# Questions tagged [analyticity]

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 of functions over the complex numbers.

783 questions
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### why is this complex function not analytic anywhere?

I know for $f(z)=z\bar z$, where $\bar z$ means the conjugate of $z$. Cauchy-Riemann equations are satisfied at $(0,0)$. Also, partial derivatives of U and V exist and are continuous everywhere. so ...
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### Proof Check: Unique Continuation of Analytic Functions

Here is the theorem I have been attempting to prove, I would love to hear some feedback on it! Theorem: Suppose $\Omega$ is a connected topological space, $f$ is a real analytic function and $f=0$ ...
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### Unique continuation of real analytic functions [on hold]

How does one prove this property? This is for my final year project and I'm finding it very difficult.
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### If an analytic map $f$ has “many” values in a negligible set $B$, does $\text{Image}(f) \subseteq B$?

Let $k>n$ be positive integers, and let $f:\mathbb R^n \to \mathbb R^k$ be a real-analytic map (i.e. every component of $f$ is a real-analytic function). Suppose that we have a measurable ...
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### Is a multivalued function analytic over its branch cut?

I want to evaluate the integral $\int_Cf(z)dz$, where C is the unit circle centered at the origin, and $f(z)$ = log(z+2). I know that the integral of a function over a simple, closed contour is $0$ ...
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### Are semianalytic sets Lebesgue measurable?

Let $f:\mathbb{R}^n\to\mathbb{R}$ be an analytic function, and consider the semianalytic set $E:=\{\vec x\in\mathbb{R}^n|f(\vec x)=0\}$. Is the set $E$ automatically Lebesgue measurable? Note: My ...
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### Analytic function vanishes in an open unit disk. Show it vanishes identically.

$f$ is an analytic function that vanishes in a unit disk which is a subset of domain $D$. Show that it vanishes all over $D$. This was one of our quiz questions and I think we need to assume that $D$ ...
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### For which subsets of $\mathbb{C}$ can a non-constant analytic function have an accumulation point of zeros?

Define $f$ to be complex analytic on some subset $S$ of $\mathbb{C}$ if $f$ is given locally by a power series (I'm not requiring $S$ itself to be open.) If $S$ is connected and open, then if there ...
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### Show that $e^{-1/x^2}$ is not analytic around $x=0$.

I have been working on the following question. Define a function \begin{align*} f(x)= \begin{cases} e^{-1/x^2}&\text{ for }x>0,\\ 0&\text{ for }x=0 \end{cases} \end{align*} Prove that $f$ ...