Chapter 14.8, Problem 1P

Let $f(z)$ be expanded in the Laurent series that is valid for all $z$ outside some circle, that is, $\left|z\right|>M$ (see Section 4 ). This series is called the Laurent series "about infinity." Show that the result of integrating the Laurent series term by term around a very large circle (of radius $>M$ ) in the positive direction, is $2\pi i{b}_1$ (just as in the original proof of the residue theorem in Section 5 ). Remember that the integral "around $\infty$ " is taken in the negative direction, and is equal to $2\pi i$ : (residue at $\infty$ ). Conclude that $R(\infty )=-b_1.$ Caution: In using this method of computing $R(\infty )$ be sure you have the Laurent series that converges for all sufficiently large z.