For questions about integration methods that use results from complex analysis and their applications

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6
votes
0answers
223 views

Complex contour integral: How does the stationary point method used in this case?

I was reading a paper which has the following integral in order to do the inverse Laplace transformation: $$ I=\frac{1}{2\pi i}\int_{-i\infty+\gamma}^{i\infty+\gamma} ...
4
votes
0answers
83 views

Double contour integral in terms of real integrals

Let $\gamma$ be a curve in $\mathbb{C}$, and let $\gamma_0$ be a circle in an open connected set $A \subset \mathbb{C}$ around $z_0 \in A$. Suppose the interior of $\gamma_0$ lies in $A$. Let $z$ be ...
4
votes
0answers
56 views

Integrate complex function over $\mathbb{C}^2$

I have a question in mind and I would appreciate your help. Usually in complex analysis we consider integrals of the form $\int_\gamma f(z) dz$ where $\gamma $ is a contour and ...
3
votes
0answers
79 views

Integral of $e^{ix^2}$

How does one evaluate $$\int_{-\infty}^{\infty} e^{ix^2} dx$$ I know the trick how to evaluate $\int_{-\infty}^{\infty} e^{-x^2}dx$ but trying to apply it here I get a limit which does not converge: ...
3
votes
0answers
24 views

Complex line integration with assumptions

Let $f: \mathbb{C} \to \mathbb{C} $ be a holomorphic function with $$ \lim_{\lvert z \rvert\to\infty} \frac{f(z)}{z^{n-1}} = 0$$ for some $n\in\mathbb{N}$. How can I prove that $$ \lim_{r\to\infty} ...
3
votes
0answers
49 views

Inverse Mellin of the exponential of the digamma function

(Cross-posted from mathoverflow: No answers yet; bounty there expires in less than 24 hours) I'm looking for a function $f_p(x)$ with real parameter $p>0$ satisfying $$ \int_0^\infty ...
3
votes
0answers
3k views

Integrate complex conjugate

I'm doing some exercises on complex integration, and stumbled over this: $$\int_\gamma \bar{z} dz,$$ Where $\gamma$ is any closed $C^1$ curve which is the boundary of a bounded, connected, open $U ...
2
votes
0answers
54 views

Showing that if $\int_C f(z) \, dz=0$ for every circle $C$ in $\mathbb{C}$, then $f$ is holomorphic.

I understand that this is the difficult direction of Morera's proof, applied to disks, rather than triangles. However, the trick of defining $$F(z)=\int_{\gamma(t)}f(w)\,dw, \text{ with } ...
2
votes
0answers
14 views

Choice of branch cut

I'm trying to rewrite an integral of the form $$\int_{-∞}^∞ (x^2+k^2)^{-s/2}e^{i(xc_1+kc_2)}dx$$ ($s>0,c_1,c_2,k\in\mathbb R$) in such a way that it is positive (for some $s$). To do so I'd like ...
2
votes
0answers
46 views

Evaluating Improper Integrals with Residues - don't think I'm calculating the residues properly

I have to evaluate the integrals $\displaystyle \int_{-\infty}^{\infty}\frac{dx}{x^{2}+p^{2}}$, for $p > 0$, and $\displaystyle \int_{-\infty}^{\infty} \frac{dx}{(x^{2}+p^{2})^{2}}$, for $p > 0$ ...
2
votes
0answers
96 views

Difficulty evaluating complex integral

The integral along the path $\gamma(t)=e^{2ti},\;t\in[0,2\pi]$ is $\begin{equation*} \int_{\gamma}\frac{1}{z^{2}-1}dz \end{equation*}$. I approached this like a real integral in the hopes things ...
2
votes
0answers
108 views

Fourier transform of a Gaussian times a rational function

I am trying to compute the following Fourier transform $$\int_{-\infty}^\infty\text{d}x\,e^{i k x}e^{-x^2/a^2}\frac{P(x)}{Q(x)}$$ where $\text{deg}P(x)+1\leq\text{deg}Q(x)$, and the roots of $Q(x)$ ...
2
votes
0answers
31 views

If $f(z)=\sum_{n=0}^{\infty}c_{n}(z-z_{0})^{n}$, then $c_{n}r^{n}=\frac{1}{2\pi}\int_{0}^{2\pi}f\left(z_{0}+re^{-2nt}\right)dt$

