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

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1answer
22 views

Complex var. integral: $\oint_{|z|=1} \frac{z^2\ dz}{\sin^3{z}\cos{z}}$

Integrate $\displaystyle\oint_C \dfrac{z^2\ dz}{\sin^3{z}\cos{z}}$; $C \rightarrow |z|=1$ I already know that $|z|=1$ is a circumference with $r=1$ and center at $(0,0)$. I also know there are ...
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0answers
43 views

Complex integration misconception?

Playing around with the complex integretion I encountered the following: Consider a holomorphic function $f(z)$ on $\Omega$. Let's say this holomorphic function has a primitve $F(z)$ such that $F'(z) ...
2
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1answer
57 views

Some inequalities for an entire function $f$

Let, $f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$ be an entire function and let $r$ be a positive real number. Then, which is(/are) correct? (a) $\sum_{n=0}^{\infty}|a_{n}|^{2}r^{2n}\le sup_{|z|=r} ...
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1answer
42 views

Cauchy Residue Theorem and Cauchy integral formula

Is it true that you can use the Cauchy Residue Theorem and the Cauchy integral formula interchangeably? I believe that the functions that satisfy the conditions of one, will indeed satisfy the ...
2
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1answer
40 views

Function holomorphic except for real line and continuous everywhere is entire

1) I've already shown that if $f:\mathbb{C}\rightarrow\mathbb{C}$ is holomorphic everywhere except for a single point and if it continuous on whole $\mathbb{C}$ then it is entire. It was quite easy. ...
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0answers
29 views

How can I solve $\int \left| \sum_{i=0}^{N} f_i(x) \right|^2 d x $ for complex functions $f_i$?

The $f_i(x)$ are complex functions and I know that $\int_{-\infty}^{\infty} f_i (x ) \, d x=c_i$. How can solve this integral? $$\int_{-\infty}^{\infty} \left| \sum_{i=0}^{N} f_i(x) \right|^2 d x $$ ...
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1answer
40 views

A complex integration arround the boundary of a rectangular region

Let, $u(x,y)$ be thereal part of an entire function $f(z)=u(x,y)+iv(x,y)$ for $z=x+iy\in \mathbb C$. If $C$ is the positively oriented boundary of a rectangular region $R$ in $\mathbb R^{2}$ then the ...
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1answer
27 views

Integrate $\int_C{\tan{z}\ dz}; C: y=x^2$ (complex numbers)

Integrate $$\int_C{\tan{z}\ dz}$$ $C$ is the parabola arc $y=x^2$ that connects the points $z=0$ and $z=1+i$. This is what I've done so far: I know that $\tan{z}=\dfrac{\sin{z}}{\cos{z}}$ And ...
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2answers
47 views

integration of a complex function

i want to solve this integral $\displaystyle \int_{-2}^{+2}\sin{ (π|x|/2)e^{-i2πkx}}\,dx$ in order to find the fourier transform of a function g(x) that is $0$ outside $(-2;2)$; i have already ...
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0answers
41 views

Evaluating complex integral along each side of rectangle

The first part of the question asks me to work out the integral of $\ell$ around the rectangle between the lines $x=-6$, $x=4$, $y=0$ and $y=8$ by evaluating the integral along each side of the ...
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1answer
104 views

Proving an Integral with Cauchy Residue Theorem

I need help proving this. The clue given is that Cauchy residue theorem can be used: $${1 \over {2\pi j}}\int_{c\ -\ j\infty}^{c\ +\ j\infty} x^{-s}\sigma^{s-1} ...
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1answer
29 views

complex analysis fundental theorem of caculus

Can anyone please explain how $$\int \frac{1}{(z-2)^3}dz $$ evaluated about the closed continuous path $$1+3e^{i2t\pi}$$ is 0 by the fundamental theorem of calculus?
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1answer
76 views

Evaluate Integral $\int_{c\ -\ j\infty}^{c\ +\ j\infty} \left({\sigma\,x^{-1}}\right)^s{\Gamma(\beta_1-1+s)\over \Gamma(\beta_1+\beta_2-1+s)}\,ds$

