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Questions tagged [contour-integration]

Questions on the evaluation of integrals along a locus in the complex plane.

3
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1answer
34 views

Contour integration and the central binomial coefficients

I am trying to compute the integral $$\int_{-\infty}^\infty \frac{x^{2n}}{(x^2 + 1)^{n + 1}}\ dx.$$ From computational evidence, it's very obvious that $$\int_{-\infty}^\infty \frac{x^{2n}}{(x^2 + 1)^{...
1
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1answer
26 views

integral of $e^z \log z$ over parabola in complex plane

I am studying for a Complex analysis exam that I have coming up in the next few weeks and I am working through some practice problems. I happen to have gotten stuck on the following integral; $$ \...
0
votes
1answer
23 views

Calculate complex curve integral along rectangle

Determine the line/contour integral of: $$\int_{\gamma}\frac {z}{z^3+1}dz$$ where $\gamma$ is the boundary of a rectangle defined for $0\leq x\leq 2$ and $-2\leq y\leq 2$. I am almost certain we ...
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0answers
18 views

Prove the contour integral is either 0 or 1 based on the value of w

Background This is part two of a two-part problem. In the first part, we proved that, given $\gamma:[a,b] \to \mathbb{C}$ is a piecewise differentiable path, $\psi:$ range $\gamma \to \mathbb{C}$, ...
2
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0answers
25 views

De forming the contour and showing that the contributions of contours of infinitely small lengths go to zero

I am considering an integral around the path $ \Gamma = C_1 \cup C_{\varepsilon_{1}} \cup C_2 \cup C_{\varepsilon_{2}}$ of a function $f(z)$ that has a pole in every cross in the images below. In ...
2
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0answers
29 views

How do I evaluate the following integral for a complex-valued function?

So, I am given the following task: Compute: $$\int_{+\gamma} \frac{2z}{(z^2 - i)^3} dz$$ when $+ \gamma$ is any curve $z = z(t)$ in $\mathbb{C}$ with $|z(t)| > 2$ for every $t$, with start point ...
1
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1answer
55 views

Certain Complex Integral

I was trying to generalize the Riemann's prime number formula for $\pi(x)$ to a general algebraic field $K$, and came across the integral: $$f(x)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\frac{d}{...
2
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3answers
54 views

Contour integral of $\frac{\sin z}{(z^2+1)^2}$.

I have the next contour integral in complex analysis: $$\oint_{\gamma} \frac{\sin (z)}{(z^2+1)^2} dz$$ With $\gamma:z=2e^{it}+1, 0\leq t \leq 2\pi$. I have tried to use the Cauchy integral formula, ...
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1answer
32 views

If $f(z)$ is analytic inside and on the circle $|z| = 2$, is $\oint_{|z|=2}f(z) = 0?$

I know if $f$ is analytic in a simply connected domain D then for any closed loop in D this integral is 0. Is that enough to say yes to this question?
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22 views

Contour Deformation in the Laplace Inversion Formula

Following Szpankowski - Average Case Analysis of Algorithms on Sequences the exponential generating function of $g(n)$ which is thought to be analytic in $n$ is defined as $$ G(z)=\sum_{n=0}^{\infty} ...
2
votes
1answer
55 views

How to compute this definite improper integral using the Residue Theorem??

I have been trying to compute this integral: $$\int_0^\infty \frac{\log^2|\tan x|}{1+x^2}\,dx$$ using the Residue Theorem but have not been successful. I tried to use the classical upper semi-...
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3answers
390 views

Evaluating $\int_{0}^{\infty} \frac{1}{a_{n}x^{n} + … + a_{2}x^{2} + a_{o}}dx$ via Residue Theory?

In the text "Functions of a Complex Variable" by Robert E. Greene and Steven G.Krantz I'm having trouble verifying my solution to $\text{Problem (1)}$ $\text{Problem (1)}$ Using Calculus of ...
0
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1answer
35 views

Evaluate $\int_{C} (z-z^2)dz$ where $C$ is the (i)Upper half of the circle $|z|=1$

Evaluate $\int_{C} (z-z^2)dz$ where $C$ is the (i)Upper half of the circle $|z|=1$ (ii)Lower half of the circle. Where $z$ is a complex number. How can I approach this problem? What ...
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1answer
40 views

Complex Analysis Question - Cosine and quadratic combined

i was doing some questions in my complex analysis booklet and have came across the following question that i don't seem to be able to get the answer for. Hoping someone on here can help! So it's the ...
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1answer
33 views

