By Terrence Napier, Mohan Ramachandran

ISBN-10: 0817646922

ISBN-13: 9780817646929

ISBN-10: 0817646930

ISBN-13: 9780817646936

This textbook provides a unified method of compact and noncompact Riemann surfaces from the viewpoint of the so-called L2 $\bar{\delta}$-method. this system is a robust process from the idea of numerous complicated variables, and offers for a different method of the essentially varied features of compact and noncompact Riemann surfaces.

The inclusion of continuous workouts operating in the course of the publication, which result in generalizations of the most theorems, in addition to the workouts incorporated in every one bankruptcy make this article excellent for a one- or two-semester graduate path. the must haves are a operating wisdom of ordinary issues in graduate point genuine and complicated research, and a few familiarity of manifolds and differential forms.

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**Extra info for An Introduction to Riemann Surfaces**

**Example text**

Thus the local inverses provide a multiple-valued C ∞ function z = reiθ → θ , which is denoted by z → arg z and is called the argument function. = (ρ, α) : → ( ) ⊂ (0, ∞) × R of the map For any C ∞ local inverse (r, θ ) → reiθ on a domain ⊂ C∗ (in particular, ρ : z → |z|), the composition 22 1 Complex Analysis in C of the exponential function with the C ∞ function z → L(z) = log |z| + iα(z) is the identity. Thus L is a holomorphic branch of the logarithmic function. Conversely, given a holomorphic branch L = τ + iα of the logarithmic function on a domain ⊂ C∗ with τ = Re L and α = Im L, we have z = eτ (z) · eiα(z) for each z ∈ .

Let S0 = S. Breaking up S into four congruent rectangular regions, one sees that for one of the rectangles S1 , M1 ≡ | ∂S1 f (z) dz| ≥ M40 . Proceeding inductively, one gets a decreasing sequence of open rectangular regions {Sν } such that for each ν ∈ Z>0 , Sν has side lengths 2−ν (b − a) and 2−ν (d − c) and f (z) dz ≥ 4−ν M0 . Mν ≡ ∂Sν There exists a point z0 in the intersection of the closures, and the function g(z) ≡ f (z)−f (z0 ) z−z0 − ∂f ∂z (z0 ) if z ∈ S \ {z0 }, if z = z0 , 0 is continuous.

Prove that for every sufficiently small > 0, f maps the disk (0; ) biholomorphically onto a convex neighborhood of 0. Hint. 1 to obtain a local holomorphic inverse g in a neighborhood of 0, show that the function |g|2 is convex near 0. 2), f ∈ O( ), and let C1 and C2 be positive constants with |f | ≥ C1 and |f | ≤ C2 on . (a) Prove that |f (z2 ) − f (z1 )| ≥ C1 |z2 − z1 | − (b) Prove that if diam a domain in C. C2 |z2 − z1 |2 2 < 2C1 /C2 , then f maps ∀z1 , z2 ∈ . 6 Examples of Holomorphic Functions In this section, we consider some examples of holomorphic functions.

### An Introduction to Riemann Surfaces by Terrence Napier, Mohan Ramachandran

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