I am at this point of integration where: $$\int_{c\ -\ j\infty}^{c\ +\ j\infty} \left({\sigma\,x^{-1}}\right)^s{\Gamma(\beta_1-1+s)\over \Gamma(\beta_1+\beta_2-1+s)}\,ds$$ whereby $\beta_1$, ...
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1answer
93 views

Cauchy's Residue Theorem for Integral $\int_{c\ -\ j\infty}^{c\ +\ j\infty} \left({\sigma \over x}\right)^s{{1-\beta^{s+1}}\over s(s+1)}\,ds$

This is a similar problem to the one I posted here. I am at this point of integration where: $$\int_{c\ -\ j\infty}^{c\ +\ j\infty} \left({\sigma \over x}\right)^s{{1-\beta^{s+1}}\over s(s+1)}\,ds$$ ...
4
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1answer
193 views

Evaluate Complex Integral with $\frac{\Gamma(\frac{s}{2})} {\Gamma\big({\beta +1\over 2} - {s\over 2}\big)}$

I am proving this integral: $$ \int_{c\ -\ j\infty}^{c\ +\ j\infty} \left(\,x^{-1}\sigma\beta^{1 \over 2}\,\right)^{s}\ \Gamma\left(\,s \over 2\,\right) \Gamma\left(\,{\beta +1 \over 2} - {s \over ...
2
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1answer
63 views

Cauchy's Residue Theorem with Multiple Gamma Functions

I previously posted a similar problem here and here. This time however I am dealing with multiple gamma functions. This is the problem I am dealing with right now: $$ \int_{c\ -\ j\infty}^{c\ +\ ...
1
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1answer
68 views

An integral with the $\Gamma$ function: $\int_{c- i\infty}^{c+i\infty} u^{s}\:\Gamma(\beta +s-1) \:ds$

I previously posted a similar problem here and I have solved many of the problems from the answers given with explanations. This time however I am at this point of integration where: $$\int_{c\ -\ ...
2
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1answer
78 views

how to calculate the following integral$\int_{-\infty}^{\infty}\frac{1}{\left(t^2+\pi^2\right)^2 \cosh(t)}dt$ [closed]

calculate the following integral $\int_{-\infty}^{\infty}\frac{1}{\left(t^2+\pi^2\right)^2 \cosh(t)}dt$ I need to very hollowing steps.thank you in advance
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1answer
87 views

how to compute this complex integral with high order polynomial?

Compute $$ \lim_{R \to \infty} \frac{1}{2\pi i} \int_{|z|=R} \frac{(2z^2+z-1)P'(z)}{P(z)+3}dz $$ where $P(z)=z^{10}+2z^9+z^5+1$. It seems like I may use residue but it contains too high ...
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1answer
50 views

How to show $\int_0^1\frac{e^{e^{2\pi it}}}{e^{2\pi it}}dt=1$

I was trying to integrate the contour integral $$\int_\gamma \frac{\vert z \vert e^z}{z^2}$$ where $\gamma$ parametrizes the unit circle counterclockwise. I cannot use the Generalized Cauchy Integral ...
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0answers
28 views

Expressing principal value of integral as real/imaginary

How is it that we can express $$ \mathrm{p.v.}\int_{-\infty} ^{\infty} \frac{\cos 3x}{x^2+4}=\Re \ \mathrm{p.v.}\int_{-\infty} ^{\infty} \frac{e^{3xi}}{x^2+4} $$ while we cannot for $$ ...
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1answer
32 views

inequality in absolute value with exponential

Can you help me with this problem please ?? If $r>0$ . Show that $\left |{\displaystyle\int_{\gamma}e^{iz^{2}}dz}\right |\leq{\displaystyle\frac{\pi(1-e^{-r^{2}})}{4r}}$ where ...
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2answers
30 views

Compute $\int_\Gamma \frac{e^\frac{1}{z}}{z-1}dz$, where $\Gamma$ is the circle $|z-1|\le\frac{3}{2}$, positively oriented.

