Complex analysis deformation theorem

1. Residue theorem (10
.22) states that the integral of f ( z ) around a closed path enclosing a single pole of f ( z ) is 2 π i times the residue at the pole.
Prove the residue theorem in Eq. (10.22).

What is the principle of deformation of paths complex analysis?

This is called the principle of deformation of paths, which we describe as follows.
If a contour Γ1 can be continuously deformed into another contour Γ2 without pulling the contour over any points where the mapping f is not differentiable, then the integrals of f around Γ1 and Γ2 are the same._{Nov 18, 2019}.