Example-37. Find an analytic function f(z) such that \operatorn... | Filo

Give an example of a nonvanishing analytic function f in the unit disc |z|<1 having infinitely many zeros. | Numerade

SOLVED:Let a function f(z) be analytic in a domain D. Prove that f(z) must be constant throughout D if (a) f(z) is real-valued for all z in D; (b) |f(z)| is constant throughout D. Suggestion: Use the Cauchy-Riemann equations and the theorem in Sec. 23 to prove part (a) . To prove part (b), observe that f(z)=(c^2)/(f(z)) if |f(z)|=c(c ≠0) then use the main result in Example 3, Sec. 24 .