Math 213a: Complex analysis Problem Set #3 (8 October 2003): Differentials, residues, Laurent series

1. Let A, B ∈ C[z] be polynomials such that B has distinct roots z_{1}, …, z_{n}. Let ω be the differential (A(z)/B(z)) dz on C − {z_{1}, …, z_{n}}. Show that the residue of ω at each z_{j} is A(z_{j})/B′(z_{j}). Conclude that if deg(A) ≤ deg(B) − 2 then ∑_{j=1}^{n} A(z_{j})/B′(z_{j}) = 0. What happens if deg(A) = deg(B) − 1?

Since these identities are purely algebraic results, they must hold for polynomials over any algebraically closed field; but — as with invariance of the residue under coordinate change — a direct algebraic proof, though possible, is harder and less revealing.

2. Determine for any entire function f the residue at z = 0 of the differential f(cot(z)) dz. In particular, what is the residue at the origin of sin(cot(z)) dz?

As far as I know, the other odd-order coefficients of the Laurent expansion of sin(cot(z)) about z = 0 are not known in closed form.

3. Fix a complex number t. Define an analytic function z(·) on the open unit disc |w| < 1 by z(w) = w(1 − w)^{t}. Here “(1 − w)^{t} ” is exp(t log(1 − w)), using the branch of log(1 − w) that vanishes at w = 0. Since z′(0) = 1 ≠ 0, this function has a local inverse: an analytic function w(·) on a neighborhood of the origin such that z = w(z)(1 − w(z))^{t}.

i) Determine the coefficients a_{n} in the power-series expansion w(z) = ∑_{n=0}^{∞} a_{n}z^{n}. ii) For t = ±1 the function w(z) may be evaluated in closed form and expanded in a

power series directly. Check that in these two cases your answer agrees with your formula from (i).

iii) For any integer β > 0 or β < 0, determine the Taylor or Laurent expansion of w^{β}

in powers of z. Can you make sense of this for arbitrary complex β? iv) From your answer to (i), recover Abel’s formula for the coefficients of the solution

of ye^{−y} = z. What are the coefficients of y^{β}?

4. [Cf. Ahlfors, p.148, Ex.1] For any s_{1}, s_{2} such that 0 < s_{1} < s_{2}, let Ω_{1} be the complement of the closed square {x + iy : |x|, |y| ≤ s_{1}}, and let Ω_{2} be the open square {x + iy : |x|, |y| < s_{2}}. Show that any analytic function on the region Ω_{1} ∩ Ω_{2} can be written as f_{1} + f_{2} where each f_{j} is an analytic function on the corresponding Ω_{j} and f_{1}(z) → 0 as |z| → ∞.

This problem set is due Wendesday, October 15, at the beginning of class.