Math 213a: Complex analysis
Problem Set #3 (8 October 2003):
Differentials, residues, Laurent series
1. Let A, B ∈C[z] be polynomials such that Bhas distinct roots z1, . . . , zn. Let ω
be the differential (A(z)/B(z)) dz on C− {z1, . . . , zn}. Show that the residue
of ωat each zjis A(zj)/B0(zj). Conclude that if deg(A)≤deg(B)−2 then
Pn
j=1 A(zj)/B0(zj) = 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 fthe 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(tlog(1 −w)), using the branch of
log(1 −w) that vanishes at w= 0. Since z0(0) = 1 6= 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 anin the power-series expansion w(z) = P∞
n=0 anzn.
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 wβ
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 yβ?
4. [Cf. Ahlfors, p.148, Ex.1] For any s1, s2such that 0 < s1< s2, let Ω1be the comple-
ment of the closed square {x+iy :|x|,|y| ≤ s1}, and let Ω2be the open square
{x+iy :|x|,|y|< s2}. Show that any analytic function on the region Ω1∩Ω2can
be written as f1+f2where each fjis an analytic function on the corresponding Ωj
and f1(z)→0 as |z| → ∞.
This problem set is due Wendesday, October 15, at the beginning of class.