Questions tagged [complex-integration]

This is for questions about integration methods that use results from complex analysis and their applications. The theory of complex integration is elegant, powerful, and a useful tool for physicists and engineers. It also connects widely with other branches of mathematics.

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

I am trying to show that $$\int_{-\infty}^\infty \frac{x^2}{\left(x^2 + a^2\right)^2} dx = \frac\pi{2a}$$ for $a > 0$ using the Residue Theorem. The formula I am using says $$\int_{-\infty}^\infty ...
Clyde Kertzer's user avatar
2 votes
1 answer
62 views

Complex fundamental theorem of calculus

The fundamental theorem of calculus for complex functions (as well) states that $\int_{\gamma} f(z)dz := \int\limits_{a}^{b} f(\gamma(t)) \gamma'(t) dt = F(\gamma(b)) - F(\gamma (a)) = \int\limits_{\...
ICOR's user avatar
  • 23
0 votes
1 answer
35 views

How can this expression be interpreted [closed]

How may the following expression be interpreted? $I=\oint_{C} f(z)|dz|$.. My approach is to simplify $I$ as $I=\oint_{C}f(z)\sqrt{(dx)^2+(dy)^2}$ but it seems inconclusive.
Mayank Kashyap's user avatar
2 votes
0 answers
101 views

Is $\int_{-\infty}^\infty dx\,e^{-ikx}H_0^{(2)}(a\sqrt{b^2-x^2})=2i\frac{e^{-ib\sqrt{a^2+k^2}}}{\sqrt{a^2+k^2}}$ true? (Gradshteyn and Ryzhik 6.616.4)

After encountering a tricky integral of the Hankel function of the second kind, I was happy to find it almost exactly in Gradshteyn and Ryzhik as entry 6.616.4, namely $$\int_{-\infty}^\infty\mathrm{d}...
Caesar.tcl's user avatar
0 votes
0 answers
32 views

Trouble with a Fourier Transform

I'm getting some conflicting results when trying to take this integral : \begin{equation} \int^{L/2}_{-L/2}xe^{i\frac{2\pi}{L}(n-n')x}dx \end{equation} where $n$ and $n'$ are both integers. I'm having ...
aelon's user avatar
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2 votes
2 answers
77 views

Which zeros of $z^4+6z^2+13=0$ are in the upper half plane?

I want to calculate $\int_0^{+\infty}\frac{x^2}{x^4+6x^2+13} dx$ using residue theorem. Consider $f(z)=\frac{z^2}{z^4+6z^2+13}$, so I need to find the zeros of $z^4+6z^2+13=0$ which are in the upper ...
Rogan's user avatar
  • 311
0 votes
0 answers
51 views

Closed form expression of complex plane integral [closed]

Let $\Delta_1, \Delta_2$ be two complex numbers and $z, w_1$ and $w_2$ three points on the complex plane. Is there a closed-form expression for the integral $$ \int d^2z \frac{1}{|z - w_1|^{2\Delta_1}}...
NoName's user avatar
  • 31
1 vote
0 answers
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definite integral involving u-sub with parameterized curves

While proving a pretty simple argument about reverse parametrization of a complex-valued function, I have an integral of the form: $$ \int_a^b f\bigl(\gamma(a+b-t)\bigr)\cdot\bigl(-\gamma'(a+b-t)\bigr)...
giorgio's user avatar
  • 461
3 votes
2 answers
83 views

What is the integral $\int_{-\pi}^{\pi} \frac{1}{2\pi}\exp{(z_1 \cos\theta + z_2 \sin\theta)}\, d\theta$?

