1
vote
0answers
39 views

implicit non-linear equations with complex variables

I am trying to understand a methodology for solving implicit non-linear equations with complex variables. I would like to solve for z1 below where z2 is known. Also both z1 and z2 are complex ...
6
votes
1answer
72 views

Solution of $\exp(z)=z$ in $\Bbb{C}$.

I have posted a related question here. I thinkg this one is more interesting: What about the solution of $\exp(z)=z$ in $\Bbb{C}$? My try : $z \mapsto e^z - z$ is entire non-constant. Perhaps ...
0
votes
1answer
19 views

Numerical evaluation of a complex integral

I have to evaluate numerically $f(z)$ via the Cauchy representation (so via a complex integral), in other words, I have to calculare $f(z)$ performing a complex integral: $\dfrac{1}{2\pi ...
1
vote
1answer
120 views

Numerical approximation of trigonometric polynomial

I have the following problem: Let $g$ be a trigonometric polynomial of degree n (there are complex coefficients $c_k$ with $k = -n, ..., n$ such that $g(t) =\sum\limits_{k = -n}^n c_{k}\exp(ikt). $ ...
0
votes
0answers
23 views

Schwarz–Christoffel-like mapping on differentiable simple cubic spline boundary

For a concept of a computer game I have in mind I came to need that. I have a 2D pond, which has a boundary that is a simple differentiable cubic spline. There are ducks floating around, looking at ...
1
vote
0answers
47 views

Runge Kutta stability region for forward euler and explicit midpoint

The interval of absolute stability is the intersection of the region of absolute stability in the complex plane with the real axis.Show that Runge Kutta forward Euler and RK explicit midpoint have the ...
0
votes
0answers
44 views

How I can evalute numerically this improper complex integral?

I need a hand with the numerical evaluation, in Mathematica, for this integral: $$f(t)=\int_{-\infty}^\infty Exp\{it(\omega_H-\omega_l-\omega_k) - \sum _{j\neq(l,k)} S_j [1-e^{-it\omega_j}]\}\, dt$$ ...
0
votes
1answer
104 views

Analytical Formula for Hilbert Transform of a Ricker Wavelet

I am attempting to validate some numerical code I have to compute Hilbert transforms. As I am interested in the Hilbert transforms of functions with rapid decay, I wanted to unit test my code with the ...
0
votes
0answers
31 views

Question about picking value large enough so that an inequality holds for all values larger than said value

This question makes me wonder about more general inequalities, but I have a particular example. Let $C$ be a positive fixed constant, $0<\epsilon<1$ be given, and assume $\alpha,\beta\in ...
2
votes
0answers
30 views

Accurate computation of arcsec near branch points

The direct numerical implementations of the usual definitions of the complex $\mathrm{arcsec}(z)=\arccos(1/z)$ and similar for $\mathrm{arccsc}(z), \mathrm{arcsech}(z), $ etc are not accurate near ...
0
votes
0answers
60 views

How can I explicitly construct a *nice* conformal mapping from a triangle to a square in MATLAB?

I know the basics of the Riemann mapping theorem, SC maps, etc. I can look up formulae for the maps from the half-plane to a triangle or rectangle. But I want a particularly nice explicit map--easily ...
0
votes
0answers
48 views

How to find an error estimate for integral of curvilinear surface triangle when using quadrature

I would like to find a way to estimate the error due to the calculation of the normal when one tries to find the volume of a volume composed of quadratic surface triangles using numerical gauss ...
2
votes
1answer
57 views

Can I switch the order of integration and “Real(z)” operation?

Let $f(z,\eta)$ be an entire function. I need to calculate (numerically) the integral: $$\int\limits _{0}^{\pi}\mbox{Re}\left(\int\limits _{0}^{\pi}f\left(z,\eta\right)d\eta\right)dz$$ Can I switch ...
2
votes
1answer
109 views

Techniques for removing removable singularities (without resorting to series expansion)?

Suppose $f: \mathbb{C} \supset U \to \mathbb{C}$ is a meromorphic function with a removable singularity at the point $z_0 \in U$. Then $f$ can be extended to a holomorphic function over all of $U$. ...
1
vote
1answer
94 views

Analysis for Engineers: Where Do You Start?

