复变函数证明,

题目

复变函数证明,
Q是n阶多项式,有不同的n个解 a1,a2,a3.an ,P是小于n阶多项式,证明P(z)/Q（z)=P(a1)/Q'(a1)(z-a1)+P(a2)/Q'(a2)(z-a2)+P(a3)/Q'(a3)(z-a3)+.P(an)/Q'(an)(z-an)

答案

显然Q(z)=A(z-a1)(z-a2)...(z-an),A是其最高次项系数.按照求导的乘法规则,有Q'(z)=A(z-a2)...(z-an) + A(z-a1)(z-a3)...(z-an) + ...+A(z-a1)(z-a2)...(z-a_{n-1})所以 Q'(a1) = A(a1-a2)(a1-a3)...(a1-an)Q'(a2) = ...