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

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42 views

evaluation of fourier transform of electric potential

I would like to ask how to evaluate equation 7? I have spent hours and still have no idea how to get a(k).
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1answer
52 views

Integration over complex plane

I have a problem with the following integral $$\int_{-\infty}^{\infty}\frac {x\sin x}{x^4+1}$$ Can someone please help me with the way the solution goes? I would highly appreciate it Thanks in ...
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2answers
62 views

the integral a complex variable

$\int\limits_{0}^{2\pi}e^{\cos\varphi}(\cos\varphi-\sin\varphi)d\varphi$ I think $e^{i\varphi}=z$ $\to d\varphi=\frac{dz}{iz}$ $\cos\varphi=\frac{z^2+1}{2z}$ $\sin\varphi=\frac{z^2-1}{2iz}$ ...
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1answer
66 views

Computing the integral $\int_{-\infty}^{\infty}\frac{z^4}{1+z^8}dz$

I need help to compute the following integral: $$\int_{-\infty}^{\infty}\frac{z^4}{1+z^8}dz$$ I need to use Cauchy's residue theorem. I can write that $z^8+1=z^8-i^2=(z^4-i)(z^4+i)$. How do I ...
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1answer
33 views

the function of a complex variable

$\int\limits_{0}^{\pi}\frac{cos^2{\varphi}}{2-sin^2{\varphi}}d\varphi$ I think $e^{i\varphi}=z$ $\to d\varphi=\frac{dz}{iz}$ $cos\varphi=\frac{z^2+1}{2z}$ $sin\varphi=\frac{z^2-1}{2iz}$ ...
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2answers
37 views

Find $P(Y_1≤3/4,Y_2≥1/2)$ of a joint probability density function.

Let Y1 and Y2 have the joint probability density function given by: $ f (y_1, y_2) = 6(1−y_2), \text{for } 0≤y_1 ≤y_2 ≤1$ Find $P(Y_1≤3/4,Y_2≥1/2).$ Answer: $$\int_{1/2}^{3/4}\int_{y_1}^{1}6(1− ...
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0answers
20 views

Residue of $\frac{\text{cot}(\pi z)}{z^6}$ at $0$

I am trying to compute $\zeta(6)$ = $\sum_1^{\infty} \frac{1}{n^6}$; I generally know how to do this using a residue-based proof, but I am stuck at the last bit, namely calculating the residue of ...
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2answers
38 views

Cauchy Riemann equation and Harmonic Condition

Question: If harmonic functions $u$ and $v$ satisfy Cauchy-Riemann equations, then $u+iv$ is an analytic function. Am a bit confused here as we already have a theorem which says that if a function ...
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2answers
72 views

Integrating $\int_{-1}^{1}\frac{dx}{(x-a)\sqrt{1-x^2}}$

I'm asked to find the value of $$\int_{-1}^{1}\frac{dx}{(x-a)\sqrt{1-x^2}}$$ where $a$ is complex and $a\not\in[-1, 1]$. I think I should use Cauchy's integration formula but don't know how to ...
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1answer
20 views

What is P(Y1−Y2>3) of a given joint density function?

Let $Y_1 $and $Y_2$ have joint density function: $$f (y_1, y_2) = e^{-(y_1+y_2)}, \text{for all } y_1 >0,y_2 >0 $$ What is $P(Y_1−Y_2>3)$? My attempt: $$P(Y_1−Y_2>3) = P(Y_1−3> ...
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2answers
196 views

Choosing parametrization for complex integration with two branch cuts

I am particularly interested in how Ron Gordon came up with the parametrization in his anser to this question: Inverse Laplace transform $\mathcal{L}^{-1}\left \{ \ln \left ( ...
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0answers
53 views

integral of $e^{-\imath x}$ from $a$ to $\infty$ and characteristic functions

In the following result: $$\frac{1}{2 \pi} \int_{-\infty}^{\infty} \Phi(u) \left[\int_a^{\infty} e^{-i u x}dx\right] du = \frac{1}{2} + \frac{1}{ \pi} \int_0^{\infty} \Re\left[\frac{e^{-i u a} ...
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1answer
55 views

Complex Integrate $\int_{-\infty}^{\infty}e^{-|\lambda t|}e^{itx}dt$

I'm working through Big Rudin's (Real and Complex Analysis) Fourier Transform chapter, and the following complex integral is part of a discussion on the Inverse Transform that Rudin mentions briefly ...
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0answers
41 views

Complex integral with roots

Integral on $C$ of $\int g(z)\,\mathrm dz$ $C$ is: (those points are $1$ and $e$) $$\begin{align} g(z) &= z^{1/4}\\ g(1) &= i\\ \end{align}$$ How do I evaluate this using Anti-derivative ...
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2answers
47 views

Integrate a function over a contour including infinitely many poles, such as $\int_{|z|=1}1/\sin(1/z)\,dz$

We can find complex integration of a function over a closed contour by residue theorem if there are only finite many singularity inside the contour. But my question is how to find the integration if ...
2
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0answers
25 views

Evaluating functions similar to the Bessel functions

In the problem there are two integrals, and one is asked to evaluate them by taking an integral over a unit circle of some chosen function. The integrals are $$\int_0^{2\pi}e{^{\sin n\theta}}\cos ...
3
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2answers
63 views

Given $f(z)$ entire function and $\left| f(z) \right| \le 1 + \left| z \right|^3$ for all $z \in \mathbb{C}$ show that $f$ is a polynomial

I'm learning about complex analysis and need some help with this problem : Given $f(z)$ entire function and $\left| f(z) \right| \le 1 + \left| z \right|^3$ for all $z \in \mathbb{C}$ show that ...
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2answers
33 views

Integral on a contour curve

Find the line integral along curve $C$ of $[f(z)]^2=z$ where $f(1)=1$. Here is curve c: https://imgur.com/uOSLwdt (Sorry for the blur, the points are $1$ and $e$) How can I solve this? I am lost. Is ...
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1answer
44 views

Evaluate the complex integral: $\int_{|z|=1}xdz$

$\int_{|z|=1}xdz$ I ended up with $2\pi$ as my final answer, can anyone confirm and/or give me a shorter way to do it? Mine involved lots of sines & cosines.
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1answer
42 views

Complex Integrals (No Residue allowed)

http://imgur.com/uTNwe4b I've made the best attempt at making my questions clear but it seems its still blurry. I will rewrite them down here: a) integral on curve c (1/z) dz b) integral on curve c ...
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1answer
31 views

If $f(z)$ is entire and $|f(z)| \le \log(2+|z|)$ for every $z \in \Bbb C$ show that $f$ is constant

I'm learning about complex analysis and need to verify my work to this problem since my textbook does not provide any solution: If $f(z)$ is entire and $|f(z)| \le \log(2+|z|)$ for every $z \in ...
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0answers
21 views

Integral with imaginary limit

I need to show some integral identities with imaginary limits: $\lim_{T \to (1-ie)\infty} \int_{-T}^{T} e^{-iw|t|} dt = \frac{2}{iw}$ and $\lim_{T \to (1-ie)\infty} \int_{-T}^{T} dt_1 dt_2 ...
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1answer
48 views

Using Cauchy's Integral Formula to show that $f(z) = e^z$ for every $z$ with $|z| \lt 1$

I'm learning about complex analysis and need some help with this problem : Given $f : \Bbb C \rightarrow \Bbb C $ analytic with $f(z) = e^z$ for every $z$ with $|z| = 1$. Show that $f(z) = e^z$ ...
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4answers
65 views

Use Complex Integrals/ Residue to evaluate $\int_0^\infty \frac{dx}{(x+1)^3 + 1}$

Use Complex Integrals/ Residue to evaluate $\int_0^\infty \frac{dx}{(x+1)^3 + 1}$ I'm not sure how to do this integration. It looks like partial fractions but I'm unsure.
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3answers
130 views

Cauchy's Theorem - Prove that $\sum_{n=1}^\infty \frac{1}{\lambda_{n}^2} $ = $\frac{1}{10}$

I seek to prove that $$\sum_{n=1}^\infty \frac{1}{\lambda_{n}^2} = \frac{1} {10},$$ by applying Cauchy Theorem to $$ f(z) = \left(\frac{z\tan(z)}{z-\tan(z)}+\frac{3}{z}\right) \frac{1}{z^2},$$ ...
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1answer
35 views

Finding the zeros of a complex function on a disc

I have encountered the following problem: Find the number of zeros of $f(x)$ on the disk $|z|$ < $1/2$ where $f(x)$ = $z^2$+$cosh(iz)$ How would one compute the solution?
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14 views

Branch cut for $\oint_{|z-3|=1} \frac{Log \, z }{(z-3)^5} dz$.

