Questions tagged [contour-integration]

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

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54 views

Computing $\int_{-\infty}^{+\infty} e^{-iky} e^{-c|k|^{\alpha}} |k|^{\beta} dk$

I want to solve the above integral, where $c >0$, $\beta = 0,2$, $0<\alpha<2 $, $y \in \mathbb{R}$, $k \in \mathbb{R}$. I tried to use the Residue Theorem adding the semi-circle contour of ...
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2answers
84 views

Integrate $\frac{\log(x^2+4)}{(x^2+1)^2}$.

Using residue calculus show that $$\int_0^{\infty}\frac{\log(x^2+4)}{(x^2+1)^2}dx=\frac{\pi}2\log 3-\frac{\pi}6.$$ I was thinking of using some keyhole or semi-circular contour here. But the problem ...
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0answers
48 views

On evaluating $\rho(s)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\Gamma(z)P(sz)\,dz.$

How do you evaluate the following integral? $$\rho(s)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\Gamma(z)P(sz)\,dz.$$ $P(\cdot)$ is the prime zeta function. I got the integral from taking $\rho(s)=...
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1answer
17 views

Breakdown of Inverse Transform from Contour Integral at short times

I need to invert a Laplace transform to obtain the temperature variation in a particular problem in time and (1-D) space. In the frequency domain, I have $$F(s)=\frac{1}{s}e^{H(s)\xi}$$ where $$H(s)=\...
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1answer
52 views

How to obtain the equality between two integrals with different integrands?

I am looking into the proof of Ramanujan's approximation formula for the partition function $p(n)$ by Stein and Shakrachi. I am confused about one step towards the end of the proof just before we ...
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27 views

A partial converse of Cauchy's theorem

Let $V\subset \mathbb{C}$ be open and $f\colon V\to \mathbb{C}$ a continuous function, and assume $$ \int_{\gamma} f(z) \, dz =0 $$ for any closed contour $\gamma \in V.$ We need to show that $f$ has ...
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23 views

complex integral of a logarithm in a square contour [closed]

Can I choose any branch of logarithm to use the Cauchy theorem to do this exercise or exist some branch where the Cauchy theorem is not valid? $$ \int_{C} \frac{\log(z-2i)}{z+2} dz$$ and $C$ is the ...
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0answers
69 views

$\zeta( \frac{z}{z-1} )$ has infinitely many zeros when |z|=1. [duplicate]

Evaluate $$ \oint_{\left\vert z\right\vert\ =\ 1} \log\left(\left\vert \zeta\left(\frac{z}{z - 1}\right)\right\vert\right) \,{\mathrm{d}z \over z} $$ where $\zeta(s)$ is the analytic continuation of ...
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1answer
63 views

Evaluate$ \oint _{|z|=1} \frac{\log\ |1-z|}{z}dz $

Evaluate $$ \oint _{|z|=1} \frac{\log\ |1-z|}{z}dz $$ My Attempt $$ I=\oint _{|z|=1} \frac{\log\ |1-z|}{z}dz $$ $$z=e^{i\theta} \Rightarrow dz =i e^{i\theta}d\theta$$ $$I=i \int_{0}^{2\pi} \log\ |1-e^...
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Compute $\int_{\gamma} f$

Let $\gamma_1 = S_1 + L - S_2 - L$ and $\gamma_2 = S_1 + L + S_2 - L$, $$S_1(t) = e^{it} , t\in [0,2\pi] $$ $$S_2(t) = 2e^{it} , t\in [0,2\pi] $$ $$ L = [1,2] $$ Let $f(z) = (\cos z)/z$. By writing ...
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1answer
45 views

Find all the possible values of $\int_{\gamma} \frac{1}{z^2+1}dz$

a) Let $D = \{z\in\mathbb{C}: z \neq \pm i\}$ and let $\gamma$ be a closed contour in D. Find all the possible values of $\int_{\gamma} \frac{1}{z^2+1}dz$. b) If $\sigma$ is a contour from 0 to 1,, ...
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Writing the representation of Inverse Laplace Transform with Trapezoidal's Rule.

