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

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

The authors' goal is to offer an exact and concise therapy of these components of complicated research that are meant to be primary to each study mathematician. They stick to a course within the culture of Ahlfors and Bers by means of dedicating the booklet to a really specific objective: the assertion and evidence of the basic Theorem for features of 1 advanced variable. They talk about the various similar methods of figuring out the idea that of analyticity, and supply a relaxation exploration of attention-grabbing outcomes and functions. Readers must have had undergraduate classes in complex calculus, linear algebra, and a few summary algebra. No history in complicated research is needed.

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That is, if yes, exhibit the holomorphic function(s); if not, prove it. 6. Prove the Fundamental Theorem of Algebra: If a0 , . . , an−1 are complex numbers (n ≥ 1) and p(z) = z n + an−1 z n−1 + . . + a0 , then there exists a number z0 ∈ C such that p(z0 ) = 0. Hints: (a) Show there is an M > 0 and an R > 0 so that |p(z)| ≥ M for |z| ≥ R. (b) Show next that there is a z0 ∈ C such that |p(z0 )| = min{|p(z)| ; z ∈ C} . (c) By the change of variable p(z + z0 ) = g(z), it suffices to show that g(0) = 0.

It has a jump discontinuity on the negative real axis (it is not defined at 0). We define the principal branch of the logarithm by the formula Log z = log |z| + ı Arg z . 2Having defined π, polar coordinates now rest on a solid foundation and can be used in proofs. 3Thus it is not a function. 3. THE EXPONENTIAL AND RELATED FUNCTIONS 41 It is a continuous function on C − (−∞, 0]; it is C 1 on this set. Properties of the complex logarithm 1. elog z = elog|z|+ı arg z = |z| eı arg z = z for all z ∈ C=0 .

POWER SERIES Proof. With notation as in the last definition, if N1 < N are positive integers, then N |SN (z) − SN1 (z)| ≤ Mn for all z ∈ A n=N1 +1 (needed for the uniform convergence argument), and N | |SN (z)| − |SN1 (z)| | ≤ Mn for all z ∈ A n=N1 +1 (needed for the absolute convergence argument). Given any > 0, we can find a positive integer N0 such that N > N N1 > N0 implies Mn < . n=N1 +1 n We shall be mostly interested in series of the form ∞ n=0 an z with an ∈ C (these are known as power series) and the associated real-valued n series ∞ n=0 |an |r where r = |z|.

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