10.1.33 Integral of a complex variable

Problem: Using the Cauchy theorem for the multiply connected domain and the Cauchy integral formula for the analytic function and its derivatives, calculate the integral: \[ \int_{L} f(z) d z \] where \( f(z)=\frac{2 z^{2}-z+1}{(z-i)^{2}(z+i)}, \quad L:|z+i|=3 \).