Saturday, 9 April 2016

complex analysis - entire function with ist imaginary part positive





f is an entire function, and it satisfies f(C){zCImz>0}. Show that f is constant.




I want to take advantage of the Liouville's Theorem, but I just can't figure out the relationship between its image part with its module.


Answer



Consider g(z)=1f(z)+i.

Clearly it is entire, and |g(z)|=1|f(z)+i|11=1.
Thus, g(z) is constant. Applying Liouville's theorem, g(z) is constant, implying f(z) is constant.



NOTE: This argument can be used to show that f(C) is dense in C by using a proof by contradiction.


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