Compute $\int_\Gamma \frac{e^\frac{1}{z}}{z-1}dz$, where $\Gamma$ is the circle $|z-1|\le\frac{3}{2}$, positively oriented. The numerator is not analytic in $\Gamma$ so we can't use Cauchy ...
1
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1answer
21 views

Contour integration over a circle

$$\int_C \frac{\cos(\ z)}{(z)^2} dz$$ where C is any circle enclosing the origin and oriented counter-clockwise. z0 = o of order 2 , f(z) = cos z $$\int_C \frac{\cos(\ z)}{z^2} dz$$ = $2 \pi i ...
2
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1answer
39 views

evaluating a contour integral where c is $4x^2+y^2=2$

Consider the integral $$\oint_C \frac{\cot(\pi z)}{(z-i)^2} dz,$$ where $C$ is the contour of $4x^2+y^2=2$. The answer seems to be $$2 \pi i\left(\frac{\pi}{\sinh^2 \pi} - \frac{1}{\pi}\right)$$ but ...
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2answers
41 views

Evaluate the complex integral [closed]

Evaluate the below integral: $$ \int_{0}^{\infty}{x^{\alpha - 1} \over 1 + x}\,{\rm d}x $$ How to start ?.
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1answer
24 views

complex integral over a line

The value of line integral over C of dz/(z^2+4) along the line x+y=1 in the direction of increasing x is ___ The answer is pi/2 I am not sure how to arrive at this answer .i had a guess that maybe i ...
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0answers
32 views

Could anyone help me to solve this integral question?

Could anyone help me to solve this integral question ? $$ \ \int_a^b t^{k-z-1}(1+mt^{-z})^{-(n+1)}dt $$
22
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2answers
482 views

How to solve $\int_0^{\infty}\frac{\cos{ax}}{x^3+1}dx$?

QUESTION. It is looked for a closed solution for following real integrals $\displaystyle\int_0^{\infty}\displaystyle\frac{\cos{ax}}{x^3+1}dx$ and ...
3
votes
3answers
86 views

Find the principal value of $\int_{-\infty}^{\infty}\frac{1-\cos x}{x^2}\,\mathrm dx$

How to find the Cauchy principal value of the following integral $$\int_{-\infty}^{\infty}\frac{1-\cos x}{x^2}\,\mathrm dx$$ How to start this problem?
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1answer
42 views

Find the contour integral around unit circle.

Evaluate the below integral by turning it into a contour integral around a unit circle: $$\int_{0}^{\pi}\frac{\cos2\phi}{1-2a \cos\phi + a^2} d\phi$$ $where\;a\neq \pm1$
2
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1answer
29 views

evaluate the complex integration

How to evaluate the below integral $$\oint_{c} \frac{dz}{e^{z}-1}$$ where $C$ is the circle $|z|=1$ Can this be done by Cauchy's formula? If yes how? Or do I need to do something else in order to ...
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2answers
29 views

Evaluate the contour integration

Evaluate the below integral: $$\oint_{c}\frac{e^{2z}}{(z+1)^4}dz$$ where $C$ is the circle, $|z|=3$
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1answer
40 views

Evaluate the contour integral

How to evaluate this, $$\oint_{c} \frac{\sin\pi z^2+\cos\pi z^2}{(z-1)(z-2)}dz$$ where $C$ is the circle, $|z|=3$ I tried below things I believe 1 and 2 are simple poles here and the equation can be ...
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2answers
16 views

finding poles for a complex rational function

So in working out the details of a trig integration with complex integrals problem, I have ended up with an integrand of $$\frac{z}{z^4+6z^2+1}$$ I need to find the roots of $z^4+6z^2+1$ to use the ...
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2answers
26 views

Question about how to get the residue for infinite amount of poles

So I am asked to find the residue for each pole such as $$ f(z) = \frac{z}{1-\cos(2z)} $$ I understand pole of order 2 with $z= 2\pi k$ excluding zero. I also understand that residue equals to ...
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1answer
56 views

Why does this function residue equal 0?

$$ f(z) = \frac{e^{2z}}{(z-1/2)^{2013}} $$ Why does this residue equal 0? If I expand Laurent series, the right side will have $\dfrac{a_{2013}}{(z-1/2)^{2013}}$ $$ + \frac{a_{-2012}}{(z- ...
2
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1answer
27 views

solve a complex integral

I stumbled on this integral, the problem says to solve it with contour integration. Any insights on how to solve this in function of $n$? \begin{equation} \int_{0}^{2\pi}\cos^{2n}(\theta)d\theta ...
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0answers
60 views

Calculate $\int_0^3 \frac{x^{\frac{3}{4}} (3-x)^{\frac{1}{4}}}{5-x}\,dx$ using principal branch

I would like to calculate the following integral $$ I = \int_0^3 \frac{x^{\frac{3}{4}} (3-x)^{\frac{1}{4}}}{5-x}\,dx $$ using contour integration but using principal branch of the function, i.e. ...
1
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1answer
38 views

How can I calculate this complex integral?