Let $U\subseteq \mathbb{C}$ be an open set, $z_{0}\in U$ and $R>0$ such that $\mathcal{B}_{R}(z_{0}) \subseteq U$. Let $f:U\rightarrow \mathbb{C}$ be a holomorphic function with Taylor's serie ...
2
votes
0answers
22 views

Complex integration by substitution

Integrate $ f(z) $ counterclockwise around the unit circle. $$ f(z) = 1/(4z-3) $$ My solution C(contour) : $ z(t) = \cos{t} + i\sin{t} = e^{it}, 0<t\leq 2\pi $ $$ \oint_C \frac{1}{4z-3} dz = ...
2
votes
0answers
73 views

Why is this integral finite?

I am looking at this integral $\displaystyle\int_{-\infty}^\infty dx \, \frac{e^{iax}}{\sinh^2{bx}}$. Now dividing the integral into real and complex parts, respectively $\int_{-\infty}^{\infty} dx ...
2
votes
0answers
32 views

Evaluating functions similar to the Bessel functions

In the problem there are two integrals, and one is asked to evaluate them by taking an integral over a unit circle of some chosen function. The integrals are $$\int_0^{2\pi}e{^{\sin n\theta}}\cos ...
2
votes
0answers
106 views

Why does this example of global residue theorem not work?

This question is related to and inspired by a previous question What is the residue obtained from this integral? , but note that the appearing functions are slightly different. Consider the following ...
2
votes
0answers
30 views

Complex integral difficulty

We need some sort of analytic expression for the integral: $$\int^\infty _{-\infty} \frac{\sqrt{\alpha^2 + 1}}{(\mathrm{i}\alpha)^{\frac{3}{4}}}\mathrm{e}^{\mathrm{i}\alpha \chi} \mathrm{d}\alpha$$ ...
2
votes
0answers
61 views

Moving limit inside a contour integral

I'm trying to compute this integral as part of a larger problem I'm working on. I'm trying to solve the integral $\int_0^\infty \frac{\sin(x)}{x}dx$ and to do it I'm using the method where you ...
2
votes
0answers
210 views

Complex line integral for closed homotopic curve.

I try to show that the following analogous result for $\textbf{closed curve}$ : Let $\gamma_0, \gamma_1$ be continuous paths in a domain $D \subseteq \mathbb{C}$ and $\omega$ be a closed form on $D$. ...
2
votes
0answers
38 views

How can I solve $\int \left| \sum_{i=0}^{N} f_i(x) \right|^2 d x $ for complex functions $f_i$?

The $f_i(x)$ are complex functions and I know that $\int_{-\infty}^{\infty} f_i (x ) \, d x=c_i$. How can solve this integral? $$\int_{-\infty}^{\infty} \left| \sum_{i=0}^{N} f_i(x) \right|^2 d x $$ ...
2
votes
0answers
67 views

Imaginary part of Log laplacian

I'm confused about how to calculate $\nabla^2 \log z$, where $z=re^{i\theta}$ is a complex number. My calculations return $$ \nabla^2 \log z = 2\pi\frac{\delta(r)}{r} [\delta(\theta) + i ...
2
votes
0answers
35 views

$ g(u;s_1,s_2)=\int_{(\alpha+\beta)^{-1}}^{\alpha+\beta}b^{s_2}(u^2-b)^{-\dfrac{s_1+s_2}{2}}(u^2b-1)^{-\dfrac{s_1+s_2}{2}} \mathrm {d}b$

Do you have any idea how to compute this function? $$ \displaystyle g(u;s_1,s_2)=\int_{(\alpha+\beta)^{-1}}^{\alpha+\beta}b^{s_2}(u^2-b)^{-\dfrac{s_1+s_2}{2}}(u^2b-1)^{-\dfrac{s_1+s_2}{2}} \mathrm ...
2
votes
0answers
304 views

Laplace transform of a product of functions

While trying to compute the Laplace transform of a certain product, part of the calculation leaves me with a Bromwich integral which has the form: ...
2
votes
0answers
139 views

Show Smoothness by Morera

I'm trying to show smoothness on $(0,\infty)(\Re)$ of the following function: $$ f(t,x)=\sum_{n=-\infty}^\infty e^{-\large \frac{(x-2\pi n)^2}{2t}}\frac{1}{\sqrt{2\pi t}} $$ The function is ...
1
vote
0answers
36 views

Exchange series and integral in a complex context.

