Question

How to evaluate contour integrals without residue theorem?

Answer 1

Let, the path of contour be along a curve γ parameterized by:

$z = \Re( \gamma(t) ) + i \Im( \gamma(t))$

such that,

$t \in [a,b]$

Then,

$\\ \boxed{ \oint_{ \gamma}f(z)dz = \int _{a}^{b} f( \gamma(t)) \gamma ^{ \prime}(t)dt }$