# A problem in complex analysis

Let $P(z)$ be a polynomial with degree $n$, and all the zeros of $P(z)$ are contained in the domain $D$, here the boundary $\partial D$ of $D$ is a smooth simply closed curve. $f(z)$ is holomorphic on a nbd of $D$.

(1) Let $$R(z)=\frac{1}{2\pi i}\int_{\partial D}\frac{f(\zeta)}{P(\zeta)}\frac{P(\zeta)-P(z)}{\zeta-z}d\zeta\ \ (z\in D),$$ $$Q(z)=\frac{1}{2\pi i}\int_{\partial D}\frac{f(\zeta)}{P(\zeta)}\frac{d\zeta}{\zeta-z}\ \ (z\in D).$$ Prove that: $R(z)$ is a polynomial with degree at most $n-1$, and $Q(z)$ is holomorphic in the domain $D$.

(2)Prove that: for any $z\in D$, $$f(z)=P(z)Q(z)+R(z).$$ If there exists holomorphic function $Q_1(z)$ and polynomial $R_1(z)$ with degree at most $n-1$ satisfying $$f(z)=P(z)Q_1(z)+R_1(z),$$ then $$Q(z)\equiv Q_1(z),R(z)\equiv R_1(z).$$

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