Related with my previous question, i want to ask about how to represents the numerical inverse laplace transform using Trapezoidal's rule. Please see here to revisit my previous question: Writing The ...
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1answer
26 views

Are contours allowed to contain branch points / branch cuts on the contour (not inside)

I just wanted to verify something, since I saw some notes online that confused me. They were doing some contour integral and $0$ was the only branch point of the function, and they chose the branch ...
4
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51 views

Writing the Real Part of Complex Integrand

I'm having a hard time to understand how's Eq. $(6.73)$ become Eq. $(6.75)$. It's taken from Numerical Methods for Laplace Transform Inversion by Cohen. Here's the problem: [...]. The basis of their ...
4
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1answer
65 views

Integrating $\log(-ix)\exp(-ix)/x^2$

I would like to compute a few integrals like $$\int_{-\infty}^\infty\frac{\log(-ix)\exp(-ix)}{x^2}\,dx$$ To be clear, here the path of integration is really $z = \epsilon i + x$, so that it avoids the ...
2
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2answers
96 views

Matsubara sum arising from QFT and contour integral

In the lecture of E. Fradkin on quantum field theory, an example of Matsubara sum is performed using contour integration (see eq. 5.214 in the lecture). It reads $$ \sum_{n=-\infty}^{\infty} \frac{e^{...
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1answer
64 views

Prove that $\oint_{|z|=1}f(z)dz = \oint_{|w|=1}f(w)dw $ [closed]

Prove that$$\oint_{|z|=1}f(z)dz = \oint_{|w|=1}f(w)dw $$ My try- $$I= \oint_{|z|=1}f(z)dz$$ Substitute $z=w$ $\Rightarrow $ $dz=dw$ $$I= \oint_{|w|=1}f(w)dw$$
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Using Stokes theorem on a discontinuous function

I've a vector function whose curl is well defined inside and outside,except on the boundary $\mathbf{\nabla} \times \mathbf{H}=\mathbf{J}_{f}$ I'm interested in using stokes theorem across the ...
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40 views

Real integral with inverse square root equals contour integral

I want to prove that $$ \oint_C \frac{1}{(\zeta-z)\sqrt{\zeta^2-a^2}}d\zeta = 2i \int_{-a}^a \frac{1}{(\zeta-z)\sqrt{\zeta^2-a^2}} d\zeta,$$ where $C$ is a closed contour such that $[-a,a]$ is inside ...
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Evaluating $\int_{-\infty}^{+\infty} dx \, e^{i x t a} f(x) f(x-t b)$, where $f(x) = \frac{1}{\sqrt{1+x^2}}$, with $t$, $a$, $b$ real

I am trying to compute $$\int_{-\infty}^{+\infty} dx \, e^{i x t a} f(x) f(x-t b)$$ where $f(x) = \frac{1}{\sqrt{1+x^2}}$ and $t, a, b \in \mathbb{R}$. Mathematica for instance doesn't know how to do ...
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1answer
37 views

I'm using the formula below and want to find the length of γ

I want to find the length of the smooth curve $γ(t)$ from $t = 0$ to $t = \pi/4$ I'm using the formula for γ: $$l(γ) = \int_a^b|\gamma'(t)|\,dt$$ apologies I can't work out how to format that ...
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47 views

Contour integral $\oint \frac{\mathrm{d}z}{2 \pi i z} \log(w+z)$

Let $\log:\mathbb{C} \to \mathbb{R} \times i (-\pi,\pi]$ denote the principal branch of the complex logarithm. For $\omega \in \mathbb{C}$, I would like to calculate the contour integral \begin{...
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1answer
71 views

An integral representation of the Mittag-Leffler function

I met the following interesting identity of the Mittag-Leffler function: $$E_\beta (-\lambda t^\beta)=\frac{\lambda}{\pi}\int_0^\infty e^{-t r}r^{\beta-1} \frac{\sin(\beta \pi)}{(r^\beta\cos(\beta \pi)...
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53 views

Compute $\int_0^{2\pi} \frac{\cos \theta}{5+4\cos\theta}\,d\theta$ using contour integration

This question was asked in my complex analysis quiz and I was unable to solve it there. Show that $\displaystyle \int_0^{2\pi}\frac{\cos (\theta)}{5+ 4\cos(\theta)}\,d\theta = -\frac{\pi}{3} $. 5 + ...
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Calculate the line integral $\oint_{C} \frac{2z^3+z^2+4}{z^4+4z^2}dz$, where $C$ is the circle $|z-2|=4$ clockwise watch.

