0
votes
1answer
49 views

Pair of functions $F(x)$ (transcendental),$A(x)$ (algebraic) with expanded series of positive integer coefficient linked by derivative

$$F(x)=\sum_0^{\infty}b_k x^k,b_k\in \mathcal{N} \bigcup 0,\exists M \space b_k \leq M^k$$. $$A(x)=\sum_0^{\infty}a_k x^k,a_k\in \mathcal{N} \bigcup 0,\exists M \space a_k \leq L^k$$ where $F(x)$ is ...
6
votes
1answer
43 views

Divergence set at radius of convergence

I came up with this question on my own while I was musing around reviewing notes. After unsuccessful Google search (thwarted by a deluge amount of webpages on basic calculus), I decided to ask here. ...
1
vote
1answer
28 views

what would be an example of a function such that $\int_{|z|=r} f(z)=0$ for all $r>0$, but not analytic everywhere?

what would be an example of a function such that $\int_{|z|=r} f(z)=0$ for all $r>0$, but not analytic everywhere. I cannot think of one..
0
votes
0answers
47 views

maximum modulus principle analytic function

I am trying to show that: Let $f$ be analytic on a given closed unit disc $D$ then prove that for every $k\in\mathbb N$ there is $w\in Bd(D)$ such that $|f(z)-w^{-k}|≥1.$ where z is in the unit disc ...
1
vote
1answer
63 views

Counterexample for Hartogs' Extension Theorem

I'll refer to Hartogs' Extension Theorem as it is stated in Wikipedia (https://en.wikipedia.org/wiki/Hartogs%27_extension_theorem#Formal_statement). I am trying to find a counterexample to show that ...
1
vote
1answer
95 views

Why doesn't the identity theorem for holomorphic functions work for real-differentiable functions?

I've been fascinated by the idea of analytic continuation and I came across the identity theorem for holomorphic functions. (http://en.wikipedia.org/wiki/Identity_theorem) On wikipedia it states: ...
0
votes
1answer
53 views

Analouge of the Mean value theorem for holomorphic functions

Let $f:\mathbb{C}\rightarrow \mathbb{C}$ be entire. Let $w_1,w_2$ be any two distinct complex numbers. Must there exist $c\in \overline{B_{|w_2-w_1|}(w_1)}$ such that ...
0
votes
1answer
122 views

Explicitly showing cokernel of exponential sequence is not a sheaf

In the classical example of short exact sequence of presheaves $$0\rightarrow \mathbb Z \rightarrow \mathcal O_X \rightarrow \mathcal F \rightarrow 1,$$ it is well known that $\mathcal F$ is not a ...
3
votes
3answers
87 views

Counterexample for a complex analysis proof

I'm having troubles coming up with a counterexample for the following: If $|f(z)|$ is continuous at $z_0$, then the function $f(z)$ is continuous at $z_0$ for complex numbers. I know I need a $f(z)$ ...
8
votes
2answers
2k views

What's the difference between Complex infinity and undefined?

Can somebody please expand upon the specific meaning of these two similar mathematical ideas and provide usage examples of each one? Thank you!
1
vote
1answer
540 views

Derivative of complex-valued function and partial derivatives.

Let $f(x+iy)=u(x,y)+i\,v(x,y)$ Cauchy-Riemann Equations are satisfied at $z_0$ $u, v, u_x, u_y, v_x, v_y$ are defined on some open neighbourhood of $z_0$ $u, v, u_x, u_y, v_x, v_y$ are continuous at ...
3
votes
2answers
175 views

finding a function with predetermined $f^{(n)}$s at $0$ and $1$.

Let $a_n,b_n$ be 2 arbitrary sequences of real numbers. Is there any $C^\infty$ function $f:\mathbb{R}\to\mathbb{R}$ such that for any $n$: $$f^{(n)}(0)=a_n,\quad f^{(n)}(1)=b_n$$
7
votes
2answers
527 views

True/False Questions for Complex Analysis

I am studying for my (introductory) complex analysis final exam tomorrow. I am practicing an old final exam, which unfortunately has no answer key. Here is a link: ...
11
votes
4answers
454 views

Counterexamples in complex analysis

In contrast to other topics in analysis such as functional analysis with its vast amount of counterexamples to intuitively correct looking statements (see here for an example), everything in complex ...
2
votes
3answers
314 views

Analytic Function with positive integers as zeros?

Do you know any nontrivial analytic function f(z) with zeros only at positive integer values of the argument z = 1, 2, 3, 4, ... ? If yes, please give some example. PS: I already thought of ...
4
votes
1answer
194 views

Explicit counter-example to corona problem

The corona problem is known to fail for the complex polydisk, for dimension greater than 2. Does anyone has an explicit example of such functions?
1
vote
1answer
105 views

Is there any complex-valued $C^\infty$ function $f(z)$ coincide with $z^{3/2}$ infinitely many times?

I want a $C^\infty$ function $f:U\rightarrow\mathbb{C}$, where $U\subset\mathbb{C}$ is a neighborhood of $0$, such that there exists a sequence of points $z_n\in U-\{0\}$ and ...
0
votes
3answers
135 views

Analytic function with assigned zeros

Is there an example of an analytic function in the unit disc whose zeros are only the points $z_n=1-1/n$?