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assume f(z) is an entire function [analytical on the whole plane], that holds for every n (natural) ==> if |z|=n^2 then |f(z)|<13. prove that f(z) is a constant function.

If f is not constant, then . But remains bounded, a contradiction.

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assume f(z) is a Meromorphic function [its finite singular points are only poles] on C, that holds for every z (complex) ==> f(z)=f(2f(z)). find all the functions f(z) [general form].

Consider the function near a pole: , where and are finite. Write and show that poles change order.

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ssume f(z) is periodical and analytical on a domain D that contains the point z=infinity. prove that f(z)=const on D.

Again, we must have . But f is periodical, and thus remains bounded.

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assume f(z) is analytical on a bounded domain D, and continuous on D* (with the "boundary"). assume also that f is not constant on D, but |f| is constant on the "boundary". prove that f has a zero on D.

Suppose not. Let . Consider a fixed and the function , which has a zero at z=a. Now . Apply Rouche's theorem on the curve .

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prove that f(z) and h(z)=f(z+z^2) have the same "kind" of singular point on z=0, and if it is a zero or a pole, then the order is also the same.

Near zero, let . Note that

,

where w is analytic at 0.