Math 536 Homework #3 Spring 2010 1. (A) State the Monodromy Theorem. Consider F(Z) = ∫ Ew2 Dw. (B) Prove F Is Entire and Maps

Math 536 Homework #3 Spring 2010

1. (a) State the Monodromy Theorem. Consider z 2 f(z)= ew dw. Z0 (b) Prove f is entire and maps the plane onto the plane. (Hint: f 0 is even) (c) Show that f is locally one-to-one but not one-to-one in the plane. (d) Explain which statement(s) is (are) wrong in the following argument:

If γ is any curve in the plane, then a local inverse of f can be defined in a neighborhood of each point of γ. If γ is a curve from 0 to z then γ is compact and can be covered by finitely many discs, such that f has an inverse on each. This provides an analytic continuation of f −1 from 0 to z along γ. Since the plane is simply connected, a global inverse f −1 of f can be defined in the plane by the monodromy theorem.

2. Suppose f and g are entire and f 2 + g2 = 1. Prove there exists an entire function h so that f(z) = cos h(z) and g(z) = sin h(z).

3. Suppose Ω is doubly connected with each boundary component containing at least two points.

If z0 Ω prove that there is exactly one r > 0 and exactly two conformal maps of Ω onto an ∈ 0 annulus of the form z : 1 < z < r with f (z0) > 0. { | | }

4. Find a function f, meromorphic in the whole plane with simple poles at n i√n (n a positive − integer), and no other poles, such that

∗ ff 1, ≡

where f ∗ is the meromorphic function

∗ f (z)= f(z).

Hint: find a holomorphic function with zeros at n i√n. − 5. Let Log denote the principal branch of the logarithm, and let t (0, 1). The discontinuities of ∈

Log(z) Log(z t) − − cancel on ( , 0), so the above function has an analytic continuation Ft(z) to C [0,t]. Let −∞ \ Γ be the unit circle with positive orientation. Compute

Ft(z)dz. ZΓ

6. Suppose f and g are nonconstant entire functions and suppose f g is a polynomial. Show ◦ that both f and g are polynomials.

7. Find a conformal map from the region

z : z + i√3 < 2 z : Imz > 0 { | | } ∩ { }

onto the unit disk. You may write your answer as a sequence of explicit conformal maps. Justify your answer.