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1answer
28 views

complex integration-how to solve the given problem

how do we calculate the value $\frac{1}{2\pi i}\int\frac{\sum_{n=0}^{15}z^n}{(z-i)^3}dz$ in $C$:|z-i|=2 ? the answer for this is 1+15i.. how to get it? can someone please explain?
1
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1answer
28 views

Estimate of complex integral

Prove that $$ \left|\int_c (2-\frac{e^z}{z-\log 2}) dz \right| <\frac{2}{3} $$ when C is the part of circle $\left| \frac{z}{\pi} -1 \right|^2 =2$ where $Re(z)\geq 0$. ($\log$ means natural ...
2
votes
1answer
38 views

Can use residue theorem for this integral

I need to compute $$I=\int_C \dfrac{e^{\sqrt{1+u}}\cdot\sqrt[4]{1+u}}{\sqrt{u}} \,\mathrm {d}u$$ where $C$ is the unit circle. I am confused about whether I can use the residue theorem to compute it? ...
1
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2answers
43 views

Contour expression explanation

$$ \begin{align} & \int_\gamma zdz \end{align} $$ $$\\ \gamma = [e,1]+[1,-1+\sqrt3] $$ contour $\gamma$ is defined as above and I can't understand it. Could someone please explain it to me and ...
3
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1answer
38 views

Is this Integral calculation correct?

Can someone confirm if my solution is right or if I have done something that is not permitted $$ \begin{align} & \int_\gamma e^{\pi z}=\int_\gamma \left( \frac{ e^{\pi z}}{\pi}\right)' \, dz ...
0
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1answer
39 views

Image of a parametrised curve and its geometrical meaning

$\gamma_1(t)= t^2 + i\, t^4 , t\in [ -1, 1]$ which is the Image of this curve and / or what is the geometrical description of the set of points $ z : z=\gamma_1(t)$ I am helping a friend study ...
0
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1answer
42 views

How to find the area where $\frac{1}{z^2-4}$, $z \in \mathbb{C}$ is holomorphic?

Suppose that you are given a problem of finding the following complex integral: $$\int_\tau \frac{1}{z^2-4} dz$$ where $\tau = \{z \in \mathbb{C}: |z|=4 \}$. My question is (in the context of this ...
1
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2answers
41 views

Contour integral of complex logarithm

Evaluate $$\int_C Log(z) dz$$ where $Log(z)$ is the principle branch of the complex logarithm (Arg$(z)\in(-\pi,\pi)$) and $C$ is the contour given by the horizontal line connecting $z=i$ to $z=i+1$, ...
2
votes
1answer
124 views

How do you integrate Gaussian integral with contour integration method?

How do you integrate $$\int^{\infty}_{-\infty} e^{-x^2} dx$$ with contour integration method? I do not even know how to setup the problem.
2
votes
1answer
85 views

Complex integral involving logarithm

I've been working on this integral for quite a while and I think I've been able to progress but now I'm stuck. So I have to prove that $$\int_C f(z)\ dz =\int_C\frac{2z}{(1+z^2)\log(2+z^2)}dz =\pi ...
2
votes
1answer
39 views

can't follow the steps of a specific complex integration

Hi: I already asked this question on the complex analysis tag but nobody answered it so then I found this complex-integration tag and was hoping that someone might be able to answer it here. It is ...
0
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0answers
62 views

What does this complex contour integral represent?

How would one evaluate the following complex contour integral in "Integral and Series Representations of Riemann’s Zeta function, Dirichelet’s Eta Function and a Medley of Related Results." The ...
1
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4answers
85 views

Help with Complex integration

I have to calculate the following integral $$\int_{-\infty}^{\infty} \frac{\cos(x)}{e^x+e^{-x}} dx$$ Anyone can give me an idea about what complex function or what path I should choose to calculate ...
1
vote
1answer
37 views

Mean value theorem for harmonic

In Problems and Solution in Mathematics by Ta-Tsien, exercise 5123, the mean value theorem is used as: \begin{equation} \text{log} |F(0)| = \frac{1}{2 \pi} \int_0^{2\pi} \text{log}|F(re^{i\theta})| ...
0
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2answers
29 views

Cauchy integral formula or something else?

