Questions tagged [complex-analysis]
The complex-analysis tag has no usage guidance.
17
questions
2
votes
3answers
209 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 ...
1
vote
1answer
52 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
$$
\text{Res}(f(z),z_i)= \lim\limits_{z\...
1
vote
0answers
69 views
Computation of a functional for large values
Consider the following function :
$$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$$
I need values for ...
1
vote
0answers
74 views
Plot of ratio of two integrals:
Consider the following integrals
$$
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$$
Where, $ F(z) = \sin^2[π\Gamma(z)/...
1
vote
0answers
28 views
Plot FDE Amplification Factor in Complex Plane
I am reviewing stability of finite difference methods. I am using the von Neumann method to study a linear PDE and an explicit finite difference scheme. The goal is to develop an expression for an ...
5
votes
1answer
147 views
Complex differentiation of linear solvers
I have a linear system $$Ax=b$$ which I'm solving approximately, and I need to take the frechet derivative of x with respect to z. Were I solving the problem exactly (either analytically or to machine ...
5
votes
0answers
151 views
MATLAB: Compute the Schwarz-Christoffel transformation symbolically
Suppose we have a conformal mapping from the unit disk in the $\omega$ plane onto the exterior of a polygon in the $z$ plane.
The Schwarz-Christoffel mapping in this case is defined as:
$$f(u) = A - ...
11
votes
1answer
200 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 ...
4
votes
3answers
129 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 $\arg(Z)$.
...
6
votes
0answers
72 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 ...
2
votes
1answer
896 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 ...
1
vote
1answer
146 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
votes
1answer
106 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
0answers
63 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
214 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.
$$
f(x,y) = \Re\left[\int_0^{x + iy}...
0
votes
1answer
216 views
Drawing 3d projection of complex surface
I have a complex surface (real dimension 2) in $\mathbb{C}^2$ with coordinates $(z,w)$ given explicitely: for any $\xi \in \mathbb{C}$ I know points $w(\xi)$ of intersection of surface with complex ...
15
votes
1answer
1k views
How to numerically calculate residues?
I need to calculate the following integral:
$$
{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 ...
