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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12.1. APPLICATIONS OF THE RESIDUE THEOREM TO REAL INTEGRALS75<br />
Pro<strong>of</strong>. Fix z ∈ D \{γ}, and define<br />
g: D \{z} → C, w ↦→ f(w)<br />
w−z .<br />
Then g is holomorphic with an isolated singularity at z. Let<br />
∞�<br />
f(w) = an(w −z) n<br />
n=0<br />
be the Taylor series expansion <strong>of</strong> f near z, so that<br />
∞�<br />
g(w) = an+1(w −z) n ,<br />
n=−1<br />
and thus res(g,z) = a0 = f(z). The Residue Theorem then yields:<br />
� �<br />
f(ζ)<br />
2πiν(γ,z)f(z) = 2πiν(γ,z) res(g,z) = g(ζ)dζ =<br />
ζ −z dζ.<br />
12.1 Applications <strong>of</strong> the Residue Theorem to Real<br />
Integrals<br />
Proposition 12.1 (RationalTrigonometric Polynomials). Let p and q be polynomials<br />
<strong>of</strong> two real variables such that q(x,y) �= 0 for all (x,y) ∈ R 2 with x 2 +y 2 = 1. Then<br />
we have � 2π<br />
where<br />
0<br />
f(z) = 1<br />
iz ·<br />
p(cost,sint)<br />
q(cost,sint)<br />
�<br />
1<br />
p<br />
q<br />
dt = 2πi� res(f,z),<br />
γ<br />
z∈D<br />
�<br />
z +<br />
2<br />
1<br />
�<br />
,<br />
z<br />
1<br />
�<br />
z −<br />
2i<br />
1<br />
��<br />
z<br />
� �<br />
1<br />
z +<br />
2<br />
1<br />
�<br />
,<br />
z<br />
1<br />
�<br />
z −<br />
2i<br />
1<br />
��.<br />
z<br />
Pro<strong>of</strong>. Just note that, by the Residue Theorem,<br />
2πi �<br />
�<br />
res(f,z) = f(ζ)dζ<br />
z∈D<br />
=<br />
=<br />
∂D<br />
� 2π<br />
f(e<br />
0<br />
iθ )ie iθ dθ<br />
� 2π<br />
0<br />
p(cosθ,sinθ)<br />
q(cosθ,sinθ) dθ.<br />
γ