# Tagged Questions

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### 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 ...
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### 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. ...
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### 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..
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### 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 ...
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### 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 ...
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### 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: ...
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### 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 ...
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### 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 ...
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### 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 $f(z)$ ...
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### 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!
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### 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 ...
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### 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 $n$: $$f^{(n)}(0)=a_n,\quad f^{(n)}(1)=b_n$$
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### 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: ...
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### 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 ...
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### 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 ...
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### 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?
### 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 ...
Is there an example of an analytic function in the unit disc whose zeros are only the points $z_n=1-1/n$?