(Partly in response to Gareth): Given a differential equation dz/dt = f(z) where f is complex analytic in some open set of C, we know from the existence theorem for such differential equations that for any (z,t) where the function F(z,t) is defined, there is an open set N of CxC = C^2 such that (z,t) lies in N and such that F(z,t) is analytic jointly in z and t (hence in each one separately, holding the other one fixed) in N. --Dan On 2013-05-21, at 1:30 AM, Gareth McCaughan wrote:
Dan Asimov wrote:
In the complex plane C, consider the differential equation
dz/dt = i(z^3-z) ... F(z,t) : = the solution z(t) of [dz/dt = i(z^3-z)] for which z(0) = z. ... (*) for all z in U, we have F(z,pi) = z ... (**) for all z in V, we have F(z,pi) ≠ z ... But wait, there's more. The Identity Theorem (sometimes known as the Permanence Theorem) in complex variables implies that if two analytic functions (like F(z,pi) and z) are equal on a nonempty open set in C, then they are equal everywhere.
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Why should we assume F(--,pi) is analytic?
-- g
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