We have the integral $\oint_{|z-3|=1} \frac{Log \, z }{(z-3)^5} dz$. I want to use Cauchy Integral Formula, so I let $f(z) = Log \, z$. Since $Log \, z$ is multivalue, I need to worry about the ...
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1answer
19 views

Integrating 0 over a simple complex area containing 0

This should, I think, by rights, be easier for me to do. That said, I can't. $$\iint_D0 dA$$ $D$ is a simple area over a complex plane, with simple closed, positively oriented boundary, $D'$, where ...
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2answers
36 views

Does the contour integral of a rational fraction function in the complex plane vanish in large radius limit?

Let $f(z)=\frac{z^m+az^{m-1}+\cdots+b}{z^n+cz^{n-1}+\cdots+d}$ be a rational fraction function of complex variable $z$, where the integers $n-m\geqslant 2$. Is the following integration limit ...
0
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1answer
23 views

Find a sequence which uniformly converges f(z), and is of the form $\displaystyle\sum_{i=1}^{\infty} \frac{c_i}{w_i-z}$

Let $f\colon\mathbb{C}\rightarrow \mathbb{C}$, analytic in the closed disk $D=\{z:|z-z_0|\leqslant R\}$. Is there a way of defining a sequence of the form $\displaystyle\sum_{i=1}^{\infty} ...
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2answers
45 views

How to evaluate the contour integral $\int_{C(0,1)} \frac{z e^z }{\tan^2 z}dz$ over the unit circle?

Let $C(0, 1)$ be the unit circle centered at the origin with radius $1$. Then I need to evaluate the following complex contour integral: $$ \int_{C(0,1)} \frac{z e^z }{\tan^2 z}dz$$ I know the ...
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1answer
65 views

Evaluate the integral $\int_0^\infty \frac {x^{1/2} dx}{x^2 + 1}$ using method of residues

I am trying to evaluate the integral $\int_0^\infty \frac {x^{1/2} dx}{x^2 + 1}$ using method of residues. I can solve this very easily without the $x^{1/2}$ on top, but I do not know what to do when ...
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0answers
42 views

Contour integral of $\int_{0}^\infty \frac{\sinh(kx)}{\sinh(x)}dx = \frac{1}{2}\tan{\frac{a}{2}}$

From Calculating the Fourier transform of $\frac{\sinh(kx)}{\sinh(x)}$ In the case of zero $\omega$ and integral starts as 0, how do I prove that using contour integral $\int_{0}^\infty ...
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1answer
25 views

Evaluating integral using invalid substitution

I was trying to show that for suitable t: $$ 2\pi(1+t/(\sqrt{(1-t)(3-t)})=\sum_{0}^{\infty}(t^n\int_0^{2\pi}1/(2-cos(\theta))^nd\theta $$ By uniqueness this is clearly the Taylor series about $0$ ...
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1answer
45 views

Help with understanding $\int_C \frac{e^z-1}{z^4}dz$ = $\frac{\pi{i}}{12}$ where C is the unit circle

I am referring to a solved example from Brown and Churchill's Complex variables and applications (ninth ed. page 231). If I simply use the Maclaurin series (which I think is a specific case of Laurent ...
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2answers
37 views

Complex integration and Gauss mean value theorem

I'm trying to show that $\frac{1}{2\pi} \int_0^{2\pi} \log | 1-ae^{i\theta}|d\theta=0$ for $|a|<1$ implies $\frac{1}{2\pi} \int_0^{2\pi} \log | a-e^{i\theta}|d\theta=0$
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1answer
15 views

Showing that $\overline{\int_{\gamma}f(z)dz}=\int_{\overline{\gamma}}\overline{f(\overline{z})}dz$

I am trying to show that Let $\gamma$ be a piecewise-$C^1$ curve, and let $\gamma$ be its image under the mapping $z\mapsto \overline{z}$ (symmetry in the real axis). Then if $f(z)$ is continuous on ...
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3answers
99 views

Show that $\int_{0}^{+\infty} \frac{\sin(x)}{x(x^2+1)} dx = \frac{\pi}{2}\left(1-\frac{1}{e}\right) $

I'm trying to show that $$\int_{0}^{+\infty} \frac{\sin(x)}{x(x^2+1)} dx = \frac{\pi}{2}\left(1-\frac{1}{e}\right) $$ using Jordan's lemma and contour integration. MY ATTEMPT: The function in ...
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0answers
26 views

How to change the equation to polar form?

