Past exam of the algebra, analysis and differential equations II course of the University of Oxford / 2009 / Analysis / 6 / b / Solution

For a positively oriented simple closed piecewise smooth curve, a point $a$ inside it and a function $h$ holomorphic on a neighborhood of the curve and its interior, Cauchy's differentiated integral formula isThe printed denominator uses $n+1$ while the derivative and factorial use $m$. The standard formula requires these indices to agree; the intended exponent in this display is $m+1$. The private transcription preserves the mismatch exactly.

For the following task, write $f(z)=zq(z)$ near zero with $q(0)=f^{\prime }(0)\ne 0$. Thus $H(z)=(z/f(z){)}^n=q(z{)}^{-n}$ extends holomorphically at zero, and is holomorphic inside the specified contour because $f$ has no other zeros there. Now $f(z{)}^{-n}=H(z)/z^n$. Apply the correctly indexed formula with derivative order $n-1$ to getPositive winding one supplies the orientation and multiplicity used in this expression.