The integral is the following: $$\int_{|z|=r} \frac{z+1}{z(z^2+4)} dz , r>0, r \neq 2 $$ I'm a little bit lost, I know that its partial fraction expansion is $$ \frac{z+1}{z(z^2+4)} = ...
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0answers
25 views

Integration of a complex integration

Let $C$ be the contour $|Z|=2$ oriented in the anti-clockwise direction.What is the value of the integral $\oint_{C}{ze^{\cfrac{3}{z}}}$$dz$ ? I don't know how to start. Please tell me which formula ...
2
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1answer
28 views

Conformal mappings to polygons: why is my integral conformal?

I'm learning about conformal mappings into polygons in a class,(undergrad complex analysis) and am having trouble understanding one of the examples given in my book. (Stein & Shakarchi) Here it ...
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3answers
18 views

Not sure how this inequality is formed - $\bigg|\int_0^{2\pi}\frac{e^{p(R+iy)}}{1+e^{R+iy}}idy\bigg| \le \frac{e^{pR}}{e^R - 1}2\pi$

I have the following inequality in my notes - $$\bigg|\int_0^{2\pi}\frac{e^{p(R+iy)}}{1+e^{R+iy}}idy\bigg| \le \frac{e^{pR}}{e^R - 1}2\pi$$ We can start as follows ...
3
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1answer
93 views

Evaluating $\int_{-\infty}^{\infty}\frac{1}{(x^2+b^2)^2}dx$

Find $$\int_{-\infty}^{\infty}\frac{1}{(x^2+b^2)^2}dx$$ We see that the only poles are at $x=\pm bi$. Integrating over the semicircular contour implies that it is equal to $2\pi i*Res_{(+bi)}$ ...
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2answers
51 views

Integral of $((x^2+1)((x-1)^2+1))^{-1}$

Find $$\int_{-\infty}^{\infty}\frac{1}{(x^2+1)(2-2x+x^2)}dx$$ So I am going to integrate this using a semicircular contour. Is it safe to say that on the curved part, the integral vanishes? because ...
3
votes
1answer
47 views

Contour Integral of $\sin(z)/(z^2-z)$

Find the integral $\int_{\lambda}\frac{\sin(z)}{z(z-1)}$ where $\lambda(t) = 10e^{it},t\in[0,2\pi]$ We notice that there are poles at $z = 0$ and $z=1$. So we can use residue theorem but I am ...
1
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0answers
33 views

Complex integration, limits, arctan

$\left.\frac12i\;\text{Log}\frac{1-(-i+e^{it})}{1+(-1+e^{it})}\right|_0^{2\pi}=\frac12i\left(\log\left|\frac{i}{i}\right|+i\arg 1-\log|1|-i\arg1+2\pi ik\right)$ could someone explain how this is ...
0
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3answers
114 views

Geometrical Interpretation of the Cauchy-Goursat Theorem?

The Cauchy-Goursat theorem is really non-intuitive and is very astounding. Can someone geometrically explain to me why its true? I'm specifically talking about this version of the theorem: For ...
2
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1answer
42 views

Limit of an integral that arose in Fourier Analysis

$$\lim_{\alpha\to\infty} \int_{0}^{\delta}\frac{\sin(\alpha x) \sin(\lambda\alpha x)}{x^2}dx$$ where $\lambda,\delta \in \mathbb{R}, \lambda,\delta > 0$. Appreciate your help in finding this limit. ...
2
votes
2answers
49 views

How could I evaluate the integral of type $\int_{-\infty}^{\infty} e^{iax}\cdot e^{-(bx^2 + cx)}$?

I have an integral of type: $$\int_{-\infty}^{\infty} e^{iax}\cdot e^{-(bx^2 + cx)}~ dx$$ And I have no clue on how to integrate that properly. What I've tried so far is writing everything above ...