1. Determine if each statement is true or false. (a) If u is open and connected,

Question

1. Determine if each statement is true or false. (a) If u is open and connected, f is analytic on u, and moreover there is zo E u such that for all n 2 0, (n) 30, then f is the zero function. (b) If f and g are analytic functions on an open connected set u, and there is an open disk DCu such that f(z) g(z) for all z E D, then f(z) g(z) for all z E u. (In other words, if f and g are analytic on an open connected set u, and moreover they coincideon an open disk in u, then they coincide on u.) (c) If f is analytic on u and the open disk centered at zo with radius Ris contained in u, then the radius of convergence of the power series expansion of f at zo is at least R. (d) If u C C is open and connected, and f is analytic on u, then f has an anti-derivative on u, i.e. there is an analytic function F on u such that F'(z) f(z) for all z E u. (e) If f is an entire function such that for all z E C, lf(z) cos2(Izl), then f is the zero function. (f If f is analytic on u, has an isolated singularity at zo, and lim f(z) exists and is L, then the function f(z) f(z) zo) is analytic on uutzol.

Explanation / Answer

(a) The statement is false because f(z) is constant

(b) Let h(z) = f(z)-g(z) then h'(z) = f'(z)-g'(z) = 0 for all z in D => h(z) is analytic and constant in an open disc D in U and hence it is constant and analytic in the open and connected set U => h(z) =0 (by theorem)

=> f(z) - g(z) =0 in U => f(z) = g(z) in U

So the statement is true

(c) The statement is true

(d) The statement is false because U should be simply connected

(e) The staement is false because every bounded and entire function must be a constant but it need not be zero

(f) The statement is true

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