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math185f09-hw3 - MATH 185 COMPLEX ANALYSIS FALL 2009/10...

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MATH 185: COMPLEX ANALYSIS FALL 2009/10 PROBLEM SET 3 For a real-valued function of two real variables, u : Ω R R , we say that u is twice continuously differentiable if all second-order partial derivatives u xx , u yy , u xy , u yx exist and are continuous on Ω R . The set of all twice continuously differentiable functions on Ω R is denoted C 2 R ). 1. We mentioned Tauberian theorems in class. Here is an example of an easy one (easy relative to other Tauberian theorems). Let n =0 a n z n be a power series with radius of convergence 1 and suppose lim n →∞ na n = 0 . (a) Show that lim m →∞ m n =0 n | a n | m = 0 . ( Hint: Problem 4 (a), Problem Set 3 , Math 104 , Spring 2009.) (b) Define a function f by f ( z ) = X n =0 a n z n for all | z | < 1 . Let x be a real variable and suppose the following left limit exists lim x 1 - f ( x ) = A. Show that the series n =0 a n converges to A . 2. Recall that C is both a real vector space of dimension 2 and a complex vector space of dimen- sion 1. A function ϕ : C C is called R -linear if ϕ is a linear transformation of real vector spaces, ie. ϕ ( λ 1 z 1 + λ 2 z 2 ) = λ 1 ϕ ( z 1 ) + λ 2 ϕ ( z 2 ) for all λ 1 , λ 2 R and z 1 , z 2 C .
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