I need to determine the function $\;f(z)$ if $$f''(z)=\oint_{\partial C_1(0)}{\sin^2\xi \over\left(\xi-z\right)^3}\mathbb{d}\xi$$ with $C_1(0):\left|z\right|<1$ positive. Additionally ...
3
votes
2answers
73 views

How to calculate $\int_{\partial B_2(0)}\frac{2z^2+7z+11}{z^3+4z^2-z-4}\;dz$?

I want to calculate $$\displaystyle\int_{\partial B_2(0)}\underbrace{\frac{2z^2+7z+11}{z^3+4z^2-z-4}}_{=:f(z)}\;dz\tag{0}$$ Partial fraction decomposition yields ...
3
votes
1answer
67 views

If $f$ has pole of order $m$, then $\text{res}\left(f,z_0\right)=\lim_{z\to z_0}\frac{1}{(m-1)!}\left\{(z-z_0)^mf(z)\right\}^{(m-1)}$

Statement: Let $$f(z):=\sum_{k=-\infty}^\infty a_kz^k$$ have a pole of order $m$ at $z_0$. Then $$\text{res}\left(f,z_0\right)=\lim_{z\to z_0}\frac{1}{(m-1)!}\left\{(z-z_0)^mf(z)\right\}^{(m-1)}$$ ...
1
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2answers
39 views

Show that $\sum_{k=1}^{n}a_ke^{2 \pi ikx}$ has a root in $\left[ 0,1 \right]$

Let $a_1, \dots ,a_n$ be arbitrary complex numbers. Define: $f \left( x \right)=\sum_{k=1}^{n}a_ke^{2 \pi ikx}$. I wish to show that there exists an $x\in\left[ 0,1 \right]$ s.t. $f \left( x ...
6
votes
2answers
91 views

The integral $\int_0^\infty \dfrac{x \sin(x)}{x^2+1} dx$

How to calculate: $$\int_0^\infty \frac{x \sin(x)}{x^2+1} dx$$ I thought I should find the integral on the path $[-R,R] \cup \{Re^{i \phi} : 0 \leq \phi \leq \pi\}$. I can easily take the residue in ...
0
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0answers
40 views

Solving an ordinary differential equation with two functions

Please I am trying to solve this differential equation but I do not seem to get the correct answer. Please help thank you. My equation is; $\frac{dc(x)}{dx} = 1+ b\frac{y(x)-y(x-1)}{y(x-1)}$ where b ...
1
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2answers
73 views

Finding $\int_0^{2\pi}\frac{\sin t + 4}{\cos t + \frac{5}3} dt$

I'd like to ask something about the following integral: $$ \int_0^{2\pi}\frac{\sin t + 4}{\cos t + \frac{5}3} dt $$ I rewrote and took another variable. $$ ...
4
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1answer
73 views

Complex Integration: theorem of Residues

I am trying to prove a theorem that is doing my head in a bit. I have tried to simplify the problem as much as possible and leave out the details, even though it might look a bit too big. The ...
0
votes
1answer
33 views

Complex exponent integral - prove $\int_a^b{e^{\lambda x} \text{dx}}=\frac{1}{\lambda}\left(e^{\lambda b}-e^{\lambda a}\right) $

How to prove the exponent integration rule: $$\int_a^b{e^{\lambda x} \text{dx}}=\frac{1}{\lambda}\left(e^{\lambda b}-e^{\lambda a}\right) $$ In the complex version of it - that is, when $\lambda \neq ...
4
votes
2answers
144 views

Complex integration, any ideas?

