exsheet4a - Sivakumar Example Sheet 4a M407 1. Let m be a...

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Unformatted text preview: Sivakumar Example Sheet 4a M407 1. Let m be a fixed positive integer. Suppose that a0 , . . . , am are complex numbers and am = 0. m Define P (z ) := k=0 ak z k , z ∈ C. Show that lim P (z ) = ∞ . z →∞ 2. Define T (z ) := z−i , z+i z = −i . (i) Show that T is a one-to-one function (i.e., T (z1 ) = T (z2 ) implies z1 = z2 ). (ii) Show that T is continuous at every point in its domain. (iii) Show that lim T (z ) = ∞. z →−i (iv) Show that T (H↑ ) = D(0; 1). (This involves proving two statements: for any z ∈ H↑ , T (z ) ∈ D(0; 1). Conversely, given any w ∈ D(0; 1), there is a z ∈ H↑ such that T (z ) = w.) Definition. Let S be a nonempty subset of the complex plane, and suppose that f : S → C is a function. Given a subset A of C, the inverse image of A under f is defined as follows: f −1 (A) := {s ∈ S : f (s) ∈ A} . 3. Suppose that f : C → C is a function. Show that the following are equivalent: (i) f is continuous on C, that is, f is continuous at every z0 ∈ C. (ii) f −1 (V ) is an open set for every open set in C. 4. (i) Show that ez1 /ez2 = ez1 −z2 for every pair of complex numbers z1 and z2 . (ii) Verify that e0 = 1 and conclude, via (i), that 1/ez = e−z for every complex number z . (iii) Use mathematical induction to show that (ez )n = enz for every z ∈ C and every positive integer n. (iv) Show that the formula in (iii) also obtains for every negative integer n. 5. Find all complex numbers z such that E (2z − 1) = 1. 6. Show that |exp(z 2 )| ≤ exp(|z |2 ) for every complex number z . 7. Prove that |exp(−2z )| < 1 if and only if (z ) > 0. 1 ...
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This note was uploaded on 03/08/2010 for the course MATH 407 taught by Professor Staff during the Fall '08 term at Texas A&M.

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