Functional Analysis

Download Complex Analysis: In the Spirit of Lipman Bers by Jane P. Gilman, Irwin Kra, Rubi E. Rodriguez PDF

By Jane P. Gilman, Irwin Kra, Rubi E. Rodriguez

The authors' objective is to provide an actual and concise therapy of these components of complicated research that are meant to be general to each learn mathematician. They stick to a direction within the culture of Ahlfors and Bers by way of dedicating the publication to a really special aim: the assertion and evidence of the elemental Theorem for capabilities of 1 complicated variable. They talk about the various an identical methods of knowing the idea that of analyticity, and provide a relaxation exploration of attention-grabbing results and functions. Readers must have had undergraduate classes in complicated calculus, linear algebra, and a few summary algebra. No historical past in complicated research is needed.

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B) The series an z n diverges for |z| > ρ. Proof. (a) Choose r0 with r < r0 < ρ such that |an | r0n < +∞. n Thus there exists an M > 0 with |an | r0 ≤ M for all n in Z>0 . Now apply Abel’s lemma. (b) We claim that for |z| > ρ, the sequence {|an | |z|n } is not even bounded. Otherwise Abel’s lemma (with r0 = |z|) would guarantee the existence of an r with ρ < r < |z| and |an | r n < +∞. This contradicts the definition of ρ. 8. Let an z n be a power series with radius of convergence ρ. Then the function defined by S(z) = an z n is continuous for |z| < ρ.

Sn−1 (z n−1 − z n ) + sn z n = (1 − z)(s0 + s1 z + . . + sn−1 z n−1 ) + sn z n . Now ∞ f (z) = lim Sn (z) = (1 − z) n→∞ sn z n . n=0 36 3. POWER SERIES Given > 0, choose N ∈ Z>0 such that |sn | < for n > N. Then ∞ N |f (z)| ≤ |1 − z| |sn | |z|n sn z n + n=N +1 n=0 N ≤ |1 − z| sn z n=0 n |z|N +1 + 1 − |z| N ≤ |1 − z| sn z n + M. n=0 Thus we conclude that limz→1 f (z) = 0. 21. Observe that we have not needed or used polar coordinates in our formal development thus far. 3. The exponential function, the logarithm function, and some complex trigonometric functions In this section we use power series to develop several functions.

Support your answer with either a proof or a counterexample. 11. (a) What are all the possible values of ıı ? (b) Let a and b ∈ C with a = 0. Find necessary and sufficient conditions for ab to consist of infinitely many distinct values. (c) Let n be a positive integer. Find necessary and sufficient conditions for ab to consist of n distinct values. 12. (a) Show that both the sine and the cosine functions are periodic with period 2π. (b) Show that sin z = 0 if and only if z = πn for some n ∈ Z. (c) Show that cos z = 0 if and only if z = π2 (2n + 1) for some n ∈ Z.

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