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        general >> complex analysis >> composition
        
(Message started by: Moon on Apr 20^th, 2004, 1:15pm)

Title: composition
Post by Moon on Apr 20^th, 2004, 1:15pm

Please help me to solve this problem.

Let f : D(P,r)\{P}->C be holomorphic. Let U=f(D(P,r)\{P}). Assume that U is open. Let g:U->C be holomorphic. If f has a removable singularity at P, does g(f(z)) have one also? What about the case of poles and essential singularities?

I'm trying to figure out what kind of singularity function exp(sinx/x) has at 0. If it's expanded as a Laurent expansion than it's so mess. So I don't know.

Thank you.

Title: Re: composition
Post by Icarus on Apr 20^th, 2004, 5:45pm

Let Q = lim_z[to]P f(z). The behavior of g^of at P will be the same as the behavior of g at Q. The conditions you give are enough to know that Q is a limit point of U, but not necessarily inside U. So g could have a pole or essential singularity at Q. If g has a pole, then g^of has one as well, though the order of the pole may be different. If g has an essential singularity, then so does g^of. If g is holomorphic (or has a removable singularity) at Q, then g^of has a removable singularity.

A removable singularity is not a true singularity. It is just a point where the "normal" definition of the function breaks down. When you see a function with a removable singularity, you should think of the singularity as having already been removed. The function is really holomorphic there. Therefore any theorem that applies to holomorphic functions also applies to those with removable singularities. Including that the composition of two holomorphic functions is also holomorphic.

If f has a pole at P, then the same thing applies, except now Q = [infty]. The behavior of g at [infty] is the behavior of g^of at P.

If f has an essential singularity at P, then unless g is constant, g^of has an essential singularity at P.

Concerning e^{(sin x)/x}, perhaps it would help if you recalled that (sin x)/x = 1 - x²/3! + x⁴/5! - ... . All the derivatives of (sin x)/x at 0 are there for the reading.