$\oint_{C} \frac{2z^3+z^2+4}{z^4+4z^2}$ When using partial fractions I have $\frac{2z^3+z^2+4}{z^4+4z^2}$=$\frac{1}{z^2}+\frac{2z}{z^2+4}$ Then $\oint_{C} \frac{2z^3+z^2+4}{z^4+4z^2}dz=\oint_{C}\frac{...
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81 views

Evaluation of $\zeta$ for integer values

I have recently found that $\zeta(2)$ can be found by integrating $\frac{\log(1+x)}{x}$ from $-1$ to $1$. Since $$ \int_{-1}^{1}\frac{\log(1+x)}{x}dx=\int_{-1}^{1}(1-\frac{x}{2}+\frac{x^2}{3}-\frac{x^...
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1answer
21 views

parametrization of path in contour integral and substitution

I just started studying complex analysis, and we just covered the basics of complex differentiable functions and the technical aspects of performing a contour integral. I thought the theory made ...
2
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2answers
48 views

Find $\int_{\gamma}\frac{(\sin z)^2}{(z-z_0)^2}dz$

Find $\int_{\gamma}\frac{(\sin z)^2}{(z-z_0)^2}dz$ where $\gamma$ is a linear pieces parameterization of closed polygonal chain $[w_0,w_1,w_2,w_3,w_0]$ with vertices $w_0=1+i, w_1=1-i, w_2=-1+i,w_3=2$....
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1answer
37 views

Contour integral of absolute value of a function

How is the contour integral with absolute value performed? Here $$\mathfrak{I}=\int_{C_1} \frac{a|z|}{z-\gamma}\mathrm{d}z$$ where the contour $C_1$ is parallel to the real line but passes above the ...
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69 views

Complex Contour Integral with tan z

I would like to compute the below integral $$I = \int_{S_n}\frac{1}{z + \tan{z} - i}dz$$ where $S_n$ is the square with vertices $n\pi(1+i), n\pi(1-i), n\pi(-1+i), n\pi(-1-i)$. If we consider $f(z) = ...
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2answers
49 views

Integrating $\int_{0}^{\infty} x^{-k} e^{-a(x+d)^2} dx$ where $k \in \mathbb{Z}_{+}$

I want to solve the following integral: $$I = \int_{0}^{\infty} x^{-k} e^{-a(x+d)^2} dx$$ where $k \in \mathbb{Z}_{+}$, and $a, d \in \mathbb{R}_{++}$. When $k =0$, the integral is a standard ...
3
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2answers
78 views

Help integrating the contour integral $\oint_C \frac{e^{\frac{1}{z}}}{z(1-qz)}dz$ around the unit circle

Consider the integral $$\oint_C \frac{e^{\frac{1}{z}}}{z(1-qz)}dz$$ where C is the anti-clockwise oriented unit circle and q is a complex constant. I have no clue how to integrate this. I tried using ...
2
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1answer
59 views

Logarithmic integral (Green function)

I am using a Green function to find the solution of an ODE and I have stuck in finding the solution of the following integral. Could someone please help me with this? The integral is $$\int_{0}^{r} d ...
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48 views

Approaches to analysis of contour integrals - showing the integral is positive

Let $F:\mathbf{C}\to\mathbf{C}$ be an analytic function whose restriction to $\mathbf{R}$ is positive and convex. Also assume $$\int_{\mathbb{R}}\vert{e^{F(i\theta)}}\vert \;d\theta < \infty\ \ \ \...
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2answers
126 views

Is it possible to evaluate the integral $I=\int_0^\infty \frac{\sqrt{x}\arctan(x)}{1+x^2}dx$ with residue theorem?

let's consider the following integral $$I=\int_0^\infty \frac{\sqrt{x}\arctan(x)}{1+x^2}dx$$ I know exactly how to evaluate it with sneaky methods and obtained $I=\frac{\pi\sqrt{2}}{8}(\pi+2\ln(2))$, ...
4
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1answer
65 views

Parametrization of Contour $C$

For the function $f(z)=1$ $(z\in \mathbb{C})$ and C is an arbitrary contour from any fixed point ${z}_{1}$ to any fixed point ${z}_{2}$ in the $z$ plane. Use parametric representations for $C$ to ...
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0answers
52 views

How are these contour integrals done?

$$\Phi_+(\alpha) = \frac{1}{2\pi i} \int_{C_1}\ln\left(2 e^{-\frac{c | z| }{2}} \cosh \left(\frac{c z}{2}\right) \right) \frac{dz}{z-\alpha}$$ and $$\Phi_-(\alpha) = - \frac{1}{2\pi i} \int_{C_2} \ln\...
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2answers
62 views

Compute $\oint_\gamma\sin z\, dz$

Compute $$\oint_\gamma\sin z\, dz$$ where $\gamma =z_0 +e^{i\theta}$ My attempt : I was thinking about residue theorem , but i don't know how to apply residue theorem and compute
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1answer
27 views

How can i calculate the contour integral of this trigonometric function?

