All Questions
7 questions
3
votes
0
answers
75
views
Separate holomorphicity implies holomorphicity on analytic varieties
Suppose that $M$ and $N$ are two complex analytic varities and suppose that $f\colon M\times N \to \mathbb{C}$ is a map. Further assume that $f$ is such that for every $p\in M$ the map $f(p,\cdot)\...
- 513
2
votes
1
answer
271
views
Dimension of intersection of real analytic sets
Suppose that $A,B$ are real analytic subsets of $\Omega\subseteq \mathbb{R}^n$ and $p\in A\cap B \neq \emptyset$. Does the intersection inequality from complex analysis still hold, i.e. does the ...
- 4,131
1
vote
0
answers
112
views
A problem related to analytic function
Let, $z,w\in \mathbb{C}$. Let, $f(z)$ be an analytic function in $|z|<1$. Define, $f(z)= g(w)$ where $g(w)$ is analytic function in $\Re(w)>1/2$ and $w=\frac{1}{1+z^2}$ .
Question Prove that $$\...
- 63.9k
3
votes
1
answer
179
views
Analytic or holomorphic extension of the ellipse perimeter function
Let ${\mathbb{R}^2}^+=\{(x,y)\in \mathbb{R}^2\mid x>0, y>0\}$.
Let $P:{\mathbb{R}^2}^+\to \mathbb{R}$ be the function with $P(a,b)=$ $\text{The perimeter of ellipse}\;\; \frac{x^2}{a^2}+\frac{y^...
- 41.1k
4
votes
1
answer
119
views
minimizing weighted length of closed curves
Let $\mathcal{A}$ be the family of closed smooth curves in the right half of the complex plane $\mathbb{C}$ such that any curve in the family must enlose the point $z=1$ and tangent to the $y$-axis at ...
- 1,350
6
votes
1
answer
409
views
Can the potential of a complete Kahler metric be bounded?
Let $X$ be a complex manifold and $\omega$ a Kahler form on $X$. A smooth function $\rho$ is called a potential of $\omega$ if $i\partial\bar\partial\rho=\omega$. By intuition, it seems that $\rho$ ...
- 858
4
votes
0
answers
714
views
Can one integrate around a branch-cut?
How meaningful is it to try to integrate around the branch-cut of a function?
For example lets say I have the function $\log(z^2+a^2)$ for $a>0$ and I choose my branch-cuts to be starting at $\pm ...
- 1,893