Try these questions from elementary complex analysis:
1. Evaluate the path integrals of these differentials around C, and explain your method:
where C = {z: |z| = 2}.
i) (6 z^5 -5z^4 +1)dz/(z^6 -z^5 + z + 1)
ii) (6 z^6 -5z^5 +z)dz/(z^6 -z^5 + z + 1)
iii) dz/(z^6 -z^5 + z + 1).
2. Let Aut(D) be the group of holomorhic automorphisms of the unit disc D.
i) Prove that those elements of Aut(D) consisting of linear fractional transformations preserving D, are “transitive” on D, i.e. they take any point of D to any other point of D.
ii) Let D = {z: |z| ≤ 1}, and prove every holomorphic automorphism of D fixing 0, is a rotation.
iii) Prove that Aut(D) consists entirely of linear fractional transformations.
3. Assume f = u(x,y) + i v(x,y), is a function on the complex plane with u,v, real valued functions with two continuous derivatives, and R is a rectangle in the complex plane.
Assume also for all z = x+iy, that has a finite limit as h-->0,
and basic results of real differential calculus, and prove that:
i) ∂u/∂x = ∂v/∂y and ∂v/∂x = - ∂u/∂y.
ii) ∂^2u/∂x^2 + ∂^2u/∂y^2 = 0.
iii) the integral of f(z)dz = 0 taken around ∂R, the boundary of a rectangle in the complex plane.
4. Use apropriate theorems of complex analysis to give a proof of the fundamental theorem of algebra, i.e. if f(z) is a non constant polynomial, then f has a complex root.
5. Assume f is a non constant holomorphic function on some neighborhood of the closed unit disc, such that |f(z)| is constant on the unit circle. Prove that f has at least one zero inside the unit disc.
6. i) Prove every meromorphic function on the Riemann sphere is necessarily rational.
ii) Prove a meromorphic differential on the Riemann sphere has two more poles than zeroes, each being counted with multiplicities.
7. If a function f is analytic on a neighborhood of the unit disc, is it possible for its values at the points 1, ½, 1/3, ¼,...to equal:
i) 0,1,0,1,0,1,...?
ii) 0, ½, 0, 1/3, 0, ¼, 0, ...?
iii) 1,1/4,1/4, 1/6, 1/6, 1/6, 1/8, 1/8, 1/8, 1/8,...?
iv) ½, 2/3, ¾, 4/5, 5/6,...?
Why or why not?