Complex Integral over a Square Path [closed]

I'm struggling to set up this integral as I just started learning complex analysis. Let C be the perimeter of the square with vertices at the points z = 0, z = 1, z = 1 +i and z = i. $$\int_C \bar{...
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0answers
45 views

Steepest descent without saddle point $\int_{0}^{\infty} t^n\exp(-x(t+1/t))dt$

Find the first term in the asymptotic expansion of $$\int_{0}^{\infty} t^n\exp(-x(t+1/t))dt$$ using the method of steepest descent. I considered this as a complex integral on the real axis. The ...
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0answers
24 views

How can i determine if the arc of the following contour integral vanishes as R approaches infinity?

The arc is the following integral (this arc is part of a semicircle , where the bottom part is over the real axis, and it traverses in a CCW orientation) $$ \int_0^\pi {re^{ia}ire^{ia}\over e^{r\cos(...
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0answers
12 views

Can the integral $\int_C \exp\{\alpha u -2\alpha \cosh^{-1}(e^u) -\beta u^2\}$ be written in terms of elementary functions?

I need to perform the integral $$I = \int_C \exp\{\alpha u -2\alpha \cosh^{-1}(e^u) -\beta u^2\}du$$ Here $\alpha,\beta$ are real and positive and the contour extends from $(\Re(u),\Im(u)) = (0,-y)$ ...
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2answers
36 views

How to evaluate contour integral

I have a homework problem to evaluate the integral $$ \oint_{\gamma}\frac{\cos z}{(z+i)^3}dz $$ along the curve $\gamma(t)=-i+e^{it}, t\in[0,2\pi]$. I proceeded to plug the given information into the ...
6
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1answer
158 views

Evaluation of $\frac{1}{2\pi i }\int_{c-i\infty}^{c+i\infty}\frac{\log(s)^{n}\log(1-s)^{m}}{s(1-s)}ds$

I am having trouble trying to evaluate the integral : $$\frac{1}{2\pi i }\int_{c-i\infty}^{c+i\infty}\frac{\log(s)^{n}\log(1-s)^{m}}{s(1-s)}ds$$ Where $0<c<1$ and $n,m$ are positive integers. ...
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0answers
105 views

Inverse Laplace transform of $1/\sqrt{P(s)}$ with $P(s)$ a polynomial of degree 4

I am trying to compute the inverse Laplace transform of $$F(s)=\frac{1}{\sqrt{[(s+\lambda_3)^2-(\lambda_1-\lambda_2)^2][(s-\lambda_3)^2-(\lambda_1+\lambda_2)^2]}},$$ where we can assume that $\...
2
votes
2answers
196 views

Find $\int_{\gamma} e^zz^n dz$ where $\gamma$ is the unit circle, using Cauchy's Integral Formula

I'm been banging my head against the wall trying to solve the following question which ask to solve the following integral using the Cauchy integral formula, and hence evaluating the corresponding ...
0
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0answers
22 views

Complex Integration involving Riemann zeta and Gamma functions

Let $G(x)=\frac{1}{2\pi\iota}\int\limits_{(3/4)}\frac{\Gamma(1+t)^2}{(4\pi^2x)^t}\frac{dt}{t}$. Here, integration is over line such that real part is $\frac{3}{4}$. We can prove below result by using ...
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0answers
16 views

How can we calculate the imaginary part of a fraction that has a term i0+ in the denominator (Sokhotski–Plemelj theorem)?

I have recently started dealing with thermal field theory for fermions and I am faced with a paper that, at some point, tries to calculate the imaginary part of a fraction that looks like: $$\frac{1}{...
1
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1answer
63 views

Evaluate $\int_\gamma \frac{e^{iz}}{z^2+1} \ dz$

Define the semicircular arc $\gamma_R$ by $\gamma_R(t)=Re^{it}$, where $0\leq t\leq\pi$ and $R>1$ is a real constant. Let $\gamma$ be the join of $\gamma_R$ and the line segment from $-R$ to $R$. ...
2
votes
1answer
57 views

Evaluate $\ \int_{|z|=2}\frac{z^2+1}{(z-3)(z^2-1)}\ dz$

I am trying to solve $$\ \int_{|z|=2}\frac{z^2+1}{(z-3)(z^2-1)}\ dz,$$ using Cauchy's Integral Formula. My attempt: $$I=\int_{|z|=2}\frac{z^2+1}{(z-3)(z^2-1)}\ dz=\int_{|z|=2}\frac{z^2+1}{(z-3)(z-...
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votes
2answers
42 views

Does Integration along the imaginary axis in the complex plane give an imaginary result?

