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10 points Suppose f is an entire function and there is a constant c such that...
Complex Analysis
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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]
3. 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 E C and Ref (2) is an entire function. Imf (2) [10] (b) -1 < Ref(x) < 1 for all 2 € C. [8]
3. Suppose f is an entire function. Show that any of the two criteria below imply that f is a constant function. (a) Imf(x) #0 for all 2 € C and Ref(is an entire function. [10] (b) -1 < Ref(x) < 1 for all z e C. [8]
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...
Complex Analysis (use the Liouville equation):
Suppose that f(z)u, ) (, u) is an entire function such that 7u9n is bounded. Prove that fis constant Hint: Multiply f by an appropriate complex constant.
Suppose that f(z)u, ) (, u) is an entire function such that 7u9n is bounded. Prove that fis constant Hint: Multiply f by an appropriate complex constant.
Exercise 3 Let f be an analytic function on D(0,1). Suppose that f(z) < 1 for all z € C and f() = 0. Show that G) . (Hint: use the function g(z) = f(2).)
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().)
(1) Suppose that f is entire and suppose there exists a constant M such that If(n)(z)| 〈 M for all z C. Show that f must be a polynomial of degree n.
Suppose fis an entire function such that there is a number M such that Re(f(z)2 - Imf(z))2 s M for all z. Prove that f must bie Hint: Compare to exercises #8: if f and g are both entire, then so are f-g and gof. Find an appropriate g so that you may apply Liouville's theorem to the entire function gof
Find the constant a such that the function is continuous on the entire real line. f(x) = [ 5x2, x 21 ax - 5, x < 1 a =