# Tagged Questions

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### Is there an explicit formula for the modulus of an annulus given a parameterization of the inner and outer boundries?

Every open set in the complex plane homeomorphic to an annulus is biholomorphic to exactly one annulus whose inner radius is 1 and whose out radius is $r>1$. Each value of $r$ gives a different ...
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### 2D Problems Which are Easier to Solve in 3D

It sometimes happens that 1D problems are easier to solve by somehow adding a dimension. For example, we convert linear differential equations for a real unknown to a complex unknown (to use complex ...
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### If Fatou set has a Multiply connected Fatou component implies every component of F(f) is bounded

Recently when I read a paper about Fatou component, I met the following theorem which cited in Professor Eremenko's paper "on the iteration of entire functions" If Fatou set has a Multiply connected ...
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### Almost analytic continuation

Let $f\in S^{\alpha}$ for some $\alpha \in \mathbb{R}$(which means that f is smooth and satisfies $|D^{\beta}f|\leq C(1+|x|)^{\alpha-\beta}$),a function $\tilde{f}$ on $\mathbb{C}$ is called an almost ...
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### Lambert $W_{-1}(x)$ as $x\rightarrow 0^-$: Asymptotic behavior

There are well known bounds for $W_0$, the "principal" real-valued branch of the Lambert-W function. For example, $W_0(x)$ lies between $\log x - \log\log x$ and $\log x - \frac{1}{2}\log \log x$, ...
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### Is Euler's formula a theorem or a definition? [closed]

The first time I tried to understand Euler's formula was about 2 years ago. I didn't need it, I just randomly ran across it, when trying to understand a Fourier transformation. The problem was, that I ...
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### Real function to entire functions

Let $g:[0,\infty) \rightarrow \mathbb{R}$ be an increasing function. Is there a way to construct an entire function $f(z)$ such that $f(x)=g(|x|)$ for all real $x$?
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### Characteristic Function of a Non-negative Random Variable Evaluated at a Complex Value

Suppose we have a non-negative random variable $X$ with density $p(x)$,and its characteristic function, evaluated at a complex number $z$, being $\phi(z)=E[e^{z X}]=\int_{0}^{\infty}e^{zx}p(x)dx$. It ...
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### Diametrically opposite points go to diametrically opposite points under stereographic projection [closed]

I asked this question in MSE here: http://math.stackexchange.com/questions/184524/diametrically-opposite-points-go-to-diametrically-opposite-points-under-stereogr but I didn’t get an answer. I really ...
Dear fellows, I have come to another conclusion which must be wrong. Let $f$ be a rational map and let $U$ be a connected but not simply connected open subset of the Fatou set such that $f(U)$ is ...