Past exam of the algebra and analysis I for mathematics and philosophy course of the University of Oxford / 2013 / Analysis / 6 / b / i

By the finite-order zero factorizations,with $F,G$ holomorphic near $a$. Shrinking the disk if necessary, continuity ensures $G$ never vanishes there, so $F/G$ is holomorphic and nonzero at $a$. ConsequentlyIf $n>m$, the exponent is $-(n-m)$, with a holomorphic nonvanishing factor. Hence