# Questions tagged [analytic-functions]

For questions about analytic functions, which are real or complex functions locally given by a convergent power series.

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### Can we write every $C^1$ complex function on the unit circle as the the difference of two approriate functions?

If $g$ is $C^1$ on the unit circle $C(0,1)$. Then there is a function $f^+$ holomorphic on $B(0,1)$ and continuous on $\bar B(0,1)$, a function $f^-$ holomorphic on $\mathbb{C}\backslash\bar B(0,1)$ ...
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1 vote
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### Smooth Riemannian metric is locally real analytic?

Let $U$ be an open subset of $\mathbb{R}^n$ and $g$ be a $C^\infty$ Riemannian metric on $U$. Given a point $x_0\in U$, does there exist a local neighborhood $x_0\in V\subset U$ and new coordinates ...
1 vote
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### Decomposition of analytic functions on the upper half plane

Let $$h:\mathbb{H}:=\{z\in \mathbb{C}:\text{Im}(z)>0\}\to \mathbb{H}\cup \mathbb{R}$$ be an analytic function. We decompose $h$, using partial fraction decomposition, into two parts $h_1$ and $h_2$,...
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### Real analytic function on $X$ is subanalytic on compact subsets $Q\subseteq X$.

$\textbf{Background.}$ I'm trying to apply Lojasiewicz's inequality to a specific function, $f$, and the Euclidean distance function $d$. Hence, I need to prove that $f$ and $d$ are subanalytic. ...
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### Proving that $f$ is rational if it is known that $f$ is analytic in $\mathbb{D}$ and satisfies $|f(z)| \to 1$ as $|z| \to 1$

This question was already asked here, I wouldn’t ask it again however I do not think that the question gets the point across that it’s trying to get across, and I can’t find anything else related to ...
• 2,129
1 vote
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### If an analytic function is identically zero in a domain, can it be always divided by another analytic function in that domain?

Let's say I have two functions $f$ and $g$, both analytic on the same domain (open and connected set) $D$, and suppose also I was able to prove that $f = 0$ on the entire domain. Question. Is it ...
• 583
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### Radius of convergence of a power series defined in terms of co-efficients around another point

I'm trying to answer the following question, having a hard time figuring out where to begin. Any help appreciated: Let f be analytic everywhere it is defined, and for $|z| < 2$ the following holds: ...
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### $f=\phi(v)+\textbf i v$ is analytic, prove that $f$ is constant

Statement: Complex function $f=u+\textbf i v$ is analytic in domain $D\subset \mathbb C$ and everywhere in $D$ $u=\phi(v)$, where $\phi=\phi(t)\in C^{1}(\mathbb R\to\mathbb R)$ and $\phi$ is strictly ...
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### Analytically continuing a function of two complex variables.

I am aware of the identity theorem and how it allows us to extend the definition of a complex analytic function from $A \subset \mathbb{C}$ to a larger domain $D$ that is an open subset of the complex ...
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### Characterizing the unimodular functions from the closed disk $\mathbb{C}$ to $\mathbb{C}$ with constraints

It is well known that if $f:\mathbb{D}\to\mathbb{C}$ is analytic, continuous on the boundary, and is unimodular (say with a finite number of zeros) then $f$ is a finite Blaschke product up to some ...
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1 vote
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### Meaning of "$f$ has a power series expansion around $p$"

In Complex Analysis by Donald Marshall (page 29), there is an exercise problem that starts with "Suppose $f$ has a power series expansion at $0$ which converges in all of $\mathbb{C}$. " ...
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### Computing the domain of analyticity of $f(z)=\sqrt{z^2-1}$
In this question, it is said that the domain of analyticity of the function $f(z)=\sqrt{z^2-1}$ over the branch $(0,2\pi)$ is $\mathbb{C} \setminus ((-\infty,-1) \cup (1,\infty))$. My question: I ...