# 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 ...

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### conformal mapping, regions of the complex plane marked +/-, find the function f,

The picture shows what the function f: $\mathbb{C}\to\mathbb{C}\cup\infty$ does to the planes. The values 0 at 0, 1 at $\pm$1, and $\infty$ at $\pm i$ are specified. To elaborate on the picture: ...
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### Analytic function on the whole plane, positive imaginary part, what can it be?

Part (a): The function f is analytic in the whole plane with positive imaginary part. What can it be? Part (b): What if all you know is that the imaginary part of f tends to 0 at infinity? what we ...
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### Analytic Functions: Notation? [duplicate]

Analytic functions are usually denoted by $\mathcal{C}^\omega$. What does the $\omega$ stand for? (The infinity symbols of a colleague of mine really look like omegas...)
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### A function property to guarantee that being constant on an interval implies identically constant

Let $f:\mathbb R\rightarrow \mathbb R$. Suppose we know that $f$ is a constant on some open/closed interval. Which condition does guarantee that $f$ is constant on $\mathbb R$? Clearly, continuity ...
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### Solutions of complex equation for analytic function

Suppose $g(z)$ is an analytic function on the upper plane, in fact is constructed by Hilbert transform, g(z) = g_R(x,y) + i g_I(x,y) \quad g_I = \mathcal{H}( g_R) + \text{real ...
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### Complex analysis question, maximum principle application

Let $\Omega=\{z, \text{Re}z>0\}$ Suppose that $f$ is continuous in the closure of $\Omega$ and $f$ is holomoprhic on $\Omega$ and there are constants $A<\infty$ and $\alpha<1$ such that ...
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### prove that a function is expressible as a power series

I started by rearrange f(z), and expanded the terms in summation. Then, I did not get very far. It would be great if anyone can let me know what is needed to figure out bn. Thanks in advance.
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### Function holomorphic in the neighb. of zero, bounded by exponent is equal 0

I want to prove that if $f$ is a holomorphic function in a neighbourhood of $0$ and $|f(\frac{1}{n})| \le \frac{1}{e^n}$ for $n$ sufficiently big, then $f =0$. I know that if $f$ is holomorphic in a ...
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### linear first order ODE with polynomial coeffficients

I'm wondering if anything can be said about the solution to a system of linear first order ODEs with polynomial coefficients, especially about analyticity of the solution. The equation is given as ...
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### Local Estimates for higher order homogeneous elliptic operators

For $u\in W^{2k}_2(\mathbb{R^n})$, $k\geq 1$, it is well known (see, for example, Exercise 12.9.4 in Krylov, N. "Lectures on Elliptic and Parabolic Equations in Sobolev Spaces") that the following ...
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### Preimage of the set of critical value of an analytic function between smooth manifolds

I have some problems with the following exercise, maybe due to alack of knowledge: Let $M$ be a connected smooth manifold and let $$f \colon M \to N$$ be an analytic map. Denote by $C_f \subset M$ ...
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### Suppose $f$ is entire and $|f(z)| \leq 1/|Re z|^2$ for all $z$. Show that $f$ is identically $0$.

This is a problem from my complex analysis textbook. The hint is to consider $g(z)=(z-iR)^2(z+iR)^2 f(z)$ and to show that $|g(z)| \leq 8R^2$. This is what i have tried: Consider $Re z \geq 0$, then ...
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### Approximating an L2 function by analytic functions

Spose I have a $h \in L^2(U)$ where $U \subset \mathbb{R}^3$ is open and bounded. Is it possible to approximate this by analytic functions? If so, spose now we take $U = \mathbb{R}^3$. Is this ...
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### Real analytic function in one point and zeros

Let $f:I\to\mathbb{R}$ be a continuous function and $t_0\in I$ such that $f$ is analytic at $t_0$ and $f(t_0)=0$. Is it true that there is an entire neighbourhood of $t_0$ in which $f$ has no other ...
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