Having taken none of the prerequisite rigorous treatments of mathematics during my undergrad years, I feel at a disadvantage to the people in my major what do have that analysis/abstract math ...
1
vote
2answers
117 views

Sequences and Contraction of a fixed point

Suppose that $g:\mathbb R \to $$\mathbb R$ is a contraction. Then $g$ has a unique fixed point $c$ and that for any number $x_0$, the sequence $x_0, x_1, x_2,\ldots$ given by $x_n = g(x_{n-1})$. ...
4
votes
0answers
149 views

How do zeros on the complex plane affect the real number line?

Let's say there is a real-valued "signal" that you can only measure at discrete points $f(x)$. You have a theory that this signal is the result of an analytic function $f(z)$ on the complex plane but, ...
0
votes
1answer
89 views

How can I approximate numerically the $\operatorname{Erf}(x)$ function using the Fresnel integrals?

I know that $$C(Z)+iS(Z)=(\pi/2)^{1/2}\cdot\frac{1+i}2\cdot\operatorname{Erf}(z)$$ but I do not know how to go from here, because what I want to approximate is the real value of ...
0
votes
1answer
212 views

Solution to equations involving Plasma dispersion function

I am trying to solve an equation involving a complex argument for the plasma dispersion function as: $z = x + \iota y$, $ x = \omega / \sqrt2 k v_{Ti} $ $ y = \nu_i /\sqrt{2} k v_{Ti} $ $S[z] = ...
245
votes
7answers
7k views

“The Egg:” Bizarre behavior of the roots of a family of polynomials.

In this MO post, I ran into the following family of polynomials: $$f_n(x)=\sum_{m=0}^{n}\prod_{k=0}^{m-1}\frac{x^n-x^k}{x^m-x^k}.$$ In the context of the post, $x$ was a prime number, and $f_n(x)$ ...
-2
votes
1answer
124 views

Integral question: zeroes of the primitive.

Let $z$ be a complex number. Let $f(z)$ be an elementary function but not a polynomial. Let its integral $F(z)$ be impossible to express in elementary functions. If we define $F(z)$ as $\int$ from $A$ ...
3
votes
1answer
111 views

Root bracketing in complex space

I have some function $F(\omega): \mathbb R\to\mathbb C$. The function $F(\omega)$ has both roots and singularities. Fortunately, I can calculate positions of singularities analytically. So my ...
1
vote
1answer
53 views

How to construct a polynomial with minimum deviation from zero on the complex region?

I need to compute the analog of Chebyshev polynomials (which give the minimum deviation from zero on [-1,1]) on the given region $\Omega\subset \mathbb C$. More precisely: find $P_n$ such that ...
1
vote
1answer
123 views

on the number of roots of harmonic polynomials

Maybe it is not a standard term, a harmonic polynomial is $h_{n,m}(z,\bar{z})=f_n(z)+\overline{g_m(z)}$ where $f$ and $g$ are polynomials in $z\in\mathbb{C}$ of degrees $n,m$ respectively.We may ...
1
vote
0answers
116 views

solution for related function

If we have $f(x)=(2A(A+\delta)-1)x^{2}-2Ax+1$, then the values of $x$ are equal to $\frac{1}{A+\sqrt{1-A(2\delta+A)}}$ and $\frac{1}{A-\sqrt{1-A(2\delta+A)}}$. The question is how to find the ...
4
votes
0answers
250 views

Computing complex principal value integral - sgn-function?

I currently face a less appealing integral which emerged computing the expectation of some random variable. It reads as (omitting all unnecessary constants except $\alpha\in(0,1)$) $$ PV ...
5
votes
1answer
487 views

Fixed point iteration for analytic functions on the unit disc

Suppose that $f(z)$ is complex analytic on $|z| \leq 1$ and satisfies $|f(z)| < 1$ for $|z|=1$. (a) Prove that the equation $f(z)=z$ has exactly one root (counting multiplicities) in $|z|<1$. ...
1
vote
0answers
290 views

How effective is this alternative to integration?

I have a function that is difficult to integrate. So I elect to work with power series representations. Suppose the power series representation for this function is the following: $f(x) = ...
4
votes
4answers
941 views

Finding all complex zeros of a high-degree polynomial

Given a large univariate polynomial, say of degree 200 or more, is there a procedural way of finding all the complex roots? By "roots", I mean complex decimal approximations to the roots, though the ...