What is the integral $\int_{-\pi}^{\pi} \frac{1}{2\pi}\exp{(z_1 \cos\theta + z_2 \sin\theta)}\, d\theta$? When $z_{1,2} \in \mathbb{R}$ then we get the modified Bessel function $I_0(\sqrt{z_1^2+z_2^2})...
Saurabh Shringarpure's user avatar
0 votes
0 answers
12 views

Complex valued Hamilton Jacobi equation

Let $g_{ij}(t,x)$ be a metric tensor with dependence on t,x. Consider $$\partial_t u(t,x) = i\sqrt{\sum_{i,j} g_{ij}\partial_iu\partial_ju},u(0,x)=u_0(x).$$ Where $u(t,x):\mathbb{R}\times\mathbb{R}^n\...
xinggu's user avatar
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0 answers
16 views

Properties of real representation of integral over complex weight function

Consider some positive and real-valued functions $p, \tilde p :\mathbb{R}^2\to [0,\infty)$ that are integrable and decay sufficiently fast at infinity. For any entire function $f: \mathbb{C} \to \...
thehardyreader's user avatar
3 votes
1 answer
145 views

$\int_{-\infty}^\infty \frac{1}{x^5+1}dx$ using contour integration.

I am wondering if I have correctly computed this integral, which I see in a lot of posts as being really hard. $\int_{-\infty}^\infty \frac{1}{x^5+1}dx$. Consider the following contour: The poles of $...
random_0620's user avatar
  • 2,251
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0 answers
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Trying to solve $I=\int_c^\infty {\sin \big(x+{k \over x} \big) \over x}dx$

While trying to find an answer to this problem on the forum, I came across this integral: $$ I=\int_c^\infty {\sin \big(x+{k \over x} \big) \over x}dx \tag 1$$ Where $c$ and $k$ are real numbers. I ...
FriendlyNeighborhoodEngineer's user avatar
0 votes
0 answers
90 views

complex analysis - Help with integrating $\int_0^{\infty} \frac{(\log x)^4}{x^2 + 1} \operatorname d\!x$ [duplicate]

I am trying to solve the following integral using a contour (large semi-circle connected to smaller semi-circle in the upper-half plane): I have split the contour into 4 parts - the large semi-circle, ...
CentraM's user avatar
1 vote
1 answer
55 views

a limit of a complex function of $\zeta(s)\zeta(s+1)\Gamma(s)$

$$f(s) = \zeta(s)\zeta(s+1)\Gamma(s) $$ This has a double pole at $s=0$ , from $\zeta(s+1)$ and $\Gamma(s)$ respectively, and $\lim_{s\to0}s^2f(s) = -1/2$ Then, the next step, I have difficulty with ...
David Lee's user avatar
  • 179
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0 answers
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Proving a theorem related to uniqueness of Fourier Series for complex-valued function.

I was studying Fourier Analysis: An Introduction by Stein and Shakarchi. I came across this theorem in Section 2.2, page number 39 Theorem 2.1 Suppose that $f$ is an integrable function on the circle ...
Souparna's user avatar
  • 910
1 vote
0 answers
16 views

Can't find a residue, a limit relating to $\zeta()$ (feat. Cauchy's P.V)

$$f(s) = \zeta(s)\zeta(s+1)\Gamma(s) $$ This has a double pole at $s=0$ , and I got -1/2 , and then to find the other $$ \lim_{s\to0} s(f(s) - \frac{-1/2}{s^2}) $$ (Actually, it is $-log\sqrt{2\pi} = ...
David Lee's user avatar
  • 179
1 vote
0 answers
57 views

Residue theorem integral - calculating with trigonometric functions

I am trying to solve the following integral: $$I = \int_0^{2\pi} \frac{d\phi}{a + 2 b \cos(\phi) + 2 c \sin(\phi)}$$ We can assume that the denominator is always strictly positive ($a \gg b, c$). ...
Manuel Ballester's user avatar
0 votes
0 answers
13 views

Question about the logarithm function in the Argument Principle

I have found the proofs of the argument principle from Ponnusamy and Silverman's Complex variables with applications and Brown and Churchill's Complex variables. But I am not sure how to make sense of ...
nomadicmathematician's user avatar
2 votes
0 answers
158 views

