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Questions tagged [contour-integration]

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

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Changing Integration Bounds to Compute Contour Integral

So I have an Integral of the form $\int_{0}^{\infty} \frac{f(x)\frac{1 - e^{ibx}}{x} }{E - x^2} dx$ where $f(x)$ is analytic everywhere and goes to zero as $R \to \infty$ for a semicircular contour on ...
Aziz's user avatar
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Continuation of the “Lorenzified” Riemann zeta $\zeta_{L}(x)$?

I have been studying a particular function that might be described as, the “Lorenz curve of the Riemann zeta function,” and I am curious about its possible analytic continuation to the complex plane. ...
zeta space's user avatar
-1 votes
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How to compute Contour Integral Numerically [closed]

So I am trying to compute the integral from $0$ to $a$ on the real axis, shown in the picture, for a function that is completely analytic on the upper half plane except at $E + i\epsilon$ where $\...
Aziz's user avatar
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1 vote
0 answers
57 views

Using Residue Theorem for functions with removable singularities

So for an Integral of the form $\int_{-\infty}^{\infty} \frac{e^{ixa} -1}{x(x^2 + 1)} dx$. My intuition is to use complex contour integration and use a contour that is a semi-circle on the upper-half ...
Aziz's user avatar
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0 votes
1 answer
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Inconsistency in evaluating complex integral using two different approaches

I am trying to evaluate the following integral $$\int_0^\infty\frac{\cos x}{1+x^2}$$. I know that the following formula for integrals $$\int_{-\infty}^\infty f(x)dx=\pi\iota R(X-axis)+2\pi\iota R(\...
James's user avatar
  • 45
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3 answers
88 views

Definite Integral of $\int_{0}^{2\pi}\frac{d\phi}{k-(a+bcos(\phi))^2+(a+bcos(\phi))}$ with $a,b \in \mathbb{R}$ and $k\in \mathbb{C}$

I tried solving this integral introducing the unit circle parametrization on the complex plane $$ z = {\rm e}^{{\rm i}\phi}\quad\mbox{and substituting}\quad \cos\...
efe sen's user avatar
3 votes
1 answer
101 views

Contour integration of an integral

I am trying to determine the following integral by using the residue theorem $$\int_{ - \infty }^\infty {\frac{{\gamma \left( {1/2,{\rm{i}}z} \right)\gamma \left( {1/2, - {\rm{i}}z} \right)}}{{{z^2} +...
Eric's user avatar
  • 31
6 votes
3 answers
669 views

Contour Integration yielding wrong result

I've seen a lot of similar questions to the one I'd like to put forth and already read many of the answers provided, anyways I haven't been able to identify any mistakes in my computations. I was ...
Claudio's user avatar
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37 views

Writing $\ln(x+1)e^{-ax}$ in terms of Meijer-G function

Is there any way to write $f(x)=\ln(x+1)e^{-ax}$ in terms of Meijer-G function? I tried calculating Mellin transform of $f(x)$ to no avail. Frustrated, I used Mathematica to get the following answer $$...
K.K.McDonald's user avatar
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Deriving the value of the series $\sum _{n=1}^{\infty }\left[\frac{2n}{e^{2\pi n}+1}+\frac{2n-1}{e^{\left(2n-1\right)\pi }-1}\right]$

I have been working on trying to show that $\displaystyle \sum _{n=1}^{\infty }\left[\frac{2n}{e^{2\pi n}+1}+\frac{2n-1}{e^{\left(2n-1\right)\pi }-1}\right]=\frac{\varpi ^2}{4\pi ^2}-\frac{1}{8}$ ...
Camishere 45's user avatar
4 votes
1 answer
143 views

How to prove $\sum_{n=0}^\infty (-1)^n f_n=-\frac{1}{2i}\int_{c-i\infty}^{c+i\infty}\frac{f_z}{\sin(\pi z)}dz$ in the sense of Borel summation?

