Questions tagged [contour-integration]

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

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$\int_{0}^{\infty} \frac {\sin x}{x}\ln x~dx$ using contour integration [duplicate]

I was able to calculate this using Leibniz rule or Feynman's technique and the result is $\frac{-\gamma\pi}{2}$ I used a semi circular disc punctured at $0$,outer arc has radius $\beta$ and inner arc ...
Madhav Asthana's user avatar
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calculating inverse fourier transform of $\cosh(t\sqrt{1-k^2})$

So i am trying to calculate the following integral \begin{equation} \int_{-\infty}^{\infty}e^{-ikx}\cosh(t\sqrt{1-k^2})dk \end{equation} I think this should be related to modified Bessel function of ...
Sourabh's user avatar
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Evaluating integrals in a non simply connected domain over an arbitrary contour given a function

Disclaimer: I know the Residue Theorem could be used for this, but we cannot use it as we have not proven it in class. In my complex analysis course, I am given a function $f(z)=\frac{1}{z^2-z}$ and a ...
Luk'yan Vilshansky's user avatar
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When working with multiple branch cuts, is there a way to chose the arguments so that log of a product can be opened as sum of individual logarithms?

Suppose a function $\eta (z)=log(\psi (z))$ where $$\psi (z)=\prod_{k=1}^{n} \left(z-z_k\right)$$ We know that $log(z)=log|z|+i(argz)$, this implies that $$log(\prod_{k=1}^{n} \left(z-z_k\right))=\...
Madhav Asthana's user avatar
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How helpful is $f^{-1}(z)=\frac1{2\pi i}\oint\ln(1-\frac z{f(w)})dw$, or the method to find it, in deriving integral representations of $f^{-1}(z)$?

$\DeclareMathOperator \erf{erf}$ Wolfram Alpha gives the following $\erf^{-1}(z)$ series: $$\sum_{n=1}^\infty\frac{z^n}{2\pi n}\int_0^{2\pi}e^{it}\erf(e^{it})^{-n}dt$$ which can be derived via ...
Тyma Gaidash's user avatar
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evaluate $\displaystyle \int_0^{ \infty } \frac{\log(x^{2}+1)}{x^{4}+1}dx$ using contour integration

I started off with this contour(I apologize for the software i used but I am new to this all). i used this contour https://dochub.com/m/shared-document/madhavasthana/1XEpyxzwN5JN1vZVQZGd38/null-1-png?...
Madhav Asthana's user avatar
4 votes
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548 views
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Contour Integrating

$$ I = \int_{0}^{\infty} \frac{\sqrt{2} \left( e^{-2vz - 2b\sinh(z)}\cos\left(\pi\left(\frac{1}{4} + v\right)\right) + e^{2vz - 2b\sinh(z)}\sin\left(\pi\left(\frac{1}{4} + v\right)\right) \right)}{\...
Marek's user avatar
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Strange algebra among elliptic moments $k^pK^qK^\prime{}^r$

Let us define the complete elliptic integral of the first kind as follows: $$ K(k)=\int_{0}^{1} \frac{1}{\sqrt{1-t^2}\sqrt{1-k^2t^2} } \text{d}t, $$ where we restrict the modulus $k$ to $|k|<1$ ...
Setness Ramesory's user avatar
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Proving the identity $\frac{1}{\sqrt{(x-a)^{2} + (y-b)^2 + (z-c)^2}} = \frac{1}{2\pi} \int_{0}^{2\pi}\frac{du}{(z-c) + i(x-a)\cos(u) + i(y-b)\sin(u)}$

$$\frac{1}{\sqrt{(x-a)^{2} + (y-b)^2 + (z-c)^2}} = \frac{1}{2\pi} \int_{0}^{2\pi}\frac{du}{(z-c) + i(x-a)\cos(u) + i(y-b)\sin(u)}$$ I am trying to understand the first part of E.T Whittaker's "On ...
Ajay Srinivasan's user avatar
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How to choose the right contour in complex integration?

I'm having some conceptual difficulty. In evaluating integrals via contour integration, the choice of the right contour seems rather like trial-and-error. For example in the integration of cosh(ax)/...
Aniruddha Bhattacharya's user avatar
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Can real integrals that are computed using complex contour integration depend on the choice of contour?

