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

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

The authors' goal is to offer an actual and concise remedy of these elements of complicated research that are meant to be widespread to each examine mathematician. They stick with a course within the culture of Ahlfors and Bers by means of dedicating the ebook to a truly specified aim: the assertion and evidence of the basic Theorem for services of 1 complicated variable. They speak about the numerous identical methods of figuring out the concept that of analyticity, and provide a relaxation exploration of attention-grabbing effects and purposes. Readers must have had undergraduate classes in complex calculus, linear algebra, and a few summary algebra. No heritage in advanced research is needed.

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For example, fn (ζ) usually means the ∞ fn (ζ). Other similar abbreviations are used. 1. i) We say that the infinite series n=0 verges at a point ζ ∈ A if {SN (ζ)} converges. In this case, we write ∞ fn (ζ) = lim SN (ζ). n=0 n→∞ ∞ fn (z) converges pointwise in A if ii) We say that the infinite series n=0 {SN (ζ)} converges for every ζ ∈ A. ∞ fn (z) converges absolutely at a iii) We say that the infinite series n=0 ∞ |fn (ζ)| converges. point ζ ∈ A if the infinite series n=0 ∞ fn (z) converges uniformly in A iv) We say that the infinite series n=0 if the sequence of partial sums {SN (z)} converges uniformly in A.

A) Given two points z1 , z2 such that |z1 | < 1 and |z2 | < 1, show that for every point z = 1 in the closed triangle with vertices z1 , z2 and 1, |1 − z| ≤ K, 1 − |z| where K is a constant that depends only on z1 and z2 . (b) Determine the smallest value of K for z1 = 1+ı and z2 = 1−ı . 13. Deduce the analogs of the CR equations for anti-holomorphic functions, in rectangular, polar, and complex coordinates. 14. Let D be an arbitrary (nonempty) open set in C. Describe the class of complex-valued functions on D that are both holomorphic and anti-holomorphic.

30. A continuous function f on a domain D not containing the origin is called a branch of the logarithm on D if for all z ∈ D, we have ef (z) = z. Later we will establish that under appropriate conditions on D a branch of the logarithm always exists. 31. Let D be a domain in C with 0 ∈ / D. Suppose f is a branch of the logarithm on D. Then g is a branch of the logarithm in D if and only if there is an n ∈ Z such that g(z) = f (z) + 2π ı n for all z in D. Proof. If g = f + 2π ı n with n ∈ Z, then for all z in D, eg(z) = ef (z) e2π ı n = z.

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