Closed form for $\int_{-\infty}^{\infty} \frac{\sin(b x)\sin(\sqrt{a+x^2})}{x \sqrt{a+x^2}}$

I am trying to evaluate $$\int_{-\infty}^{\infty} \frac{\sin(b x)\sin(\sqrt{a+x^2})}{x \sqrt{a+x^2}} dx$$ for $a>0$ and real $b$. When $|b|>1$, we can split $\sin(bx)$ into complex exponentials, ...
Wouter's user avatar
  • 7,673
2 votes
0 answers
46 views

Would the following not be a correct proof of Cauchy's Integral Theorem?

I'm somewhat confused as to how the general version of Cauchy's Theorem does not follow (almost) immediately from its version in a disk. At least Ahlfors as well as Stewart & Tall prove the ...
Sam's user avatar
  • 4,792
1 vote
0 answers
54 views

Integration with residue theorem

I am trying to calculate the following integral: $$ \int\limits_{-\infty}^{\infty}\frac{x \arctan(x-a)}{(x-b)^2+y^2}d x $$ where $a$, $b$ are positive constants and also y is a positive variable. (...
Banx's user avatar
  • 135
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0 answers
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Does being homotopic to $0$ imply being homologous to $0$ or viceversa?

For context, the question popped up while studying the homotopical version of Cauchy's Integral Theorem, and comparing it with the homological version. Definition: two $γ_0, γ_1 : [0, 1] → U$ be two ...
Sam's user avatar
  • 4,792
2 votes
0 answers
147 views

Can contour integration produce mascheroni or Catalan's constant?

I am currently learning contour integration and I noticed that certain integrals are not mentioned at all when discussing contour integration and some of those are the following : $$\int_{0}^{\infty} {...
Super killer's user avatar
1 vote
1 answer
35 views

Verifying Fourier inversion for a rectangular function

I define the Fourier transform of a function $g$ to be $ \hat{g}(\lambda) = \int_{-\infty}^\infty g(x)e^{-i\lambda x} \, \mathrm{d}x $ with an associated inverse formula $$ g(x)= \frac{1}{2\pi} \int_{...
Randall's user avatar
  • 515
1 vote
1 answer
26 views

Harmonic conjugated functions and simply connected domain in wikipedia

In wikipedia (link), it says: (i) Therefore, a harmonic function $u$ admits a conjugated harmonic function if and only if the holomorphic function $g(z)\colon=u_{x}(x,y)-iu_{y}(x,y)$ has a primitive $...
studyhard's user avatar
  • 161
0 votes
0 answers
23 views

Lebesgue Integral on Rough Paths

In a nutshell: how can we integrate rough paths using Lebesgue integration? What I mean by a rough path is a continuous map $[a,b]\to\mathbb{R}^n$, or $[a,b]\to\mathbb{C}$ in the complex case. (The ...
Sam's user avatar
  • 4,792
1 vote
0 answers
58 views

Understanding how to compute $\int_{0}^{+\infty} \frac{\sin t}{t}dt $ via complex integration. [duplicate]

Question: Understanding how to compute $\int_{0}^{+\infty} \frac{\sin t}{t}dt $ via complex integration. Let us first define $ I(r)=\int_{(|z|=r)} \frac{e^{iz}}{z}dz. $ (a) Show that $I(r) \to 0 $ as $...
Confused's user avatar
1 vote
0 answers
58 views

How to evaluate Lorentz invariant integral with complex parameters

For context: there are Schwartz distributions called Pauli-Jordan (or Schwinger) functions which come up in quantum field theory. Given $m>0$, $x_0\in\mathbb{R}$ and $\vec{x}\in \mathbb{R}^3$, $D_m^...
TheEmptyFunction's user avatar
2 votes
0 answers
120 views

Integration by parts does not work for this complex integral. Why?

