Tagged Questions

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an inequality derived from conformal automorphisms of unit disk

Let $f$ be a holomorphic function on $D(0,1)$ such that $|f(z)|<1$ for all $z\in D(0,1)$. I have obtained $$\frac{|f(0)|-|z|}{1+|f(0)||z|}\leq |f(z)|\leq \frac{|f(0)|+|z|}{1-|f(0)||z|}.$$ Is it ...
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The functions $\{f_n(x) = n\}$ are analytic and each miss the points $-2, -3$. But, they are not a normal family. So what am I missing. Thanks.

Here is a theorem of Montel: Let $\mathcal{F}$ be a family of analytic functions defined on a domain $\Omega$ . If there are two fixed complex numbers $a$ and $b$ that are omitted from the range of ...
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Prove that A is both open and closed. [closed]

Let $X = \{ z : | z | \leq 1 \} \cup \{ z : | z - 3 | < 1 \}$ be a subset of $\mathbb{C}$. Let the metric be the usual metric $d(x,y) = | x-y |$. Prove that the set A = $\{ z : | z | \leq 1 \}$ ...
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How to get a function if you have the Fourier coefficients

So I have $$H(e^{i\omega})=\sum_{n=-\infty}^\infty C_ne^{i\omega n}$$ and I know that: $$C_n = \frac{2}{\pi n}\sin^2\left(\frac{\pi n}{2}\right)$$ How can I work out the function that this makes? I ...
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If $f(z):=\sum_{n=0}^\infty a_nz^{-n}$ is compact convergent, then $f$ is holomorphic

Let $\left(a_n\right)_{n\in\mathbb{N}}\subset\mathbb{C}$ such that $$f(z):=\sum_{n=0}^\infty a_nz^{-n}$$ is compact convergent on $B_r(0)\setminus\left\{0\right\}$. I want to show: $f$ is ...
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value of an integral depending on a parameter in complex plane

For each $z\in\mathbb{C}$, evaluate the integral $$\int_0^1\int_0^{2\pi}\frac{1}{re^{i\theta}+z}d\theta dr.$$ How to evaluate it? Thanks.
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Cauchy's Integral Formula: conditions vs singularities

I'm sure this is a simple misunderstanding but it was annoying me. So using the version of Cauchy's Integral Formula given on Wikipedia http://en.wikipedia.org/wiki/Cauchy's_integral_formula, it is ...
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$\int _{C^{+}(0,3)} \frac {dz}{2-\sin z}$

$$\int _{C^{+}(0,3)} \frac {dz}{2-\sin z},$$ $z$ is complex. I have no idea how to solve $2-\sin z$. I will be really grateful for any help
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Is residue may be equal to infinity?

Is residue may be equal to infinity? Is it possible?
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Solving the ODE $[(1-x^2)\frac{\partial}{\partial x} - \lambda]f = [\frac{\partial}{\partial x} - \frac{\lambda}{a}]g$

I want to solve $f(x)$ in terms of $g(x)$ in the following ODE $$\left[(1-x^2)\frac{\partial}{\partial x} - \lambda\right]f(x) = \left[\frac{\partial}{\partial x} - \frac{\lambda}{a}\right]g(x),$$ ...
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