All Questions
4 questions
3
votes
1
answer
200
views
Complex analytic function $f$ on $\mathbb{C}^n$ vanish on real sphere must vanish on complex sphere
I'm considering a complex entire function $f$: $\mathbb{C}^n\to \mathbb{C}$. Suppose $f=0$ on $\{(x_1,\cdots,x_n)\in\mathbb{R}^n:\sum_{k=1}^n x_k^2=1\}$. I want to prove
$$f=0\textit{ on } M_1=\{(z_1,\...
- 3,599
11
votes
1
answer
1k
views
Dual of the space of all bounded holomorphic functions
Let $\mathbb{B}$ be the open unit ball in $\mathbb{C}^n, n\geq 1$ and let $H^\infty (\mathbb{B})$ be the space of all bounded holomorphic functions on $\mathbb{B}$. It is well known that $H^\infty (\...
CommunityBot
- 1
9
votes
1
answer
3k
views
If $f$ is $C^{\infty}$ and $f^2$ is analytic, then $f$ is analytic
Assume that $f:\mathbb{C}^n\rightarrow \mathbb{C}$ is a $C^{\infty}$ function such that $f^2$ is (complex) analytic. Then one can show that $f$ is analytic. (Note: Liviu Nicolaescu and Alexandre ...
- 91.8k
3
votes
1
answer
2k
views
Extensions of Real Analytic to Holomorphic Functions in One & Several Variables: References?
A problem I'm working on requires the application of Cauchy's estimate for the modulus of the coefficients of a holomorphic function's power series representation, but the original functions with ...
- 91.8k