There is a longer integral for which integration by parts $\displaystyle\int udv=uv-\int vdu$ was attempted as it came across in research: $$\frac i{2\pi}\int_0^{2\pi}\underbrace{\ln\left(1+\frac{e^{-...
Тyma Gaidash's user avatar
1 vote
1 answer
120 views

Branch cut integral $\int_{-1}^1\left(1-x^2\right)^{\frac{1}{2}} d x$

Define the branch of $f(z)=\left(1-z^2\right)^{\frac{1}{2}}$ by the branch cut $(-\infty,-1] \cup [1,\infty), f(0)=1$. Use this branch and a suitably chosen semi-circular contour (with finite radius $...
ThetaOmega's user avatar
2 votes
0 answers
34 views

Practical integration along vertical lines

I want to ask a practical integration and I hope the answer could be a line by line form to for me to understand. Consider complex integration. Assume the integration is consist of a vertical line $K: ...
user39511's user avatar
0 votes
1 answer
100 views

Is $\int_0^{\pi}\log \sin \theta d\theta$ not well-defined?

Is $\int_0^{\pi}\log \sin \theta d\theta$ not well-defined? On the third edition of Ahlfors' Complex Analysis, page 160 it states: As a final example we compute the special integral \begin{equation*} \...
studyhard's user avatar
  • 161
0 votes
1 answer
37 views

Prove the following complex integral satisfies $\mid \int_\ell \frac{z^3}{z^2+1}dz \mid \leq \frac{9\pi}{8}$

Prove $\mid \int_\ell \frac{z^3}{z^2+1}dz \mid \leq \frac{9\pi}{8}$ where $\ell$ is $|z|=3,\Re(z)\geq 0$. My attempt: $$\mid \int_\ell \frac{z^3}{z^2+1}dz \mid \leq \int_\ell \frac{\mid z\mid^3}{\...
Dr. John's user avatar
  • 453
1 vote
0 answers
33 views

Complex integral from quantum mechanics [duplicate]

I got this integral in one quantum mechanics problem: $$\int_{-\infty}^{\infty} e^{-i\cdot p_y y/\hbar}e^{-\beta y^2}dy$$ and the solution was apparently $\sqrt{\frac{\pi}{\beta}} e^{-\frac{p_y^2}{4\...
Ivy's user avatar
  • 65
2 votes
1 answer
93 views

Confusions about the definition of "residue" in Ahlfors' Complex Analysis

On the third edition of Ahlfors' Complex Analysis, page 149 it states: (i) Draw a circle $C_j$ about $a_j$ of radius $<\delta_j$, and let $P_j=\int_{C_j}f(z)dz$ be the corresponding period of $f(z)$...
studyhard's user avatar
  • 161
2 votes
2 answers
222 views

Mistakes in proving $\int_{\gamma}\frac{dz}{z-a}=2k\pi i$ in Ahlfors' Complex Analysis

Thanks again for Martin's new extension What a great extension! But I still need to check $\int_{\alpha}^{\beta}w(s)ds=\int_{\gamma}\frac{1}{z-a}dz$ . The integrands are different only at "...
studyhard's user avatar
  • 161
0 votes
1 answer
88 views

How to calculate $\int_{C_a} \frac{e^z}{(z-a)(z-b)}dz$?

How to calculate $\int_{C_a} \frac{e^z}{(z-a)(z-b)}dz$. Here, $C_a$ is a "small" circle about $a$ so that $b$ is outside of $C_a$. Actually, if I can prove the following statement, I can ...
studyhard's user avatar
  • 161
1 vote
2 answers
104 views

Proof of Cauchy's theorem for punctured domains.

