Let \(P,Q\in\CC[z]\) with \[ \deg Q\ge\deg P+2. \] Prove that there exists \(r>0\) such that if \(C\) is a closed curve lying outside the circle \(|z|=r\), then \[ \int_C\frac{P(z)}{Q(z)}\,dz=0. \]
Solution
If \(P=0\), the conclusion is immediate. Assume henceforth that \(P\ne0\), and write \[ m\coloneqq\deg P, \qquad n\coloneqq\deg Q, \qquad d\coloneqq n-m\ge2. \]
1. There is a neighborhood of \(w=0\) on which\[ \frac{P(1/w)}{Q(1/w)}=w^dH(w) \]for a holomorphic function \(H\).
Define \[ \widetilde P(w)\coloneqq w^mP(1/w), \qquad \widetilde Q(w)\coloneqq w^nQ(1/w). \] These are polynomials in \(w\). Moreover, \[ \widetilde Q(0) \] is the leading coefficient of \(Q\), hence is nonzero. Therefore \[ H(w)\coloneqq\frac{\widetilde P(w)}{\widetilde Q(w)} \] is holomorphic near \(0\), and for \(w\ne0\) sufficiently small, \[ \frac{P(1/w)}{Q(1/w)} = w^{n-m} \frac{\widetilde P(w)}{\widetilde Q(w)} = w^dH(w). \]
2. There exists \(r>0\) such that, for \(|z|>r\),\[ \frac{P(z)}{Q(z)} = \sum_{j=0}^\infty c_j z^{-(j+d)} \]with locally uniform convergence.
By step 1, \(H\) has a Taylor series \[ H(w)=\sum_{j=0}^\infty c_jw^j \] for \(|w|<\rho\) for some \(\rho>0\). Choose \(r>1/\rho\). Substituting \(w=1/z\) into the identity from step 1 gives, whenever \(|z|>r\), \[ \frac{P(z)}{Q(z)} = z^{-d}H(1/z) = \sum_{j=0}^\infty c_jz^{-(j+d)}. \] Taylor-series convergence is uniform on compact subsets of \(|w|<\rho\), so the resulting Laurent series is uniform on compact subsets of \(|z|>r\).
3. The rational function \(P/Q\) has a holomorphic primitive on theexterior domain \(|z|>r\).
Because \(d\ge2\), every exponent in step 2 is at most \(-2\). Define \[ F(z) \coloneqq -\sum_{j=0}^\infty \frac{c_j}{j+d-1} z^{-(j+d-1)}. \] The differentiated Laurent series has the same radius of convergence as the series in step 2, so \(F\) is holomorphic for \(|z|>r\) and termwise differentiation gives \[ F'(z) = \sum_{j=0}^\infty c_jz^{-(j+d)} = \frac{P(z)}{Q(z)}. \] The condition \(d\ge2\) is precisely what prevents a \(z^{-1}\) term, whose antiderivative would require a logarithm.
4. If \(C\) is any closed piecewise \(C^1\) curve lying in \(|z|>r\), then\[ \boxed{ \int_C\frac{P(z)}{Q(z)}\,dz=0. } \]
By step 3, the integrand is \(F'\) on the whole exterior domain. The integral of a holomorphic derivative over a closed curve is zero.
5. Q.E.D.
Step 4 proves the assertion for the radius chosen in step 2.