so f is constant
Let f(z) be entire and such that Im f(z)S cfor all z. By considering the function...
Let f(z) = ee^z . Find Re(f), Im(f) and |f|
3. Let f be an entire function whose modulus is contant on a circle centred at a. Show that f(z) = c(z - a)" for some integer n > 0 and a constant ceC.
Let f: C→C be an entire, one-to-one function. (a) Explain why g()-f() f(0) is an entire 1-1 function (b) Explain why there exists0 such that B(O,e) C g(B(O, 1)). Hint: Open Mapping thm.] (c) Explain why Ig(z)2є if 221 . [Hint: g is 1-1.] (d) Since g(0)=0, g(z)=2h(z) for some entire function h(z). Explain why h(z) is never 0 (e) Show that there is a constant C>0 such that 1/h2)l C if21 (f) Deduce that 1/h (z) is a constant...
question3 3. * Let f be an entire function which restricts to a real function f:R R on the real axis. Show that for all z e C, f(z) = f(z). (Hint: refer to Problem set 2 Qn 1.)
Q1. Let S = {z € C: Im z = 1}. Find the interior points, exterior points, boundary points and accumulation points of S. Is Sopen? Is S closed? Justify your answer. Let D be a domain in C and f:D → S be a function such that f is analytic everywhere in D, prove that f is constant throughout D. Give an example of a sequence (2n) of distinct points in that converges to i.
10 points Suppose f is an entire function and there is a constant c such that Ref(z) < c for all z. Show that f is constant. (Hint: Consider exp(f(z)).]
7. Let f be an entire function. Suppose there exists € >0 such that f(2) > € for every 2 E C. Show that f is constant. (Hint: Apply Liouville's theorem to the function g(2) = 1/f().)
Problem 8. Let f(z) = u(x, y) iv(x, y) be an entire function with real and imaginary parts u(x, y) and v(x, y). Assume that the imaginary part is bounded v(x, y) < M for every z = x+ iy. Prove that f is a constant 1
Complex Analysis Need it ASAP Suppose f is an entire function. Show that any of the two criteria below imply that f is a constant function. (a) Imf(z) #0 for all z EC and Ref(z) is an entire function. [10] (b) -1 < Ref (2) <1 for all z E C. [8]