Let $a_{n},b_{n},c_{n}$ complex sequences and let $k>0$ a real parameter. Assuming that $$\sum_{n\geq1}\sum_{m\geq1}\left|\frac{a_{m}b_{n}}{c_{m+n+k}}\right|<\infty\tag{1} $$ if $k>1/2 $ ...
1
vote
0answers
24 views

Find the complex integral over a path

I have to integrate the complex function $$z+1/z$$ which is parameterized by $\gamma(t), 0 \le t\le 1$ and satisfies $Im\gamma(t) > 0$, $\gamma(0) = -4+i$ and $\gamma(1) = 6+2i$. Can I assume the ...
1
vote
0answers
21 views

Evaluating a complex integral using Cauchys Integral Formula

I need to evaluate the following complex integral: $ \int_{\phi}\frac{z^3}{z^2+i} dz$ where $\phi$ is the circle centered at $0$ with radius $2$ I know that there is a singularity at $z = ...
1
vote
0answers
12 views

Cauchy-Riemann equation analogue but for the quaternions

given a function over the quaternions $$ U(x,y,z,t)+iV(x,y,z,t)+jW(x,y,z,t)+kR(x,y,z,t)=f(x,y,z,t) $$ what are the analogues of the Cauchy Riemann equation for the quaternionic plane so the function ...
1
vote
0answers
29 views

Determine the number of zeros using the Argument Principle

I'm tasked with finding the zeros of $f(z)=z^3+1$ that lie inside the first quadrant using the Argument Principle, which I have simplified below: $$N=\frac{1}{2\pi}[arg(f(z))]_C$$ where N represents ...
1
vote
0answers
33 views

Integrating a complex function

I have the integral $ \int_{C(0;2)} \frac{e^z}{z^3+9}dz $ I was told I could use Cauchy's Integral formula but I'm still stuck, I'm not sure how to apply it. Any help would be great!
1
vote
0answers
23 views

Complex Integral evaluation.

I am completely stuck with the evaluation of the following integral : $I = \int_{-\infty}^{\infty} \frac{\sinh(x)}{\sinh(ax)}dx, a>1.$ I am supposed to use a rectangle, such that the bounds are ...
1
vote
0answers
57 views

Can I switch to polar coordinates if my function has complex poles?

You can think this of the following as a 3d QFT where we try to calculate the self-energy of two fields. $I$ is a this external self-energy and let us assume it does not depend on the loop momenta ...
1
vote
0answers
22 views

Hyperelliptic Curves and Period Integrals

I've been thinking about the period integrals on a hyperelliptic curve of the form $y^{2}=F(x)$, where $F(x)$ is a polynomial of degree-$2n$ with complex coefficients. Of course, a hyperelliptic ...
1
vote
0answers
23 views

Integral around a square in the complex plane

Let $f(z)$ be any continuous function defined in the complex plane with the property that $$\bigg|\int_{R_n}f(w)dw\bigg|\leq n^2\log(n),$$ for any $n>1$ and any square $R_n$ with side length $n$. ...
1
vote
0answers
43 views

Show that $\int_{C} \frac{e^{az}}{\sin(\pi z)} \,dz \rightarrow 0 \quad as \quad R \rightarrow \infty$

Given the straight line in the complex plane: $b+iR$ to $b+1+iR$ where $0<b<1$ and $|Im(a)|<\pi$, show the following: $$\int_{C} \frac{e^{az}}{\sin(\pi z)} \,dz \rightarrow 0 \quad as \quad ...
1
vote
0answers
39 views

Theory of complex integration

I don't know if this is a pedestrian question or not, but here goes. I am working through Ahlfors' complex analysis, which has been a great (and challenging) text to get a grasp of the basics of the ...
1
vote
0answers
36 views

Let $f:U\rightarrow \mathbb{C}$ holomorphic no constant, then $\mbox{Re}f+\mbox{Im}f$ has no minimum in $U$.