As the title shows, I would like to prove this identity in the sense of Borel summation, $$\sum_{n=0}^\infty (-1)^n f_n=-\frac{1}{2i}\int_{c-i\infty}^{c+i\infty}\frac{f_z}{\sin(\pi z)}dz,$$ providing ...
HC Zhang's user avatar
4 votes
0 answers
76 views

Validity of Python-derived solution for contour integral $\oint f(z)f(z-\overline{z})~dz$

$\newcommand{\on}[1]{\operatorname{#1}}$ $$ \mbox{Consider the function:}\quad \on{f}\left(z\right) = \frac{{\rm e}^{tz}}{\left(1 + z^{2}\right)^{3}}\, \left(\sqrt{t} - t\right)\ \ni\ t,z \in \mathbb{...
MASTER DHRUV's user avatar
2 votes
2 answers
132 views

What I am doing wrong in contour integration?

Note: I have edited this post after reading the comments. I am learning to integrate using contours. I tried testing this technique on following integral: $$\int_{-\infty}^{\infty} \frac{\sin^{2}x}{x^...
Prince Yadav's user avatar
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1 answer
42 views

Integral of Complex logarithm makes sense?

For example, I know that the principal branch of logarithm is not defined over negative real axis. I think the integral of this logarithm along a circle doesn’t make sense. Moreover, I know that the ...
Brody's user avatar
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2 votes
2 answers
51 views

Computing residue at infinity

Let $g$ be holomorphic function in $G = \{|z| > 100\}$ where $$g (z) = \frac{z^{99}}{\prod_{k=1}^{100} (z-k)}$$ I would like to compute $Res(g,\infty)$. By definition $$Res(g,\infty) = -Res(\frac{1}...
SparklyCape290's user avatar
6 votes
3 answers
145 views

Integral of principal value of $[\tanh(x+a)-\tanh(x+b)]/x$

How do you solve this integral involving the Cauchy principal value? $$ \mathcal{P} \int_{-\infty}^{\infty} \frac{\tanh(x+a)-\tanh(x+b)}{x} dx \\ = \int_0^\infty \frac{\tanh(x+a)+\tanh(x-a)-\tanh(x+b)-...
Bio's user avatar
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0 votes
1 answer
114 views

Calculate $\displaystyle\int_0^\infty \frac{x^2}{1+x^7} \, dx$ [duplicate]

Calculate $\displaystyle\int_0^\infty \frac{x^2}{1+x^7} \, dx$ This showed up on a complex analysis qualification exam. First, I will write $$\displaystyle\int_0^\infty \frac{x^2}{1+x^7} \, dx = \lim_{...
Grigor Hakobyan's user avatar
1 vote
0 answers
68 views

$\int _0^{\infty }\:\frac{\cos \left( 3x\right)}{\left(x^2+1\right)\cdot \left(x^2+4\right)}dx$

I know how to integrate this but I'm unable to understand why do we chose $e^{i3z}$ over $\cos(3z)$ while doing the contour integration. if someone can explain the difference between these choices and ...
Shahbaz Ahmad's user avatar
1 vote
0 answers
115 views

Contour integration with 4 or more branch points

Is there a notable example, or any for that matter, of commutator contour integration of multivalued function with $4$ or more branch points? Example of the case with 2 branch points used for integral ...
ahrvoje's user avatar
  • 27
0 votes
1 answer
43 views

How to get the sign right for branch-cut contour integration of the standard free-field propagator

(Apologies for any awkwardness. This is my very first post.) This is a question about how to get the sign right for the classic integral dealt with here Keyhole Contour with Square Root Branch Cut on ...
Alred's user avatar
  • 1
0 votes
1 answer
64 views

Computing flux integral in two ways what is my mistake?

