10.1.44 Integral of a complex variable
Condition: Calculate the integral. \[ \oint_{|z|=3}\left(z \cos \frac{1}{z}+\frac{\sin \frac{\pi z}{2}}{(z-2)^{2}(z-4 i)}\right) d z \text {. } \]
Calculation of integrals from functions of a complex variable. Sub-integral functions can be either analytic or with singular points. Application of the main residue theorem, as well as the Cauchy formulas for calculating integrals of functions with poles in a simply connected domain.