I need to solve the following integral of a function analytic on an open environmemt of the unit circle, but i dont know how to get that answers $$\frac{1}{2\pi ...
2
votes
3answers
61 views

Help with equality of complex integrals

I need to prove this equality of integrals...but i dont know how to begin, so if anyone can give an idea... Let f a continuous function on $\overline{D}=\{z : |z|\leq 1\}$. Then: ...
3
votes
1answer
171 views

How to find $\int_0^{\pi}\frac{\sin n\theta}{\cos\theta-\cos\alpha}d\theta$

I was doing some work in physics and I came up with a definite integral. I tried everything I could but couldn't solve the integral. The integral is $$ \int_0^\pi {\sin\left(n\theta\right)\over ...
1
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1answer
61 views

Help with improper integral [duplicate]

I need help solving this integral: $$\int_0 ^\infty \frac{\sin(x)}{x} dx$$ I have a help that says that try to calculate the integral of $$\frac{e^{iz}}{z}$$ for a "proper path"... but I don't know ...
0
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1answer
39 views

understanding a particular step in proof of cauchy's theorem for triangles

"Hi: I am reading "complex variables" by Ash and Novinger and they prove "cauchy's theorem for triangles early in the book". Unfortunately, there's a step in their proof that I don't follow. ...
1
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1answer
25 views

Complex integral with different contours

If I have a complex integral to solve using the Cauchy Integral formula with the same point but with different contours, in which the point used is inside both contours, is the result the same? Say ...
1
vote
1answer
26 views

Parametric form of curve $\vert z+i\vert = 1$

I need to integrate a complex function through the curve $\vert z+i\vert = 1$. As far as I know I need the parametric form of this curve. I know that when I have $\vert z\vert = 1$, the parametric ...
1
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1answer
37 views

$\int _{C^{+}(0,3)} \frac {dz}{2-\sin z}$

$$\int _{C^{+}(0,3)} \frac {dz}{2-\sin z},$$ $z$ is complex. I have no idea how to solve $2-\sin z$. I will be really grateful for any help
1
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0answers
22 views

Explicit computation of integrals along loops

I am learning a bit about integration on one-dimensional complex tori. It's exciting stuff, but I have some trouble making things a bit explicit. Let's consider the elliptic curve $E = \mathbb ...
1
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1answer
56 views

Complex integral inequality

Statement Let $\gamma$ be the curve that goes through the upper unit circle counterclockwise (positive orientation). Prove that $$\left|\int_{\gamma} \dfrac{\sin(z)}{z^2}dz\right|\leq ...
1
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0answers
26 views

Find natural number to satisfy inequality given with integral

Find the number $N\in \mathbb{N} $ for which the following inequality is true, if $\sigma(t)=|z|=1 $, with $0\leq t\leq \pi/2$ and $\left | \int_{1}^{i} e^{z^{-1}} \right | \leq N \frac{\pi}{2}$. ...
1
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0answers
55 views

Calculating this integral: $\int_{\partial_{10}(1-2i)}\frac{39!\require{cancel}\cancel{z^z}z^z}{(z+4)^{42}}dz$

Please take a look at $$\int_{\partial B_{10}(1-2i)}\frac{39!\require{cancel}\cancel{z^z}ze^z}{(z+4)^{42}}dz$$ At a first glance, this looks like a case for Cauchy's differentiation formula, which ...
1
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2answers
51 views

The value of the itegral $\int_{\gamma} \dfrac{dz}{z-a}$ is a multiple of $2\pi i$

I am reading Ahlfors' proof of the lemma: Lemma If the piecewise differentiable closed curve $\gamma$ does not pass through the point $a$, then the value of the integral $$\int_{\gamma} ...
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0answers
22 views

Evaluate the bound of the following complex integral

Let $q$ be a complex analytic function such that $\left|q\right|\neq0$ everywhere in the domain, except at the boundaries ($\left|q\right|\rightarrow0$ as $z\rightarrow\pm\infty$). Is it possible ...
1
vote
1answer
49 views