So doing it this way is easy $ \int_D \nabla F dV $ which gives me $8/3$ But doing it via $ \int _{\delta D}F\cdot nds=\int _{\delta D}F\cdot n\left|r\left(t\right)\right|dt$ gives me troubles. After ...
user832075's user avatar
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0 answers
15 views

Question regarding terms with negative coefficients in multivariate Fox H function

I was studying a paper where an integral expression in terms of Fox H function of multiple variable were used. The definition of multivariate Fox H (extracted from appendix A-1 of Mathai-Saxena) is as ...
K.K.McDonald's user avatar
  • 3,263
6 votes
0 answers
178 views

Finding mistake in contour integral. $f(z)=\frac{\exp{(-1+i)z}}{z \cdot z^{1/2}}$

I'm trying to calculate the following real integral: $$I=2\int_{0}^{\infty}\frac{e^{-x^2}\sin(x^2)}{x^2}\mathop{\mathrm{d}x} = \int_{0}^{\infty}\frac{e^{-t}\sin{t}}{t^{3/2}}\mathop{\mathrm{d}t}.$$ I ...
Josemi's user avatar
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0 answers
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Question about integration of a meromorphic function

Let $f,g$ be two entire functions that are not polynomials, which satisfy the following: $f(0)=g(0)=1$ and $f'(0),g'(0)\neq 0$; On the real line, $\lim_{z\to+\infty} f(z)=\lim_{z\to-\infty}g(z)=+\...
Yining Wang's user avatar
  • 1,289
0 votes
1 answer
131 views

fourier transform $\frac{1}{\sqrt{1-(x+ia)^2}}$

I am trying to evaluate the Fourier Transform $$ f(z)=\int_{-\infty}^\infty dx \exp(-ixz)\frac{1}{\sqrt{1-(x+ia)^2}}\quad \mbox{where}\ a > 0 $$ Does anyone have any clues/hints at obtaining an ...
evening silver fox's user avatar
0 votes
1 answer
59 views

Approximating asymptotically the Laplace inverse of $\frac{\exp\left(-\sqrt{s^2 + 1}\right)}{\sqrt{s^2 + 1}}$ for larger t

I was trying to find the behavior of the inverse laplace transform of the function $$F(s)=\frac{\exp\left(-\sqrt{s^2 + 1}\right)}{\sqrt{s^2 + 1}}$$ for larger $t$. So basically here is my approach: I ...
MB17's user avatar
  • 173
1 vote
3 answers
173 views

Computing integral using complex analysis $\int_{-\infty}^{\infty}\frac{x}{(\sinh(x)-i)}dx$

I couldn't find related problem, so here I am. I have this integral: $$ \int_{-\infty}^{\infty}\frac{x}{\sinh(x)-i}dx=I $$ that i need to compute using complex analysis. I am given the answer is $\pi$...
Fabio I's user avatar
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0 votes
0 answers
20 views

Conditions for $ \int_{\Gamma}f = \lim_{\epsilon \to 0} \int_{\Gamma \epsilon} f $

Theorem 7.7 in Bak-Neumann's Complex Analysis book states: Suppose f is continuous in an open set D and analytic there except possibly at the points of a line segment L. Then f is analytic throughout ...
giorgio's user avatar
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1 answer
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$ \int_{-\infty}^{\infty} e^{ipx/\hbar} / (p^2 + \beta^2)\, dp $ has two different results.

I tried to evaluate the integral $$ \int_{-\infty}^{\infty}{{\rm e}^{{\rm i}px/\hbar} \over p^2 + \beta^2}\,{\rm d}p\quad \mbox{through}\ Contour\ Integration. $$ The integrand has two simple poles ...
Kalyan 's user avatar
1 vote
0 answers
52 views

Hörmander: Analyticity and wave front sets of parameter integral in existence proof for parametrix for real principal type operators

I'm trying to understand the proof in Hörmanders Analysis of Differential Operators Vol 1 for Theorem 8.3.7. (Existence of parametrix for real principal type differential operators) (1) Citation of ...
xajas's user avatar
  • 21
2 votes
1 answer
124 views

Help needed with computing a complex integral

I am having trouble calculating the integral $$ I \equiv \int_{0}^{\infty}\sin\left(r\right)r^{n - 2}\log\left(r\right)\,{\rm d}r\quad \mbox{where}\ n\ \mbox{is an ...
Βασίλης Γερμανίδης's user avatar
2 votes
1 answer
113 views

