(Solved): .1 The function f(z) expanded in a Laurent series exhibits a pole of order m at z=z0. Show that ...

.1 The function $f(z)$ expanded in a Laurent series exhibits a pole of order $m$ at $z=z_0$. Show that the coefficient of ${\left(z-z_0\right)}^{-1},a_{-1}$, is given by $a_{-1}=\frac{1}{(m-1)!}\frac{d^{m-1}}{d{z}^{m-1}}{\left[{\left(z-z_0\right)}^{m}f(z)\right]}_{z=z_0},$ with $a_{-1}={\left[\left(z-z_0\right)f(z)\right]}_{z=z_0},$ when the pole is a simple pole $(m=1)$. These equations for $a_{-1}$ are extremely useful in determining the residue to be used in the residue theorem of the next section. Hint. The technique that was so successful in proving the uniqueness of power series (Section 5.7) will work here also.