# meromorphic map and poles

• Apr 20th 2011, 03:32 PM
poeish
meromorphic map and poles
1) Let f be meromorphic in C. Show that if f(1/z) is also meromorphic then f is the quotient of two polynomials.

2) Let a be an isolated singularity of f. Then exp(f(z)) has no pole in a.

Regarding 1), I am quite clueless. Regarding 2) it is clear that if a is avoidable for f, then it is avoidable for exp o f, because exp o g is an holomorphic extension of exp o f, if g is one of f.
But if a is a pole of f, I cant see why it should not be one of exp o f, for isnt lim_z->a e^f(z) = oo, because lim_z->a f(z) is oo?. I guess I shoul try and prove it is an esential singularity, as happens with e^(1\z) in 0.

Thanks
• Apr 23rd 2011, 04:11 AM
FernandoRevilla
Quote:

Originally Posted by poeish
1) Let f be meromorphic in C. Show that if f(1/z) is also meromorphic then f is the quotient of two polynomials.

Hint : If f ( z ) and f ( 1 / z ) are meromorphic then, z_0 = infty is a pole or a removable singularity.