Compute $\int_{|z|=1} |z^5-1|^2 \hspace{0.1cm} |dz|$

Compute $$\int_{|z|=1} |z^5-1|^2 \hspace{0.1cm} |dz|$$ Let $z = e^{i \theta}$. Then $dz = iz \hspace{0.1cm} d\theta$ gives $|dz| = |z| |d \theta| = |e^{i \theta}| |d \theta|$ Then \begin{align} |...
Grigor Hakobyan's user avatar
5 votes
2 answers
191 views

How to solve the integral $\int_0^\infty\frac{e^{-x} \sin x}{(e^{3 x} + 1) x^{3/10}} dx$

$$ \mbox{How to solve the following integral ?}:\quad \int_{0}^{\infty}\frac{{\rm e}^{-x}\sin\left(x\right)}{\left({\rm e}^{3 x} + 1\right)x^{3/10}}{\rm d}x $$ I think it cannot be solved using ...
stephan's user avatar
  • 437
2 votes
0 answers
101 views

Contour integral $\oint_{c}\frac{1}{z \left(1-(z+1) \sqrt{\frac{r}{z}}\right)}dz$

I am new to complex analysis and am trying to figure out the following contour integral: $$ I = \oint_{\scr C}\frac{1}{z\left[1 - \left(z + 1\right) \sqrt{r/z}\,\right]}\,{\rm d}z $$ where $0<r<...
plywood98's user avatar
2 votes
1 answer
121 views

How to approach this complex integration problem

Let $\gamma(t)= e^{it}, 0\leq t \leq 6\pi$. Then the value of the integral $$\frac{1}{2\pi i} \int _ {\gamma}\frac{z^{99}}{z^{100}-1}dz$$ equals? We can't use Cauchy integral formula or residue ...
Praveen Kumaran P's user avatar
1 vote
0 answers
39 views

Convergence of integral of gamma function over hankel contour

Let ${H}$ denote hankel contour then I want to show that, $$\oint_{H}t^{z-1}e^{-t}dt$$ Converges for all $z\in\mathbb{C}$. I am familiar that this integral is used in the analytic continuation of ...
RAHUL 's user avatar
  • 1,531
1 vote
1 answer
82 views

May I find $\zeta(-1)$ using the Hankel formula for $\zeta$, but not the reflection formula?

The Reimann zeta function for $\Re(s) > 1$ is $$\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s},$$ and we may show that the analytic continuation of $\zeta$ onto $\Re(s) \leq 1, s \neq 1$ is $$\zeta(s)...
Robin's user avatar
  • 3,940
1 vote
0 answers
81 views

Singularities of $\frac{1}{1-z^n}$

I'm looking to classify the singularities of $g(z) = \frac{1}{1-z^n}$ and compute the residue at each pole. Now, $g(z)$ has singularities at the roots of unity $w^k$, where $w=\exp\{\frac{2\pi \textbf ...
CatsAndDogs's user avatar
0 votes
1 answer
81 views

Integral calculation $ \int_\gamma \frac{\sqrt{z+1/2}}{z^2+1/4}dz $ along a self-Intersecting Curve in $\mathbb{C}$

I'm trying to calculate the integral: $\gamma:[0,1]\to \mathbb{C},\gamma(t)=\sin(8\pi t)+i\cos(4\pi t)$ $$ \int_\gamma \frac{\sqrt{z+1/2}}{z^2+1/4}dz $$ where $\sqrt{\ }$ is the principle value with ...
Gao Minghao's user avatar
0 votes
0 answers
48 views

$\sum_{n=-\infty}^\infty \frac1{(n+a)^2}$ by residues

By using the theorem below, I found the sum ($a\neq 0$): $$\sum_{n=-\infty}^\infty \frac1{(n+a)^2}=-\pi Res\left[\frac{\cot\pi z}{(z+a)^2}\right]_{z=-a}=-\pi\frac d{dz}(\cot\pi z)(-a)=\frac{\pi^2}{\...
Bob Dobbs's user avatar
  • 11.9k
0 votes
1 answer
47 views

