Tagged Questions

A function is analytic if it has a converging power series expansion. Please also use one of (real-analysis) or (complex-analysis) to specify real or complex analyticity: though the definition is the same, the properties of functions analytic over real numbers is quite different from the properties ...

17 views

42 views

Application of Schawarz lemma??

Let $f$ be an analytic function defined on the unit disc $D=\{z:|z|<1$. If $|f(z)|\leq 1-|z|$ for all $z\in D$ then show that $f$ is a zero function on $D$. We have $|f(z)|\leq 1-|z|$ for all $z$ ...
16 views

Mobius transformations

Suppose f is a continuous function on the extended complex plane which is analytic except possibly at one point and maps lines and circles to lines and circles. Does it follow that f is necessarily a ...
26 views

42 views

function analytic in the entire complex plane is constant

Let the function $f$ be analytic in the entire complex plane and suppose that $\frac{f(z)}{z}\rightarrow 0$ as $|z|\rightarrow \infty$. Prove that $f$ is a constant. As $\frac{f(z)}{z}\rightarrow 0$ ...
209 views

Laurent expansion of $\frac{1}{\sin z}$

Question is a fully solved exercise in Gamelin's complex analysis. Exercise : Consider the Laurent series expansion for $\frac{1}{\sin z}$ that converges on the circle $\{|z|=4\}$. Find the ...
48 views

Why is this point analytic?

Suppose you had the function $$p(x) = \frac{\sin(x)}{x}$$ I know, from other material online, that this point is analytic at the point $x=0$. However, my understanding was that a point of a function ...
31 views

How to prove this function is entire?

Given a function which Fourier coefficient decay fast as $k^{-k}$, for example \begin{align} f(x):= \sum_{k=1}^\infty \frac{1}{k^k} \exp(2\pi \, i\,k\,x) . \end{align} How can we prove this function ...
13 views

How is it possible to find all singular (non-analytic) points of a differential equation?

Suppose we had a second order differential equation of the form $$y'' + p(x)y' + q(x)y = 0$$ I know that a point $x=x_{0}$ is said to be ordinary if $p(x_{0})$ and $q(x_{0})$ can be expressed as an ...
39 views

29 views

133 views

35 views

Is the solution of a polynomial an analytic function on the polynomial parameters?

Be $\mu(z_1, \ldots, z_L)$ the only positive real solution to the equation $$\sum_{l=1}^L z_l \mu^l = 1$$ With $z_1 = 1$, $z_l \geq 0 \forall l$. Clearly, varying the ...
23 views

Prove: Show $\frac{|f'(z)|}{1-|f(z)|^2}<\frac{1}{1-|z|^2}$ for $f(z)=z^n$ and any $n \in N$

Prove that if $f:\mathbb{D} \to \mathbb{D}$ (where $\mathbb{D}$ is the unit disk) is given by $f(z)=z^2$, the for all $z \in \mathbb{D}$, we have $$\frac{|f'(z)|}{1-|f(z)|^2}<\frac{1}{1-|z|^2}$$ ...
30 views

Holomorphical extension to the Annulus

Let $D=\{z:1<|z|<2\}$ and $f$ is holomorphic on $D$. Suppose that f has a primitive $f_1$ on D and $f_1$ also has a primitive $f_2$, etc for every $n$ $f_n$ has a primitive $f_{n+1}$ in $D$. How ...
Suppose $\{f_n\}$ is a sequence of analytic in a region $D$
suppose $\{f_n\}$ is a sequence of analytic in a region $D$ such that $|f_n(z)|\leq M_n$, where $\sum_n M_n<\infty$, and $\lim_{z\to z_0}f_n(z)=L_n$. Show that if $p=p(z)\to\infty$ as $z\to z_0$, ...