5
votes
0answers
39 views

Reference for the fact that a smooth function analytic on every line is itself analytic

Let $f \in \mathcal C^\infty(\mathbb R^p)$ ($p \geq 2$) be a smooth function such that the functions $g_d(t) := f(td)$ are all analytic for all $t \in \mathbb R$ and all $d \in \mathbb R^p.$ (i.e. $f$ ...
1
vote
1answer
15 views

Extending a holomorphic function to a radial limit function for almost every angle

I've read in several places about the "well known theorem" which states that a holomorphic function on the (open) unit disk $D=\{z\in\mathbb{C}:\ |z|< 1\}$ can be extended to its boundary on almost ...
0
votes
1answer
32 views

About a reference collecting the main properties of the modulus and the argument of $f$

Let $f$ be an analytic function in the whole complex plane. We can write $f$ in its polar form: $$f(z)=ρ(z)exp(iθ(z))$$ My question is about a reference collecting the main properties of the modulus ...
1
vote
1answer
67 views

Continuous dependence of zeros on a parameter

Let $F:I\times J\to\mathbb{R}$ be a $C^k$ (or analytic) function, with $I,J$ real open intervals. Set $f_\lambda(x):=F(\lambda,x)$ and consider the parametric equation $$f_\lambda(x)=0\,.$$ Assume its ...
11
votes
1answer
309 views

Images of compact subsets in the plane

Let $K$ be an infinite compact subset of $\mathbb{C}$. Is it true that there exists a sequence $(f_n)_{n>0}$ of functions holomorphic in some neighborhood of $K$, such that the images $f_n(K)$ are ...
0
votes
1answer
212 views

A consequence of Runge's theorem

I'd like to have a reference for the proof of the following fact of complex analysis. I think it follows from Runge's theorem, but I don't know how to prove it. Fact. Let $U \subseteq V \subseteq ...