Compute $\displaystyle\int^\infty_{-\infty} dx\displaystyle\int^\infty_{-\infty} dy\displaystyle\int^\infty_{-\infty} dz \delta\left(\sqrt{x^2 +y^2+z^2} - R\right)$.
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1answer
75 views

Differentiating a contour integral

Let $P(z, t)$ be a cubic with the parameter $t$, and consider $$\mathcal{I} = \int_{\gamma(t)} \frac{dz}{\sqrt{P(z, t)}}.$$ Here, $\gamma(t)$ is a contour in the complex plane that encloses any two of ...
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1answer
20 views

Solve for $m=0,1,2…$ and $n\in\Bbb{Z}$ the integral $\int_{\Gamma_1} z^n(1-z)^mdz$

I need to solve the integral for $m=0,1,2...$ , and $n\in\Bbb{Z}$ $\qquad\int_{\Gamma_1} z^n(1-z)^mdz$ where $\Gamma_1$ is the circle centered in $0$ with radius $1$. I'm struggling trying to see ...
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0answers
102 views

Differentiating a period of an elliptic curve under the integral sign

Let $$g = \frac{27J}{1 - J},$$ where $J$ is the absolute invariant, and define $$\Omega = \int_{\gamma(J)} \frac{dz}{\sqrt{4z^3 + g(z + 1)}}.$$ Here, $\gamma(J)$ is a contour in the complex plane that ...
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3answers
63 views

Integrate: $\int_0^{2\pi} \frac{d\theta}{1+\cos^2\theta}$

I'm trying to integrate this here fella: $\int_0^{2\pi} \frac{d\theta}{1+\cos^2\theta}$ from examples in Ablowitz I know that for $|A|^2>|B|^2$ and $A>0$, $\int_0^{2\pi} ...
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1answer
40 views

Complex Laurent Series and Contour Integral

Let $f(z) = \sin{(\frac{1}{z})}$, where $z \neq 0$. Find a Laurent Series expansion of $f$ around the annulus $D: 1< |z|<3$. Use the result to find $$\oint \limits_C ...
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1answer
30 views

Properties of a function whose has its own Fourier transform

A posted exercise is from Complex Analysis, Stein This exercise generalizes some of the properties of $e^{-\pi{x^2}}$ related to the fact that it is its own Fourier transform. Suppose ...
2
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0answers
75 views

Integral of $e^{ix^2}$

How does one evaluate $$\int_{-\infty}^{\infty} e^{ix^2} dx$$ I know the trick how to evaluate $\int_{-\infty}^{\infty} e^{-x^2}dx$ but trying to apply it here I get a limit which does not converge: ...
3
votes
0answers
23 views

Complex line integration with assumptions

Let $f: \mathbb{C} \to \mathbb{C} $ be a holomorphic function with $$ \lim_{\lvert z \rvert\to\infty} \frac{f(z)}{z^{n-1}} = 0$$ for some $n\in\mathbb{N}$. How can I prove that $$ \lim_{r\to\infty} ...
1
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1answer
42 views

Extension of Cauchy Integral Formula

I'm now taking a course in complex analysis and in wikipedia it was said that Cauchy Integral formula is true also for a function which is "holomorphic in the open region enclosed by the path and ...
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0answers
37 views

Complex integration upper semi-circle $\int f(z)dz$ when $r>1$

Question: Let $r$ be a real number, $r > 0$ and let $Lr$ be the line from the point $−r$ to $r$ in $C$. Let $γr$ be the upper half circle with radius $r$ and center in $0$. $$ \ f(z) = ...
2
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1answer
46 views

Evaluating the complex integral $\int_{0}^{\infty} \frac{\sin(x+i)}{x+ i } \, \mathscr{d}x$

Through the course of a problem I am working on I have reached two integrals that look similar to some of the trigonometric integrals. The integrals I have are the following: $$\int_{0}^{\infty} ...