Cauchy Integral Formula Example

I am a beginner at calculating Cauchy Integrals but this one didn't look familiar to examples I have found. I think it is a linear integral. Could anyone give ideas how to solve it? Thank you in ...
2
votes
1answer
25 views

Basic complex integral property

I am trying to learn complex analysis using Alfohrs textbook and I have doubts about the proof of this property: When $a\leq b$, the fundamental inequality (1) $|\int_a^b f(t)dt| \leq \int_a^b ...
2
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2answers
54 views

A Contour Integral I

What is the value of the integral \begin{align} \int_{-a}^{c} \sqrt{ \frac{a+x}{c-x} } \ \frac{dx}{(d-x)(x-b)} \end{align}
1
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1answer
54 views

Complex integration Question - Contour Method [duplicate]

I'm asked to find: $$\int_{-\infty}^\infty \frac{\ln(x^2+1)}{1+x^2} dx $$ Attempt Considering $$ \oint \frac{\ln(z^2+1)}{(z+i)(z-i)} dz $$ So first I find the branch points of the function. This ...
3
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3answers
75 views

How can an improper integral have multiple values?

Integrals like this are said to dependend on the contour of integration: $$\int^{\infty}_{-\infty}\frac{x\sin x}{x^2-\sigma^2}dx=\pi e^{i\sigma}\space \mathrm{or}\quad \pi \cos\sigma $$ How is it ...
0
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0answers
47 views

A question on particular functions in $L^\infty$

Let D be the open unit disk in the complex plane, $\mu$ be the arc length measure, and $f, g\in L^{\infty}(\partial D,\mu)$ satisfying the following equations: $$ \int_{\partial D} ...
3
votes
1answer
24 views

Analytic function with values at specific points

Problem: Show that there is no function $f$, holomorphic on a neighbourhood of $z=0$, such that its value on $z=\frac{1}{n}$ is $(-1)^n (\frac{1}{n})$. To contrast, find a function holomorphic ...
0
votes
1answer
36 views

Quick question on complex integral

For $f(z) = (1-z^2)^{\frac{1}{2}}$, how do I show that the integral of $f(z)$ from $0$ to $\pi$ is $O(R^{-2})$? $$\int f(z) dz = \int \frac{1}{z(1-z^2)^{\frac{1}{2}}} dz $$
1
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1answer
34 views

Is Cauchy's integral theorem affected by integral direction?

hello,everyone,I hava a exam question aboat the integraion~ as shown below I know 1/(Z^2-1)=1/2(1/[z-1]+1/[z+1]) the integrantion around 1 should be 2*pi*i, but i am confused if the integrantion ...
0
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1answer
85 views

an amazing complex integral along unit circle

Let $f$ be an entire function on $\mathbb{C}$. Let $z_0\in\mathbb{C}$. Let $C=\{z\in\mathbb{C}\mid |z-z_0|=1\}$. Suppose $f(z)\neq f(z_0)$ for any $|z-z_0|\leq 1$, $z\neq z_0$. Given $f'(z_0)=2$, ...
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2answers
35 views

Complex integration around a singularity [duplicate]

I am trying to integrate the function $f(z)=$$\frac{5}{z^2}$ from -3 to 3 and I am supposed to develop a closed region that avoids the origin and use the analyticity of the function in this region to ...
0
votes
1answer
44 views

Contour integral of analytic function with singularity

I am supposed to integrate $f(z)=$$\frac{5}{z}$ from -3 t0 3 but I am having trouble understanding how to do this. I've done the integration the "hard" way by using parametrizations but now I need to ...
2
votes
1answer
59 views

Integration of complex function with respect to complex variable

I was given as homework to calculate the complex integral limit $$\lim_{T\rightarrow \infty} \frac {1}{2\pi i}\int_{c-iT}^{c+iT}\frac {x^s}{s^{k+1}}ds $$ where $c>0$ and $k\geq1$ is an integer. ...