I am unable to prove the following lemma of Cauchy's integral theorem for simple closed curve. Lemma: Let $R$ be a simply connected region. If $f(z)$ be analytic in $R-\{a\}$ and is continuous on $R$ ...
General Mathematics's user avatar
0 votes
0 answers
34 views

Fourier transform $\left[\mathrm{csch}(x+i\epsilon-t)\right]^n\left[\mathrm{csch}(x+i\epsilon+t)\right]^m$

In a physics related problem, I am trying to compute the Fourier transform \begin{align} \mathcal{F}\left[\frac{1}{\sinh^{n}\left[\pi T_R\left(x+ i\epsilon-t\right)\right]\sinh^{m}\left[\pi T_L\left(...
hyriusen's user avatar
  • 117
2 votes
1 answer
115 views

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
  • 5,955
0 votes
1 answer
53 views

How to integrate this function of bessel integration form?

Thank you for reading my question! As we know, the first kind Bessel function of 0th order can be integrated as follows $$ J_0(x) =\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{-ix\sin{(a+\tau)}}d\tau =\frac{1}{2\...
Xiangyu Cui's user avatar
1 vote
1 answer
49 views

expansion of real asymmetric integral on complex plane

I would like to solve the following integral for $f(x)$ $$\int_{0}^{\infty} \frac{1}{(a+ix)(b+ix)(c+ix)} dx $$ by expanding it to the complex and then using a contourlike the half circle, i.e. if $C$ ...
Jannis Erhard's user avatar
0 votes
0 answers
37 views

Approximate inverse Mellin transform

I need to evaluate the following complex integral (which is essentially an inverse Mellin transform): $$\int_{-c-i\infty}^{-c+i\infty} \Gamma^2 (-s) \Gamma (s+1) \Gamma (a-s) \mbox{}_1F_1(s;1;-y) x^s{...
math.amuser's user avatar
1 vote
1 answer
53 views

How to solve integral with cubic polynomial at the denominator with Byrd-Friedman's handbook of elliptical integrals?

Is there an equivalent elliptic integral (with solution) in the Byrd-Friedmann's elliptical integral Handbook? This is the integral: $$ \int \frac{1}{\sqrt{(x-p)(x-q)(x-̅q)}}dx $$ where we have three ...
Mark Int's user avatar
0 votes
1 answer
47 views

Integral of $z$ over a triangle doesnt't match the result from Cauchy-Goursat theorem

Given a triangle with vertices $0,1,i$ I'm asked to evaluate the integral $$\int_{\partial \triangle} \! z \ dz$$ by the Cauchy-Goursat theorem I know that the integral is equal to zero because $f(z)=...
Noobunaga's user avatar
0 votes
0 answers
43 views

Integration using Cauchy's theorem vs numerical method

It is not a homework question. I just want to learn complex integrations. I want to evaluate the following integral $$ I = \int \frac{dE}{2\pi} E^2 \left(\frac{1}{(E-E_n+iη)^2 (E-E_m+iη) }-\frac{1}{(E-...
Luqman Saleem's user avatar
1 vote
1 answer
69 views

Confusion in working out $\mathcal{P} \int_{-\infty}^{\infty} \frac{dz}{z-i}$.

I'm trying to work out $$I=\mathcal{P} \int_{-\infty}^{\infty} \frac{dz}{z-i}$$ where $\mathcal{P}$ denotes the fact we are taking the Cauchy principal value. Let $\gamma$ be the contour such that $\...
Robin's user avatar
  • 3,227
2 votes
0 answers
56 views

Need some clarifications on the 'dogbone' contour (especially the argument system)

Related problem: Understanding Dogbone contour example Here, we need to compute $\int_{0}^3 \frac{x^\frac{3}{4}(3-x)^\frac{1}{4}}{5-x}dx$ using the dogbone contour. I am trying to understand https://...
Muses_China's user avatar
  • 1,008
0 votes
0 answers
43 views

Solve Convoluted Integral with Gauss Hypergeometric function

I'm working on an operator problem that requires solving the following complicated integral, involving the Gauss Hypergeometric function \begin{equation} \int_{-\infty}^{\infty} \int_{-\infty}^0 \...
MultipleSearchingUnity's user avatar

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