MAT 542  Assignment No. 7  Due: April 5, 2002

(1) Suppose f is holomorphic on Δ ; Δ = unit disc. If f is one- to one in Δ, then the area of f(Δ) is

π ∑_{n=1}^{∞} n|C_{n}|^{2},

where

f(z) = ∑_{n=0}^{∞} C_{n}z^{n} , |z| < 1.

Prove this.

(2) Show that any holomorphic function on the extended plane ℂ∪ {∞} must be constant.

(3) Show that any meromorphic function on the extended plane is rational.

(4) Let f be a rational function. Assume that f is not constant. Show that there is an integer n > 0 such that

f(z) = c

has precisely n solutions in ℂ ∪ {∞} (counting multiplicities) for all c ∈ ℂ ∪ {∞}. Identify n with another algebraic invariant for f.

(5) Let f be a meromorphic function on ℂ. Define the residue of f at ∞ so that the residue theorem holds for paths around ∞. Prove that the sum of all the residues of f is zero, provided f has an isolated singularity at ∞. What happens if the singularity at ∞ is \underline{not} isolated?

(6) Let D be a domain in ℂ and let f be meromorphic on D. Assume that f is holomorphic and non-zero on the range of γ, a positively oriented Jordan path in D, homotopic to a point in D. We have shown that

\frac{1}{2πı} ∫_{γ} \frac{f′(z)}{f(z)}dz = Z − P,

where Z = number of zeros of f in i(γ) and P = number of poles of f in i(γ). The above identity is called \underline{the argument principle}. Justify this terminology by proving that

\frac{1}{2πı} ∫_{γ} \frac{f′(z)}{f(z)}dz = \frac{1}{2πı} ∫_{γ} d log f(z) = \frac{1}{2π} ∫_{γ} d arg f(z).

(Show also that the use of multivalued functions does not cause any problems.)