Math 411: Honours Complex Variables - University of Alberta
Math 411: Honours Complex Variables - University of Alberta
Math 411: Honours Complex Variables - University of Alberta
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78 CHAPTER 12. THE RESIDUE THEOREM AND APPLICATIONS<br />
Problem 12.2.<br />
Let D ⊂ C be open, let f : D → C be holomorphic, and let z0 ∈ C\D be a simple<br />
pole <strong>of</strong> f. Show that<br />
res(gf;z0) = g(z0)res(f;z0)<br />
for every holomorphic function g: D ∪{z0} → C.<br />
Proposition 12.2 (Rational Functions). Let p and q be polynomials <strong>of</strong> one real<br />
variable with degq ≥ degp+2 and such that q(x) �= 0 for x ∈ R. Then we have<br />
� ∞ �<br />
p(x) p<br />
dx = 2πi� res<br />
q(x) q ,z<br />
�<br />
,<br />
where<br />
−∞<br />
z∈H<br />
H := {z ∈ C : Imz > 0}.<br />
Pro<strong>of</strong>. Since degq ≥ degp+2, the Comparison Test yields that the indefinite integral<br />
exists.<br />
For r > 0 consider the semicircle<br />
γr: [0,π] → C, θ ↦→ re iθ .<br />
Let ǫ > 0 be such that, for D := {z ∈ C : Imz > −ǫ}, we have<br />
Then D is simply connected and p<br />
q<br />
{z ∈ H : q(z) = 0} = {z ∈ D : q(z) = 0}.<br />
is holomorphic on D except at the zeros <strong>of</strong> q in<br />
H. For r large enough so that all zeros <strong>of</strong> q in D lie in the interior <strong>of</strong> [−r,r]⊕γr, we<br />
see by the Residue Theorem that<br />
�<br />
�<br />
p(ζ) p<br />
dζ = 2πi� res<br />
q(ζ) q ,z<br />
�<br />
= 2πi �<br />
�<br />
p<br />
res<br />
q ,z<br />
�<br />
.<br />
[−r,r]⊕γr<br />
z∈D<br />
Since degq ≥ degp+2, there exist numbers R > 0 and C ≥ 0 such that<br />
� �<br />
�<br />
�<br />
p(z) �<br />
�<br />
C<br />
�q(z)<br />
� ≤<br />
|z| 2<br />
γr<br />
z∈H<br />
for all z ∈ C with |z|≥ R. It follows that<br />
��<br />
�<br />
�<br />
p(ζ)<br />
� q(ζ) dζ<br />
�<br />
�<br />
�<br />
C πC<br />
� ≤ πr sup ≤<br />
ζ∈{γr} |ζ| 2 r<br />
for r ≥ R and thus<br />
�<br />
p(ζ)<br />
lim dζ = 0.<br />
r→∞<br />
γr q(ζ)