I am studying the following integral: $$\int_{-\infty}^\infty \frac{1}{2\pi i} \frac{-e^{-ix(a-b)}}{x^2 - c^2} dx$$ where $a, b, c > 0$ are real constants. The integrand has poles along the real ...
CBBAM's user avatar
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Evaluating $\int_{-\infty}^\infty \frac{pe^{ipr}}{\sqrt{p^2 + m^2}} dp$ using complex contours

Consider the following integral: $$\int_{-\infty}^\infty \frac{pe^{ipr}}{\sqrt{p^2 + m^2}} dp$$ where $r, m$ are positive constants. This integral appears in a quantum field theory textbook by Peskin &...
CBBAM's user avatar
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Integral via Contour Integration involving many-valued functions

In E.G.Phillips Functions of a complex variable it is stated that for an integral of the form $\int_{0}^{\infty}x^{a-1}Q(x)dx$ where $a \in \mathbb{R}$ is not necessarily an integer, the contour to be ...
Claudio Menchinelli's user avatar
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Integral representation for reverse Bessel polynomials

$\DeclareMathOperator ZZ$Series solutions to partly invert $e^x(x^2+a)$ and $e^xx(x+a)$ exist when more general quadratic-exponential equations are reduced. Using the reverse Bessel polynomials $p_n(x)...
Тyma Gaidash's user avatar
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Evaluation of Integral via the Residue Theorem

I was trying to solve some complex integrals via Contour Integration and found myself stuck with the following exercise: $$ I = \oint_{\gamma}z\sin\left(\frac{1+z}{1-z}\right)\mathrm{d}z,\,\gamma = ...
Claudio Menchinelli's user avatar
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If all zeros of complex polynomial (of deg. > 1) are inside a circle of radius R, then $\oint\limits_{C }\frac{1}{P(z)}\,\mathrm{d}z = 0$ [duplicate]

I need to answer this without using Residue Theorem, Maximum Modulus Principle or more advanced theorems, but rather rely on more basic results such as Cauchy's formulas for integration of analytic ...
giorgio's user avatar
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Small Question regarding contour integration of $\int_{1=|z-i|} \frac {1}{z^2+1}dz$

I am not certain if my Process of using Cauchy's Theorem is sound. $$ \int_{1=|z-i|} \frac {1}{z^2+1}dz = \int_{1=|z-i|} \frac {1}{(z-i)(z+i)}dz = \int_{1=|z-i|} \frac {\frac {1}{z-1}}{(z+i)}dz $$ ...
Pascal's user avatar
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Does Cauchy's integral formula generalize to non-analytic functions?

Cauchy's integral formula states that if the complex function $f(z)$ is analytic on a closed domain $D$ of the complex plane and $a$ is in the interior of $D$, then $$f(a) = \frac{1}{2 \pi i}\oint_{\...
tparker's user avatar
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What is the relationship between the two versions of the Sokotski-Plemelj theorem?

Wikipedia gives the follow statement of the general Sokhotski-Plemelj theorem: Let $C$ be a smooth closed simple curve in the plane, and $\varphi$ a complex-analytic function on $C$. Define $$ \phi_i(...
tparker's user avatar
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3 votes
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176 views

Fourier Transform of Gaussian with imaginary pole

I'm struggling at the moment with the following integral, which is essentially an inverse Fourier transform of a Gaussian with a single pole on the imaginary axis: $$I(Q) = \frac{1}{2}\int_{-\infty}^{\...
hedlejo's user avatar
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Fourier transformation with a square root-log term

(Note: I posted the exact same question in the physics StackExchange, but to get a breadth of people looking at the problem, I am coming to the Math StackExchange also since, well, it is just an ...
MathZilla's user avatar
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What is the correct technique to integrate logarithms on paths on the complex plane? [duplicate]

I will use the following integral as a dummy exercise. \begin{equation} \int_0^{+\infty} \frac{\ln x}{(1+x^2)^2} dx \end{equation} I thought that the correct way to compute this kind of integrals (...
propriofede's user avatar
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I need help evaluating the integral $\int_{-\infty}^{\infty} \frac{\log(1+e^{-z})}{1+e^{-z}}dz$

I was playing around with the integral: $$\int_{-\infty}^{\infty} \frac{\log(1+e^{-z})}{1+e^{-z}}dz$$ I couldn't find a way of solving it, but I used WolframAlpha to find that the integral evaluated ...
Abdullah's user avatar
2 votes
1 answer
163 views

Evaluating $\int_{0}^{\infty}{\frac{e^x}{x!} dx}$

I was trying to evaluate $\int_{0}^{\infty}{\frac{e^x}{x!} dx}$ or approximate it (WFA only gives an approximate result so maybe there is no closed form). I tried the Fourier transform : $$\mathcal{F}\...
AnthonyML's user avatar
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Contour Integral: $\int_{c} e^{(z+1/z)} dz$ where $c$ is a unit circle.