Let $U\in\mathbb{C}$ be an open and connected set and let $f:U\rightarrow \mathbb{C}$ holomorphic. Suppose that $f$ is not constant. Show that $\mbox{Re}f+\mbox{Im}f$ has no minimum in $U$. Remark: I ...
1
vote
0answers
37 views

If $J(L)=\int_{L}\frac{dz}{p(z)}$, where $p(z)$ is a polynomial with $n$ distinct roots, how many values can $J$ take on?

My question is related to this one, but it is not a duplicate, because I am not allowed to use Residues or Cauchy's Integral Formula to solve it. The only tools I have at my disposal are Cauchy's ...
1
vote
0answers
56 views

Evaluation of an improper integral with complex exponential

Are there any convenient ways to calculate an integral of the form $$ \int_{-\infty}^\infty\frac{a_1 e^{j\omega\alpha}+a_2e^{j\omega\beta}}{1 + a_1a_2e^{j\omega\gamma}}d\omega$$ where ...
1
vote
0answers
36 views

Showing a function is harmonic given some information

"Let $a ∈\mathbb C$ and $r > 0$. Let $f : S(a, r) → R$ be continuous. Let $g : B(a, r) → R$ be the Poisson integral of $f$. Then (1) $g$ is harmonic on $B(a, r)$ and (2) $f \cup g : B(a, r) → ...
1
vote
0answers
47 views

Evaluating the complex integral- Correct approach?

I'm asked to evaluate the following complex integral with C being the unit circle: $$\oint_{C}^{}{\log(z-z_0)dz}\quad |z_0|>1$$ $\log(z-z_0)$ is multivalued and has a branch point at $z=z_0$. ...
1
vote
0answers
31 views

Matusubara sum contradiction?

In many textbooks, the following fermionic Matsubara sum is given as a useful identity: $$T\sum_{\omega_n}\frac{1}{i\omega_n-\epsilon}=\frac{1}{e^{\epsilon/T}+1},$$ where $\omega_n=n\pi T$ for all odd ...
1
vote
0answers
19 views

Infinite total variation of complex measure in Feynman path integral

I am trying to understand this: If one tries to define a Feynman path integral as a Wiener integral, then the complex measure could be of infinite total variation. What exactly does this mean? How ...
1
vote
0answers
76 views

Complex integration with infinitely many poles on imaginary axis

I'm trying to integrate with a closed contour on the upper-half of the complex plane. $I = \displaystyle\int_{-\infty}^\infty \dfrac{z\,\mathrm{sech(z)}}{[(z-a)^2+b^2][(z+a)^2+b^2]} dz$ There are ...
1
vote
0answers
55 views

Integral of complex logarithm on a disk in the plane

Let $a$ be a complex number and $D$ the disk centered around $0$ and of radius $R$. I would like to compute the integral I=$\int_D \log(|z-a|)d^2z$. I am interested in particular in the case $R\gg ...
1
vote
0answers
16 views

Is there a general algorithm to determine new contours for multivariable change of integration variables

Is there a general algorithm to determine the new region of integration upon a multivariable change of variables (where the old variables are a function of all the new variables). I have to do a ...
1
vote
0answers
54 views

integral of $e^{-\imath x}$ from $a$ to $\infty$ and characteristic functions

In the following result: $$\frac{1}{2 \pi} \int_{-\infty}^{\infty} \Phi(u) \left[\int_a^{\infty} e^{-i u x}dx\right] du = \frac{1}{2} + \frac{1}{ \pi} \int_0^{\infty} \Re\left[\frac{e^{-i u a} ...
1
vote
0answers
104 views

Differentiating a period of an elliptic curve under the integral sign

Let $$g = \frac{27J}{1 - J},$$ where $J$ is the absolute invariant, and define $$\Omega = \int_{\gamma(J)} \frac{dz}{\sqrt{4z^3 + g(z + 1)}}.$$ Here, $\gamma(J)$ is a contour in the complex plane that ...
1
vote
0answers
43 views

Complex integration upper semi-circle $\int f(z)dz$ when $r>1$

Question: Let $r$ be a real number, $r > 0$ and let $Lr$ be the line from the point $−r$ to $r$ in $C$. Let $γr$ be the upper half circle with radius $r$ and center in $0$. $$ \ f(z) = ...