Q. on Mittag Leffler theorem and analytic sheaf cohomology

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In summary, the Mittag-Leffler theorem is a mathematical theorem that states the existence of a sequence of rational functions that converge to a given analytic function on a connected open set in the complex plane. Analytic sheaf cohomology is a tool used to study the topological properties of complex manifolds using sheaves. The Mittag-Leffler theorem is often used in analytic sheaf cohomology to construct cohomology classes. Both have various applications in mathematics, including complex analysis, algebraic geometry, and number theory. There are also generalizations of the Mittag-Leffler theorem for meromorphic functions on Riemann surfaces and algebraic varieties.
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See http://camoo.freeshell.org/cohomquest.pdf" [Broken]
thanks
Laura
 
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The answer is yes, you could express [tex]e^{1/sin z}[/tex] as [tex]f+g[/tex], [tex]f[/tex] analytic in [tex]Re(z)>\pi/2[/tex] and [tex]g[/tex] analytic in [tex]Re(z)<\pi[/tex]! Pretty surprising, but I looked it up in Lars Hormander's book "An introduction to complex analysis in several variables". The proof isn't all that difficult, tho I didn't read the proofs before that one.
Laura
 

1. What is the Mittag-Leffler theorem?

The Mittag-Leffler theorem is a mathematical theorem that states that for any analytic function on a connected open set in the complex plane, there exists a sequence of rational functions that converge to the given function uniformly on compact subsets of the open set.

2. What is analytic sheaf cohomology?

Analytic sheaf cohomology is a mathematical tool used to study the topological properties of complex manifolds. It involves using sheaves, which are mathematical objects that associate algebraic structures to topological spaces, to study the cohomology groups of a complex manifold.

3. How are the Mittag-Leffler theorem and analytic sheaf cohomology related?

The Mittag-Leffler theorem is often used in analytic sheaf cohomology to construct cohomology classes. It provides a way to find rational approximations to analytic functions, which can then be used to construct sheaf cohomology classes.

4. What are some applications of the Mittag-Leffler theorem and analytic sheaf cohomology?

Both the Mittag-Leffler theorem and analytic sheaf cohomology have many applications in complex analysis, algebraic geometry, and number theory. They are used to study properties of algebraic varieties, Riemann surfaces, and abelian varieties, among other mathematical objects.

5. Are there any generalizations of the Mittag-Leffler theorem?

Yes, there are several generalizations of the Mittag-Leffler theorem, including the Mittag-Leffler theorem for meromorphic functions on Riemann surfaces and the Mittag-Leffler theorem for meromorphic functions on algebraic varieties. These generalizations extend the theorem to more general classes of analytic functions.

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