Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
This is the Past Exam of Math Tripos which includes Geometric Invariant Theory, General Relativity, Galaxies, Gravitational Potential Energy, Further Character Theory etc. Key important points are: Complex Analysis, Schwarz Pick Lemma, Family of Analytic Functions, Bounded Analytic Function, Positive Imaginary Axis, Complex Number of Modulus, Hyperbolic Metric, Arbitrary Positive Constant, Conformal Maps
Typology: Exams
1 / 3
This page cannot be seen from the preview
Don't miss anything!
Monday 3 June 2002 9 to 12
Attempt FOUR questions
There are six questions in total The questions carry equal weight
1 State and prove the Schwarz - Pick Lemma.
The function f : DR โ C is analytic on some disc DR = {z โ C : |z| < R} with R > 1 and satisfies A 6 |f (z)| 6 B when |z| = 1.
The constants A and B satisfy |f (0)| < A 6 B < โ. Prove that f has a zero at some point zo โ DR and that |zo| > |f (0)|/B. Show that there are functions f for which we obtain equality in this inequality.
2 What is a normal family of analytic functions? Show that the set of all analytic functions from a plane domain D into the unit disc D is a normal family.
Let g : H+^ โ C be a bounded analytic function on the upper half-plane H+^ and suppose that g(z) โ as z tends to โ along the positive imaginary axis. Show that the functions z 7 โ g(tz) for t > 1 form a normal family. Deduce that, for each ฮต > 0, we have g(z) โ as z tends to โ in the sector
S(ฮต) = {w โ H+^ : ฮต < arg w < ฯ โ ฮต}.
Let h : D โ C be a bounded analytic function and let ฯ be a complex number of modulus 1. Suppose that h(rฯ) โ as r โ 1. Show that h(z) โ as z tends to ฯ in the region ฮฃ(k) = {z โ D : there exists r โ [0, 1) with ฯ(z, rฯ) < k}.
Here ฯ is the hyperbolic metric on the unit disc and k is an arbitrary positive constant.
3 State the Schwarz - Christoffel formula for conformal maps from the upper half- plane H+^ onto a polygonal domain D. Explain the formula when D is the domain S(ฯ ) obtained from H+^ by cutting along the straight line segment from 0 to a point ฯ โ H+.
Let g : H โ C be the map z 7 โ (z โ 1)k(z + 1)^1 โk^ where 0 < k < 1 and we choose the principal branches of the fractional powers. Show that g maps the upper half-plane conformally onto the domain S(ฯ ) for some value of ฯ.
How is g related to the Schwarz - Christoffel map onto S(ฯ )?
4 Explain how to define the hyperbolic Riemannian metric on any plane domain D that has the unit disc as its universal cover. Prove that it is well-defined and explain how it gives a metric on D. Calculate the hyperbolic metric on the annulus A = {z โ C : 0 < |z| < 1 }.
Prove Picardโs Great Theorem. (You may assume the existence of a universal cover for the 3-punctured sphere.)
Paper 5