Here's my integral $\int_0^{2\pi} \frac{cos(n\theta)}{cosha+acos\theta}d\theta$ where $|a|<1$ I tried $\int_0^{2\pi} \frac{e^{in\theta}}{cosha+acos\theta}d\theta$ but stuck here because of ...
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2answers
37 views

How can i find residues for this function?

I have a contour integral $\int_{0}^{\infty} \frac{x^3}{x^5-a^5}dx $ where $ a>0$ Here I tried to find residue, however I failed since the roots are $z=a,-(-1)^{1/5},(-1)^{2/5},-(-1)^{3/5},(-1)^{4/...
1
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2answers
77 views

A weird contour integral calculation

I have function $\int_{-1}^{1}\frac{\sqrt{1-x^2}}{1+x^2}dx$. Here to use residue thm, I rewrite the integral as $\int_{-1}^{1}\frac{\sqrt{1-z^2}}{1+z^2}dz$ with the poles $z=i$ and $-i$. However ...
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1answer
32 views

Determining the contour to use during contour integration

Let us say we want to integrate $$\int_{-\infty}^{\infty} \frac{dx}{1+x^4}$$ We do this by c contour integration of the form: $$\oint_{-\infty}^{\infty} \frac{dz}{1+z^4}$$ However my question concerns ...
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0answers
29 views

Area of Domain as.Sum of Power Series [duplicate]

Suppose that f is a one-to-one analytic function mapping the disc $|z|<1$ onto a bounded domain D. Show that the area of D is given by $$A(D)=\pi \sum_{n=1}^{\infty} n|a_n|^2$$, where $\sum_{n=0}^{\...
0
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1answer
54 views

Area of domain of a complex variable function

Suppose that f is a one-to-one analytic function mapping the disc $|z|<1$ onto a bounded domain D. Show that the area of D is given by $$A(D)=\int \int_{|z|<1} {|f'(z)|}^2 dxdy$$ This is a ...
0
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1answer
40 views

Contour integration proof explanation (term goes to 0)

Prove that $$\int_\infty^{-\infty} \frac {cos x} {e^x + e^-x} dx = \frac {\pi} {e^{\pi/2} + e^{-\pi/2}}$$. I have a question about the proof shown below: why does the third integral (the union of ...
1
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0answers
23 views

Can i use series expansion when i don't have any idea to solve ML inequality?

As already mentioned on my title i need to estimate a contour integration on a straight line and have to show the value is $0$ when $R$ goes to $\infty$. Here is my contour: I'm working on $\Psi^2$. ...
0
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0answers
14 views

Calculating $\displaystyle{ \int_{-\infty}^{\infty}\frac{\cos\left(a x\right)}{(x^{2} + 1)(x^{2} + 4)}\,{\rm d}x\, \quad}$ using complex integration [duplicate]

I want to calculate $\displaystyle{ \int_{-\infty}^{\infty}\frac{\cos\left(a x\right)}{(x^{2} + 1)(x^{2} + 4)}\,{\rm d}x\, \quad}$. I started of by rewriting the integral cos(ax) into e^{iax}+e^{-iax} ...
2
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1answer
94 views

Solving the integral of $\frac {dx} {1+x^3}$

Compute integral $$\int_0^{\infty} \frac{dx}{1+x^3}$$. I used the formula from complex variables by fisher (2.6 formula (9)) that $$\int_0^{\infty} \frac{dx}{1+x^a}=\frac {\pi} {a\sin(\pi/a)}$$ This ...
0
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1answer
28 views

Evaluate $\int_C{(z^3-e^z)dz}$ along a curve

Evaluate $\int_C{(z^3-e^z)dz}$ where C is the arc of the circle centered at $0$ from $(3,0)$ to $(0,-3)$. I tried using $\gamma(t)=3\cos(t)+3i\sin(t)$ at first, but it gets confusing. In the other ...
2
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1answer
96 views

Integrate $\int_{-\infty}^{\infty} \frac{e^{-x^2}}{(x + jp)^M (x - jp)^M } dx$

I want to solve: $$ \int_{-\infty}^{\infty} \frac{e^{-x^2}}{(x + jp)^M (x - jp)^M } dx$$ where $M \in \mathbb{Z}_{++}$ is even. To start, I create a square contour $\Gamma$ in the upper-half plane: $[-...

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