Questions tagged [complex-analysis]

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2
votes
3answers
193 views

Is the imaginary part needed in this problem?

Before jumping into my question, let me contextualize it. I'm doing numerical simulations of a Helmholtz scattering problem $$\Delta p + \kappa^2 p = 0\, .$$ The incident pressure wave $p^{inc}$ will ...
0
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1answer
43 views

Calculating residue of a rational function

I have a function $$ f(z) = \frac{1}{(z-z_1)(z-z_2)(z-z_3)(z-z_4)} $$ All of $\{z_1,z_2,z_3,z_4\}$ are simple poles. The residues for this function are given as $$ \text{Res}(f(z),z_i)= \lim\limits_{z\...
1
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0answers
66 views

Computation of a functional for large values

Consider the following function : $$f(x) = \sin^2(\frac{π\Gamma(x)}{2x})$$ Now consider the following functional : $$I(x)=\int_0^\infty \frac{f(x + iy) − f(x − iy)}{e^{2πy}-1} dy$$ I need values for ...
1
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0answers
73 views

Plot of ratio of two integrals:

Consider the following integrals $$ I_1(x) =\int_0^\infty\mathrm dy \frac{F(x + \mathrm iy) − F(x −\mathrm iy)}{\mathrm e^{2πy}-1}, $$ And $$I_2(x) =\int_1^x F(t)dt$$ Where, $ F(z) = \sin^2[π\Gamma(z)/...
1
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0answers
25 views

Plot FDE Amplification Factor in Complex Plane

I am reviewing stability of finite difference methods. I am using the von Neumann method to study a linear PDE and an explicit finite difference scheme. The goal is to develop an expression for an ...
5
votes
1answer
145 views

Complex differentiation of linear solvers

I have a linear system $$Ax=b$$ which I'm solving approximately, and I need to take the frechet derivative of x with respect to z. Were I solving the problem exactly (either analytically or to machine ...
4
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0answers
131 views

MATLAB: Compute the Schwarz-Christoffel transformation symbolically

Suppose we have a conformal mapping from the unit disk in the $\omega$ plane onto the exterior of a polygon in the $z$ plane. The Schwarz-Christoffel mapping in this case is defined as: $$f(u) = A - ...
11
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1answer
192 views

Numerically Recovering Imaginary Part of Analytic Continuation from Real Part

My situation. I have a function of a complex variable $f(z)$ defined through a complicated integral. What I am interested in is the value of this function on the imaginary axis. I have numerical ...
3
votes
3answers
128 views

How to calculate $\arg(z_1z_2\cdots z_n)$ to minimize results error?

As in title, which method is the most optimal for numerical calculating value of: $\arg(z_1z_2\cdots z_n)$? Method 1: one can first calculate $Z=z_1z_2\cdots z_n$ and then calculate $\arg(Z)$. ...
4
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0answers
68 views

Complex Integral Equation Solution in MATLAB

I need to solve an integral equation in the form: $$A(z)+\int\limits^{z_2}_{z_1}B(z') \frac{z^N}{z^N-z'^N} \frac{e^{i\beta}}{|z|}\mathrm{d}z'=0 $$ where $A(z)$ distribution is known and we are ...
1
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1answer
142 views

Applicable solvers for nonlinear coupled PDEs

I've been trying to find an applicable PDE solver for cases such as this: Although when dealing with stiff equations in the complex domain, applying existing packages has been problematic. I've ...
7
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1answer
105 views

Are there special methods for solving $f'(z)=0$ for analytic $f$?

I am trying to solve a bunch of equations for the zeros of the derivative of an analytic function, and I would like to know if there exist methods that exploit this structure to provide better ...
2
votes
1answer
844 views

Complex Numerical Integration using GSL

I want to program an integration routine in C++ using the GSL library but for complex functions. How should I split my integrand to apply the gsl_integration_qag function on it. Just integrate the ...
2
votes
0answers
60 views

Need a smart way to numerically take residues in a multidimensional integral

I'm trying to do an integral of the form $\int_C f(u,v) $, where $C$ is a set of contours in $u$ and $v$. In particular, each variable's contour starts at $-\infty+i \epsilon$, goes around a branch ...
6
votes
2answers
212 views

Continuation procedure to solve for a 2D curve that satisfies f(x,y) = 0

I have some function of $R^2$, that must be numerically computed. For instance, I might be interested in a real-valued contour integral that begins from (x,y) = 0. $$ f(x,y) = \Re\left[\int_0^{x + iy}...
0
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1answer
215 views

Drawing 3d projection of complex surface

I have a complex surface (real dimension 2) in $\mathbb{C}^2$ with coordinates $(z,w)$ given explicitely: for any $\xi \in \mathbb{C}$ I know points $w(\xi)$ of intersection of surface with complex ...
15
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1answer
1k views

How to numerically calculate residues?

I need to calculate the following integral: $$ {1\over 2\pi i} \int_C f(E) \, d E $$ $$ f(E) = {\rm Tr}\,\left(({\bf h} + E)\,{\bf G}(E) \right) $$ Where $\bf h$ is a matrix (one particle kinetic and ...