1
vote
0answers
45 views

Does a complex polynomial can be approximated by a Euler-type exponential function?

Can one uniformly approximate a complex polynomial with a Euler-type exponential function? I mean: let $E\subset\Bbb C$ be a compact set with the connected complement $\Omega:= \bar {\Bbb C}\setminus ...
2
votes
1answer
128 views

Weierstrass Approximation Theorem for $\Bbb C$

The Weierstrass approximation theorem states that any continuous function $ f : I \rightarrow \Bbb R $ on a closed, bounded, connected subset $ I \subseteq \Bbb R $ can be uniformly approximated by ...
0
votes
0answers
32 views

Is there a way to expand Re(Li(a^z)) in series?

I'm searching a way to expand $ f(z) = Re(Li(a^z)), a \in R, z \in C $ in series. The computer-friendly, quickly convergent series is a huge plus. For being 'computer-friendly' I mean a relatively ...
2
votes
1answer
138 views

Stone-Weierstrass theorem with $p(1/x)$

I am trying to prove that for a continuous $f\colon[1,\infty)\rightarrow\mathbb R$ and $f(x)\to a$ as $x\to\infty$ it could be approximated by $g(x)=p(1/x)$ where $p$ is a polynomial.
4
votes
0answers
63 views

Can we use a sum of residues to develop an asymptotic expansion for this unknown function?

In the course of solving a particular physical problem, I have derived a relationship between two unknown functions: $$ f(s) = \frac{s \sinh{\frac{\pi s}{2}}}{2 \pi i \beta} \int_{-c- i ...
1
vote
0answers
34 views

Zeta function universality: How to compute the shift parameter for simple functions?

I've come across Zeta function universality. For a nice function $f$ in a nice subset $U$ of the complex strip between real $0$ and $1$, one can find a real $t$, such the zeta function $\zeta$ shifted ...
17
votes
3answers
348 views

Approximating $1/z$ by polynomials

Let $C=\{\mathrm e^{\mathrm it}, 0\le t\le 3\pi/2\}$ and $f(z)=1/z$. By Runge's theorem, there is a sequence of polynomials $p_n(z)$ such that $$\lim_n p_n(z)=f(z)$$ uniformly on $C$. Does anyone ...
4
votes
1answer
205 views

Approximate a complex measurable function pointwisely almost everywhere by polynomials

This is Exercise 13.12 in Rudin's Real and Complex Analysis: Let $f$ be a complex-valued measurable function defined in $\mathbb{C}$. Then there is a sequence of polynomials $P_n$ such that ...
3
votes
1answer
361 views

Application of Runge's theorem

Runge's theorem states: Let $K$ be a compact subset of $\mathbb C$ and let $S\subset \overline{\mathbb C}\setminus K$, such that $S$ contains at least one point in each connected component of ...
0
votes
1answer
318 views

Discrete approximation - exponential function and integrals

Let $f$ be a complex-valued continuous function on $\mathbb{R}_+$ with compact support and let $g, h$ be two complex-valued continuous functions on $\mathbb{R}_+$ such that $g$ is bounded and ...
14
votes
1answer
349 views

A cute approximation for $\cot(2\pi x)$(!?)

Numerical calculations and some theory leads to the suggestion that $$\cot(2\pi x) \rightarrow\frac{1}{2\pi}\sum_r \frac{1}{x-r}$$ where $r$ ranges over all the roots of $B_{2n+1}$ (Bernoulli ...
2
votes
1answer
80 views

Finding the asymptotic limit of an integral.

I'm having trouble finding the asymptotic of the integral $$ \int^{1}_{0} \ln^\lambda \frac{1}{x} dx$$ as $\lambda \rightarrow + \infty$. Can anyone help? Thank you!
3
votes
1answer
270 views

Approximation with complex polynomials on $S^1$ - can it be done?

Can one uniformly approximate a function 'similar' to identity on $S^1$ with complex polynomials? I mean a function like: $f(z)=z \cdot (1+h \cdot \sin(m\cdot Arg(z)))$, for $|h| < 1,\ m \in ...
7
votes
2answers
534 views

Approximating roots of the truncated Taylor series of $\exp$ by values of the Lambert W function

To everyone: don't bother writing up another answer, i'm giving this bounty Antonio's answer. It just doesn't let me yet (24 hours delay). If you map the nth roots of unity $z$ with the function ...
19
votes
2answers
552 views

Maximum of Polynomials in the Unit Circle

Let $z_{1},z_{2},\ldots,z_{n}$ be i.i.d random points in the unit circle ($|z_i|=1$) with uniform distribution of their angles. Consider the random polynomial $P(z)$ given by $$ ...
9
votes
2answers
594 views

Complex Zeros of $z^2e^z-z$

Can anyone give me a hint on showing (in a relatively elegant way, as I know the answer from WolframAlpha), that the complex valued function $z^2e^z-z$ has at most 2 roots with norm less than 2? ...
2
votes
1answer
501 views

Uniform approximation of continuous functions by polynomials in two variables

Consider a subset $K\subset \mathbb C^2$ consisting of pairs $(z,\bar z)$ such that $|z|=1$. Is there an easy way to see that continuous functions on $K$ can be uniformly approximated by polynomials ...
1
vote
1answer
320 views

Complex-Analytic theorem similar to Runge's theorem

I'm trying to prove a result similar to Runge's theorem and Mergelyan's theorem (link at the bottom of the previous link), but without the condition of analyticity. The problem is as follows: Let γ : ...
1
vote
0answers
291 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) = ...