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

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2
votes
1answer
47 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
171 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
55 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
48 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
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 ...
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
44 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
52 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
46 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
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 ...
3
votes
3answers
59 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
48 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
143 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
40 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
29 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
28 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
338 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
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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
81 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
68 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
80 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
87 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
34 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
43 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
30 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 ...
3
votes
2answers
37 views

How to use an ML estimate to show the solution to an integral

I have a question I needed to show that $$\lim_{R\to\infty} \int_{C_R} \frac {z^2+4z+7}{(z^2+4)(z^2+2z+2)} dz=0$$ For $C_R$ the circle with radius R, center z=0 and positively oriented. Which I have ...
1
vote
2answers
48 views

Evaluate the following improper integral.

$$ \int^{+\infty}_{-\infty} \frac{x\sin 4x}{x^2-4x+8}dx \, $$ My Thoughts: I know that I should start by changing the integral to: $$ \int^{+\infty}_{-\infty} ...
1
vote
1answer
35 views

Complex Integral

I am stuck computing the following complex integral $$\int_{|z| = 1}\frac{z^2}{4e^z -z}dz$$ I do not even know if the integrating function has a pole and then using residue calculus. Using the ...
1
vote
1answer
69 views

Help With Bromwich Inversion Formula Proof

To prove(copied from handwritten notes so possibly wrong): Bromwich Inversion Formula. Fix $x_0∈ℝ $. If $F$ is complex analytic on $\{z:\Re z > x_0\} $ and for every $x>x_0$, $y↦ F(x + iy )$ ...
0
votes
1answer
37 views

Improper integral (using methods in complex variables) [closed]

Let $0<a<1$. Evaluate the integral $$\int_0^\infty \frac{x^{a-1}}{1+x} dx.$$
2
votes
1answer
42 views

Complex integration on circle

Calculate the integral of $g(z)$ along the closed path $|z-i|=2$ in the positive direction when i)$g(z)=\frac{1}{z^2+4}$ ii)$g(z)=\frac{1}{(z^2+4)^2}$ First I checked the described area ...
0
votes
1answer
26 views

Complex integration and theorems

If $C$ is a closed path oriented in the positive direction and $$g(z_0)=\int_C \frac{z^3+2z}{(z-z_0)^3}$$ show that $g(z_0)=6\pi iz_0$ when $z_0$ is in interior of $C$ and $g(z_0)=0$ when $z_0$ is out ...
0
votes
0answers
22 views

when is $\int_{t=-x}^{x} e^{iat}\operatorname{sinc}(t) \, dt = 0$, with $x \in \mathbb{R}$?

For what value of $x \in \mathbb{R}$ is $\int_{t=-x}^{x} e^{iat}\operatorname{sinc}(t)\,dt = 0$, where $a$ is some constant?
1
vote
1answer
41 views

Why is $\left| \int_{|z|=1} \frac{f(z)}{z}dz\right| \leq \int_{|z|=1} \frac{|f(z)|}{|z|}|dz|$?

Why is $$\left| \int_{|z|=1} \frac{f(z)}{z}dz\right| \leq \int_{|z|=1} \frac{|f(z)|}{|z|}|dz|$$ and not $$\left| \int_{|z|=1} \frac{f(z)}{z}dz\right| = \int_{|z|=1} \frac{|f(z)|}{|z|}|dz|?$$
0
votes
1answer
67 views

Inverse $z$ transform - contour integration

Here is my task: Find inverse $z$ transform of $$X(z)=\frac{1}{2-3z}$$if $$|z|>\frac{2}{3}$$ using definition formula. I found that $$x(n)=\dfrac{1}{3}\left (\dfrac{2}{3}\right ...
0
votes
0answers
35 views

Lebesgue integration in $\mathbb{C}$

I'm confused as to how we are supposed to integrate $$\frac{1}{\pi}\int_U\left[\frac{d}{dz}\left( \frac{z-\alpha}{1-\bar\alpha z }\right)\right]^2 \, dm$$ where $U$ is the unit disc, ...
1
vote
1answer
34 views

Cauchy integral formula for rational function, help with step

I have $P(\lambda) = (i\lambda)^m + O(\lambda^{m-1})$ a polynomial in $\lambda$, and $\Gamma$ a contour counterclockwise around the roots of $P$. I need to prove: ...
2
votes
1answer
34 views

Integrals with complex functions: integration by parts and conjugate

I am working with integrals of complex functions. I assume all terms are well-defined. If $u=u(x,t):\mathbb{R}^n\times\mathbb{R}_+ \to \mathbb{R}$ (a real function), I have \begin{equation} ...
1
vote
1answer
58 views

one complex variables (integration)

how to prove $\int_{C_R}\frac{\log^3(z)}{(1+z^2)^2}\,dz$ goes to $0$ as $R$ goes to $\infty$, with $C_R=Re^{it}$ for $0<t<\pi$, and $R>0$
1
vote
2answers
43 views

How to calculate $\int_{C(0,1)}\frac{\sin z}{z^4}dz$

$\displaystyle\int_{C(0,1)}\frac{\sin z}{z^4}\:\mathrm{d}z$, where $C(0,1)$ is the circle around $0$ with radius $1$ $$\displaystyle\int_{C(0,1)}\frac{\sin ...
0
votes
1answer
56 views

How can I solve this integral with complex number?

$n$ here is a complex number such that $n=n_r+in_i$ How can I solve this integral? $$\int_{0}^{\infty}\frac{x^4}{|x^2-n^2|^2} d x=? $$