Decide which of the following statements are true and which are false. Prove the true statements...

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Decide which of the following statements are true and which are false. Prove the true statements and give a counterexample to the false statements. (a) For any z E C, sin? z + cos? z = 1. (b) If f(2) is a polynomial of order > 1 then lim, f(2) = 0. (c) Let f be an analytic function on some domain D with f <1 throughout D. Then f is constant. (d) If f(2) is analytic on a domain 2 and |f(2)| attains its minimum at some point zo €N then f is constant. (d) The limit sin z lim 0. (e) If f(2) is a complex valued function defined on some domain 2 such that u = Re f and v = Im f are harmonic then f is analytic on 2. Decide which of the following statements are true and which are false. Prove the true statements and give a counterexample to the false statements. (a) For any z E C, sin? z + cos? z = 1. (b) If f(2) is a polynomial of order > 1 then lim, f(2) = 0. (c) Let f be an analytic function on some domain D with f <1 throughout D. Then f is constant. (d) If f(2) is analytic on a domain 2 and |f(2)| attains its minimum at some point zo €N then f is constant. (d) The limit sin z lim 0. (e) If f(2) is a complex valued function defined on some domain 2 such that u = Re f and v = Im f are harmonic then f is analytic on 2.

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