Contour integral of complex generalized function

I'm reading a derivation of the Green function for the hyperbolic PDE and got to the point where I need to evaluate the integral: $$\int_{-\infty}^{+\infty}e^{i\omega t}(\frac{1}{\omega-kc-i0}-\frac{1}...
Krum Kutsarov's user avatar
18 votes
0 answers
500 views

Estimating a complex integral for a probability problem: $HH$ is more likely to appear than $THT$ or $TTT$ in $n$ coin flips

The probability problem asks us to toss a coin $n$ times (actually $100$ times) and Alice gets a point whenever there is a $HH$ and Bob gets a point whenever there is a $THT$ or a $TTT$. Who is more ...
HackR's user avatar
  • 1,812
3 votes
1 answer
85 views

Integral over the real line of a function with a second-order pole $\int_{-\infty}^\infty \frac{e^{-(A\omega+iB)^2+C}}{\omega^2} d\omega$

I am trying to solve an integral of the form \begin{equation} \int_{-\infty}^\infty \frac{e^{-(A\omega+iB)^2+C}}{\omega^2} d\omega, \end{equation} where $A,B,C\in \mathbb{R}$, $A>0$. Attempt 1: ...
Yvonne's user avatar
  • 33
0 votes
0 answers
14 views

Indexes of nonhyperbolic equilibrium points in planar vector fields

There is a well-known theorem in dynamical systems stating that if $\gamma$ is a "sufficiently nice" closed curve (continuous, piecewise smooth, nonconstant function from $[a,b]$, say $[0,1]$...
Boris Dimitrov's user avatar
0 votes
1 answer
59 views

Principal Value Integral Using Contour Integration

$$ \mbox{The Principal Value of}\quad \int^\infty_{-\infty}\frac 1{x(x^2+1)}dx=0 $$ since the integrand is an odd function. In the complex plane, the integrand has simple poles at 0 (residue 1) at $i$...
Drooga's user avatar
  • 109
0 votes
0 answers
49 views

Is there oriented scalar field line integral, or non-oriented vector field line integral?

I'm studying complex (integral) analysis and struggling with it. BTW, this is my first math stackexchange question, so please forgive any mistakes on my part. We have definition of complex line ...
Ryu's user avatar
  • 1
2 votes
1 answer
82 views

Integration contour of $\int_0^L\frac{e^{-ikx}\sin\left(nx\right)}{\sin(x)}dx$?

As the title says, I am trying to solve the Fourier integral \begin{align} I=\int_0^L\frac{ e^{-ikx}\sin\left(n x\right)}{\sin x}dx, \end{align} where $n\in\mathbb{Z}$ and $L$ if finite. I have ...
hyriusen's user avatar
  • 147
4 votes
2 answers
120 views

Questions on integrating $\int_0^{\infty} \frac{x^{\frac{1}{3}}}{1 + x^2} dx$ using contour integrals

We have the following integral to solve $$ I = \int_0^{\infty} \frac{x^{\frac{1}{3}}}{1 + x^2} dx $$ I've managed to solve this integral. Using the substitution $u = x^2$ we can show that $I = \frac{1}...
Noud's user avatar
  • 533
20 votes
3 answers
2k views

I think I don't truly understand Cauchy's Integral theorem

Cauchy's theorem states that closed line integral of some holomorphic functions yields zero, in some good regions (i.e. simply connected domain). More explicitly, $$ \oint_\gamma f(z) d z=0. $$ Many ...
SunnyMath's user avatar
  • 309
1 vote
0 answers
90 views

How to calculate $\int_{0}^{2\pi}(a+\sin{x})^{-3/2}dx$ [closed]

I wonder if there is an analytic solution for the following equation: $$\int_{0}^{2\pi}(a+\sin{x})^{-3/2}dx$$ Here, $a$ is a constant. Would you please give an advice?
donggun's user avatar
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