is there a notion of algebraic (analytic) function?
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From: Axel Vogt on 27 Jun 2010 15:35 David Bernier wrote: > By analogy with algebraic numbers over the field Q of rational numbers, > is there a notion of a complex-analytic function that > is algebraic, that is some complex-analytic f, > f: U -> C where U is a non-empty open subset of the field C of > complex numbers which in some sense is "algebraic" ? .... Besides what already said you may search for 'meromorphic functions'.
From: Robert Israel on 27 Jun 2010 16:50
Axel Vogt <&noreply(a)axelvogt.de> writes: > David Bernier wrote: > > By analogy with algebraic numbers over the field Q of rational numbers, > > is there a notion of a complex-analytic function that > > is algebraic, that is some complex-analytic f, > > f: U -> C where U is a non-empty open subset of the field C of > > complex numbers which in some sense is "algebraic" ? > ... The standard definition: An analytic function f(z) on U is algebraic if it satisfies a polynomial equation P(f(z), z) = 0 where P(y,z) is a polynomial in two variables with complex coefficients, and degree at least 1 in y. > Besides what already said you may search for 'meromorphic functions'. You may, but it doesn't have much to do with the question. Not every meromorphic function is algebraic. Also, the only meromorphic functions on all of C that are algebraic are rational functions. All other algebraic functions, when analytically continued, have branch points. -- Robert Israel israel(a)math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada |