roots, I get $z=-3i$ and $z=3i$ I also know that $$\int^\infty_{-\infty} f(x) dx=2 \pi i \sum^m_{k=1} ...
1
vote
0answers
45 views

What is the meaning of this integral?

Does anyone know the meaning of this type of integral? $\displaystyle{\int f(z) \,\overline {dz}}$. I think this means $\displaystyle{\int u\,dx + v\,dy+i\int v\,dx -u\,dy}$ where $f=u + iv$
5
votes
2answers
73 views

Evaluate $\int_{|C|=2} \frac{dz}{z^2 + 2z + 2}$ using Cauchy-Goursat

I've split the integral around $z_1 = 1 - i$ and $z_2 = 1+ i$ using the contours $C_1$ and $C_2$: $ \int_{|C|=2} g(z) dz = \int_{C_1} g(z) dz + \int_{C_2} g(z) dz$ In this case, $g(z)$ for $C_1$ ...
1
vote
0answers
35 views

Using Multiple Branch Cuts in a Contour Integral

I have the integral $$I=\int_0^{\infty}\!\frac{x^{1/2}}{\left(x^3+a^3\right)^{1/2}\left(2x^3+a^3\right)^{1/6}}\,\mathrm dx$$ which I am trying to integrate using complex integration. I know that ...