Given is the following contour integral : The “arc-length” path vanishes according to Jordan’s Lemma. Now, is it correct to use the residue theorem, and set the Integral along the imaginary axis ...
0
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1answer
25 views

Closing Contours 'Up' and 'Down' in a Contour Integral

I am considering an integral which can be evaluated using techniques from complex variables. The first step is to simplify the integral. $$\int^{\infty}_{-\infty}dk\:\frac{e^{ikr}-e^{-ikr}}{k} = \...
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votes
2answers
38 views

How to show the loop integral of $\frac{1}{z-a}$ vanishes…

...when the loop is a positively oriented circle $C$ and $a$ lies outside of $C$? My work so far has been to show $\frac{1}{z-a}$ is contimuous when $z\neq a$. As such, every loop integral of $\frac{...
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0answers
20 views

Showing that $\exp\left((\epsilon e^{i\theta})t-(\epsilon e^{i\theta}/k)^{1/2}x\right)-1\le N\epsilon^{1/2}$

A part of a book I am reading is that the reader can easily verify that $$\exp\left((\epsilon e^{i\theta})t-(\epsilon e^{i\theta}/k)^{1/2}x\right)-1\le N\epsilon^{1/2}$$ for some positive constant $...
4
votes
2answers
55 views

How can I evaluate this complex integral $\int_{|z|=1}e^{\frac{1}{z}}\cos{\frac{1}{z}}dz$?

I'm trying to evaluate the following complex integral using the residue method. $$\int_{|z|=1}e^{\frac{1}{z}}\cos{\frac{1}{z}}dz$$ The point $z_0=0$ seems to be a singularity. I'm not sure but I ...
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0answers
104 views

Evaluating $\int_0^{\infty} \frac{\sqrt{x}}{x^3+1} \mathrm{d}x$ with contour integration

This integral has stumped me for quite a bit. $$\int_0^{\infty} \frac{\sqrt{x}}{x^3+1} \mathrm{d}x$$ I have identified poles at $x=e^{-\frac{i\pi}{3}}, e^{\frac{i\pi}{3}}, -1$. Edit: I have changed ...
1
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2answers
64 views

Parameterizing divots in contour integrals

I have recently attempted to pick up complex analysis and have been stuck on this problem for a few days: $$\int_{-\infty}^\infty \frac{\sin^2x}{x^2}\mathrm{d}x$$ Fortunately, it would seem that ...
2
votes
1answer
53 views

Compute $I(r) := \int_{C[-2i,r]}\frac{dz}{z^2+1}$ for $r\neq 1,3$

The title says it all. I'm trying to calculate $I(r) := \int_{C[-2i,r]}\frac{dz}{z^2+1}$ for $r\neq 1,3$, but I need some help. So we start by looking at three different cases, namely when $r<1$, $...
4
votes
1answer
109 views

Evaluating complex integral $\int_{0}^{\pi} \frac {x \sin x}{1+a^2-2a(\cos x)} $ via different contour

I got an complex integral $\int_{0}^{\pi} \frac {x \sin > x}{1+a^2-2a(\cos x)} $ for $a \ge 1$ and my given contour is a rectangle such that $|Re(z)|\le \pi$ and $0 \le |Im(z)| \le h \to \infty$....
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2answers
27 views

Right contour for integrating goniometric function with the $x^n$ as an argument

How would you integrate: $\int_0^\infty \sin (x^n) \,dx$ $\;$for $n \gt 1$ I mean the result is via gamma function and there exists a formula for that gamma function but I struggled with the rooting ...
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1answer
27 views

Proving pathwise independence of integral of f along any path is equivalent to Cauchy's integral theorem

Let $f: U \subset \mathbb{C}$ be analytic and $\gamma_1, \gamma_2$ be two arbitrary closed paths. Prove that: $ \int_{\gamma} f(z) \ dz = 0$ for any closed path $ \iff\int_{\gamma_1} f(z) \ dz = \...
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2answers
54 views

Help with Mellin-Barnes Integral (product of two Hypergeometrics)