From Cauchy's Residue theorem, I know that the value of integral is equal to: $$2\pi*i*\sum{R}$$ where $R$=Residues of poles inside the contour. And $z_0=0$ being the only residue inside the unit ...
санкет мхаске's user avatar
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1 answer
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Contour Integral involving a branch of $1 / \sqrt{z}$

I was asked the following question involving computing contour integral for a certain branch of the complex function $1 / \sqrt{z}$. This is mainly about testing our understanding of different ...
giorgio's user avatar
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1 answer
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Contour integral of $\sqrt{\frac{z}{z-1}}$

Which Laurent series could be used to solve $$\oint_{|z|=2}\sqrt{\dfrac{z}{z-1}}dz$$ if it has a branch cut at $y=0$, $x\in(0,1)$? I thought maybe it couldn't be solved by residue theorem because ...
Joan S. Guillamet F.'s user avatar
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Contour integral with finite limits

I have the following integral $$I=\int_a^\infty \frac{dx}{x^2-\frac{1}{x^2}},$$ where $a>0$. This integral contains a pole at $x=1$, and I rewrite the integral as $$\int_a^\infty \frac{x^2}{x^4-1}...
furious.neutrino's user avatar
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Convert a definite integral to a contour integral

There is an integral equation that shows up in scattering problems in electromagnetics where the kernel looks like this: $$G(z, z') = \frac{1}{8\pi^2} \int_0^{2\pi}d\phi' g\left(z, z', \phi'\right)$$ ...
CHNE0569 CHNE0569's user avatar
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Understand the geodesic, $\gamma$, induced by an abelian form $\omega$ and their integral

in my phd thesis, informaly I use this result: Let $(A_i,\omega_i)$, $i=1,2$, where $A_i$ is an annulus and $\omega_i$ is an abelian differential form with a translation structure on $A_i$ that ...
Framate's user avatar
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2 votes
1 answer
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Integrating a Modified Bessel function of the second kind with a singularity

Does someone know how to handle the integral $$\int_{-\infty}^{\infty} \frac{K_0\!\left(\lvert \tau \rvert \sqrt{q^2 \alpha ^2}\right)}{q^2}\cos (q x)\,\mathrm{d}q $$ $\alpha$ is a real number and $\...
Roeland van den Wildenberg's user avatar
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104 views

Integral $\int_0^\infty \frac{\cos(x)}{(2x-\pi)(x^2+\pi^2)} dx$ using semicircle contour

I’ve tried many approaches to try and solve the following integral (replacing $\cos(x)$ with $e^{ix}$, applying Feynman’s trick and much more). $$I := \int_0^\infty \frac{\cos(x)}{(2x-\pi)(x^2+\pi^2)} ...
Jan's user avatar
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$\int_{|z| = 2 }\frac{4z^7-1}{z^8-2z+1} dz= ?$

Problem: Compute $\int_{|z|=2} \frac{4z^7 - 1}{z^8 - 2z + 1}dz$. Attempt: I notice that if $w = z^8 - 2z + 1$ then $dw = 2 (4z^8 - 1) dz$ so $$I=\int_{|z|=2} \frac{4z^7 - 1}{z^8 - 2z + 1}dz = \frac{1}{...
mathlover314's user avatar
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Contour integration with polylogarithm

Starting from Bose-Einstein integral representation of the polylogarithm $$Li_{s}(z) = \frac{1}{\Gamma(s)}\int_{0}^{\infty}\frac{t^{s-1}}{e^t/z - 1}dt \quad \quad(1)$$ it's not too hard to obtain the ...
serpens's user avatar
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2 votes
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Continuity of a function defined by an improper integral

Let $c > 0$ and let the function $f : (0, \infty) \to \mathbb{C}$ be defined as $$ f(y) = \int_{c - i\infty}^{c+i\infty} \frac{y^s}{s(s+1)} \, ds. $$ I want to show that $f$ is continuous. My ...
Epsilon-Delta's user avatar
1 vote
2 answers
171 views

Solving a contour integration

I am struggling with the following integration, $$ I = \int_{-\pi}^{\pi}\frac{e^{i(a+n)(\theta-i\eta)}}{i(\theta -i\eta)}d\theta $$ where $\eta > 0$, and $i$ is the imaginary symbol, and $a, n$ are ...
user123's user avatar
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Integrating the same integral in two ways

I need to calculate the integral of $\int_{-1}^{+1}|z| dz$ in two ways: integrating along a line integrating along the arc of unit radius circle However, I have some trouble coming to the right ...
Arbatus's user avatar
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3 votes
2 answers
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Solving trigonometric integral using contour integration [duplicate]

How to solve this integral using contour integration:$$\int_{-\infty}^{\infty} \frac{\cos px - \cos qx}{x^2} dx \stackrel{?}{=} \pi (q-p)$$ Here is what I have tried: I don't know if I can use Residue ...
gujaral's user avatar
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2 votes
2 answers
183 views

