Exercise · ZW-LAURENT-SERIES-006 · ★★☆ Intermediate
If \(f\) is holomorphic on the whole disc \(|z-z_0|<R\), what does its Laurent series on \(0<|z-z_0|<R\) look like?
Choose one option.
A It has infinitely many negative terms B It equals its Taylor series: \(a_n=0\) for \(n<0\) C It has exactly one negative term D It does not exist Show answer B. It equals its Taylor series: \(a_n=0\) for \(n<0\)← Back to the question
Explanation The coefficients \(a_n\) with \(n<0\) are integrals of a function holomorphic on the disc, so they vanish by Cauchy–Goursat; the Taylor series remains.
The Laurent series of a function \(f\) holomorphic on an annulus \(r<|z-z_0|<R\) is its expansion in integer powers, positive and negative, of \(z-z_0\), with coefficients \(a_n\).
\[f(z)=\sum_{n=-\infty}^{\infty}a_n(z-z_0)^n\]
More practice problems: laurent series