I am trying to prove that $$\int_0^1 \frac{dz}{z^2} z^{h}\cdot {}_{2}F_{1}(h,h;2h;z) \cdot {}_{2}F_{1}\left(\frac{1+2a}{2},\frac{1-2a}{2};1;\frac{z-1}{z}\right) = -\frac{\Gamma(2h)}{\Gamma{(h)}^2} \...
4
votes
2answers
122 views

Evaluating the integral $\int_{0}^{\infty} \frac{\exp(-u^2)}{1+u^2} \, du$

I am trying to calculate the following integral $$ \int_{0}^{\infty} \frac{\exp(-u^2)}{1+u^2} \, du. $$ Wolfram gives a beautiful analytical answer: ${\rm e}\pi\operatorname{erfc}(1)$. I've tried ...
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0answers
15 views

Contour Integrals involving Log(-x), where does negative sign go?

\begin{align} -i \pi^2 &= \left( \int_R + \int_M + \int_N + \int_r \right) f(z) \, dz \\ &= \left( \int_M + \int_N \right) f(z)\, dz && \int_R, \int_r \mbox{ vanish} \\ &=-\int_\...
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1answer
58 views

Integration of cotangent with complex calculus

I am supposed to integrate the function $f(t^{\prime})=cot(2 w t^{\prime})$ by taking into account the singularities at $t^{\prime}=\frac{n\pi}{2w}$ via complex integration. It is quite easy to ...
3
votes
2answers
33 views

Evaluate $\int_\gamma z \ \Im(z^2) \ dz$

I am trying to find $$\int_\gamma z\ \Im(z^2) \ dz,$$ where $\gamma$ is the unit circle traversed once, anticlockwise. My attempt: let $\gamma(t)=e^{it}\implies \gamma'(t)=ie^{it} \ \ \ \ t\in[0,2\...
1
vote
2answers
86 views

Contour Integral of irrational polynomial from -1 to 1

I've been stuck at htis contour integral problem for a few hours now, and seem to be hitting brick walls. $$ \int_{-1}^1 \frac{\sqrt{1-x^2}}{1+x^4}dx\,, $$ I tried a trig substitution $x=\cos{\theta}...
1
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1answer
63 views

Evaluate the integrals $\int_{\gamma}z^{n}dz$ for all integers $n$.

Evaluate the integrals $$\int_{\gamma}z^{n}dz$$ for all integers $n$. Here $\gamma$ is any circle centered at the origin with the positive (counterclockwise) orientation. I don't know how to ...
0
votes
1answer
58 views

Looking for the value of $\int_{-\infty }^{b} {2(y^2)^a}(e^{2(b-y)}-1)^{-1/2} \, dy$ [closed]

I'm trying to calculate the following definite integral $$\int_{-\infty }^{b} \frac{2(y^2)^a}{\sqrt{e^{2(b-y)}-1}} \, dy=\int_0^{\infty}\frac{\left(\left(b-\frac{y}{2}\right)^2\right)^a}{\sqrt{e^y-1}}...
7
votes
2answers
214 views

About the proof that $\int_0^\infty\frac{dx}{x^2+6x+8} =\frac12\log2$ via residue formula

In the text "Functions of one Complex Variable" by Robert E.Greene and Steven G.Krantz is my understanding of the proof to $\text{Proposition (1.1)}$ correct ? $\text{Proposition (1.1)}$ $$\...
4
votes
6answers
149 views

Prove that $\displaystyle\int_0^1\,\frac{\ln(x)}{\sqrt{1-x^2}}\,\text{d}x=-\frac{\pi}{2}\,\ln(2)$.

I have discovered via contour integration that $$\int_0^\infty\,\frac{\exp(t\,u)}{\exp(u)+1}\,\text{d}u={\text{csc}(\pi\,t)}\,\left(\frac{\pi}{2}-\int_0^{\frac{\pi}{2}}\,\frac{\sin\big((1-2t)\,y\big)}{...
1
vote
1answer
31 views

Arguments/phases due to branch cut for following keyhole contour integral?

I was given the a contour integral problem which I am rather unsure about. First, I was asked to classify the singularities and required branch cuts for $$f(z)=\frac{1}{z^3}\log[(1-z)]^2.$$ I know ...
2
votes
3answers
62 views

Integral of $\int_{-\infty}^{\infty} \frac {dk}{ik+1}$

I came across this integral today in the context of inverse fourier transforms: $$ R(x)={1 \over 2\pi}\int_{-\infty}^{\infty} \frac {e^{ik(x-1)}}{ik+1}dk$$ I know the solution is supposed to be $$ ...