Solving integral with residues

I have to solve the integral $\int_{-∞}^\infty \frac{cos(x)dx}{(x^2+a^2)(x^2+b^2)}$ using residues. Here's my attempt to the point where I am stuck: Noting that $\cos(x)=\text{Re}(e^{ix})$. We can ...
Arbatus's user avatar
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Relationship between Cylindrical and Spherical Hankel Functions via Integration

I am looking for the cylindrical wave expansions of spherical waves. For example, for zeroth order we have the Sommerfeld identity: $\frac{e^{jkr}}{r}=\frac{j}{2}\int_{-\infty}^{\infty}\frac{e^{jk_z|z|...
M Hossain's user avatar
2 votes
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57 views

Closed form expression of the countour integral of Gamma functions combined with a Gaussian hypergeometric function?

Is there any way of writing the following contour integral as a closed form expression ? $$ \int_{-i \infty}^{+i \infty} dp \int_{-i \infty}^{+i \infty} ds \Gamma(d/2 + s + p -1) \Gamma(-\Delta - s - ...
NoName's user avatar
  • 31
-4 votes
1 answer
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Arnold's Trivium problem 69

Does anyone have solution for Arnold's Trivium problem #69? Prove that the solid angle based on a given closed contour is a function of the vertex of the angle that is harmonic outside of the contour.
Ketchounez's user avatar
2 votes
3 answers
168 views

Finding General Formulas for $\int_0^{2\pi} \frac{1}{a+b\cos(x)}$ and $\int_0^{2\pi} \frac{1}{a+b\sin(x)}$

Hello Math Stack Exchange Community, I have a quick question: How would I find the general formulas for $\int_0^{2\pi} \frac{1}{a+b\cos(x)}\, dx$ and $\int_0^{2\pi} \frac{1}{a+b\sin(x)} \, dx$ using ...
Sid Meka's user avatar
2 votes
0 answers
118 views

How to evaluate contour integral $z^i $ over a unit circle

How to evaluate $$\oint_{C}z^idz $$ over a unit circle in complex plane. I tried following. t = angle theta $$z = e^{it}$$ $$z^i = i \operatorname{Log} z = i \operatorname{Log} (e^{it})= i i t= - t$$ $...
user35553's user avatar
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0 answers
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Integrating along the closed contour crossing the branch cut

I would like to evaluate the following simple integral with complicated contour. $$ \lim_{r\rightarrow 0} \oint_{C (r)}dz~F(z) ,\quad F(z):=\left(\sqrt{z+m}+\sqrt{z-m}\right), \quad m>0 $$ where ...
user239970's user avatar
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0 answers
64 views

Analytic structure of $ \int_{0}^{1} dx [m^{2} - x(1-x)p^{2}]^{\epsilon} $

I have the following equation for which I am looking to describe its analytic structure: $$ \int_{0}^{1} dx [m^{2} - x(1-x)p^{2}]^{\epsilon} $$ where $p^{2} > 0, m^{2} > 0,$ and $|\epsilon| < ...
Jack's user avatar
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1 vote
0 answers
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Application of Cauchy residue theorem to Matsubara sums in physics

In quantum field theory (specifically when calculating free fermionic propagators via coherent state path integral), we encounter the following sum: $$I_{\beta}(\tau,\tau')\mathrel{\mathop:}= \frac{1}{...
Jamin's user avatar
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0 answers
36 views

Singularities inside closed path

Why, when calculating contour integrals, we need to avoid the singularities inside the region delimited by the closed path we're integrating over? Wouldn't we need to care only about singularities ...
Joan S. Guillamet F.'s user avatar
2 votes
1 answer
81 views

Is my solution correct? $\oint_C \frac{5\sec 2z}{e^{-3z}-1}\,\mathrm{d}z$ where $C:|z|=10^{-4}$

Problem.src) Compute $$\oint_C \frac{5\sec 2z}{e^{-3z}-1}\,\mathrm{d}z,$$ where $C:|z|=10^{-4}$. I solved the problem like this, but I don't know if this is the right way $$\oint_C \frac{5\sec 2z}{e^...
pleaseiwanttograduate's user avatar
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0 answers
106 views

Show that $\frac{1}{2\pi i}\int_{\gamma_p}z\frac{h'(z)}{h(z)}dz$ is in $\mathbb{Z}+\mathbb{Z}\tau$

This is the problem IV.3.F I found at page 127 of the book "Algebraic Curves and Riemann Surfaces" of Rick Miranda. Let $\tau \in \mathbb{C}$ such that $Im(\tau)>0$ and define the lattice ...
100nanoFarad's user avatar

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