Line integral
Example

Consider the function f (z) = 1/ z, and let the contour L be the counterclockwise unit circle about 0, parametrized by z (t) = e with t in using the complex exponential. Substituting, we find: ∮ L 1 z d z = ∫ 0 2 π 1 e i t i e i t d t = i ∫ 0 2 π e − i t e i t d t = i ∫ 0 2 π d t = i (2 π − 0) = 2 π i. {\displaystyle {\begin{aligned}\oint _{L}{\frac {1}{z}}\,dz&=\int _{0}^{2\pi }{\frac {1}{e^{it}}}ie^{it}\,dt=i\int _{0}^{2\pi }e^{-it}e^{it}\,dt\\&=i\int _{0}^{2\pi }dt=i(2\pi -0)=2\pi i.\end{aligned}}}

This is a typical result of Cauchy's integral formula and the residue theorem.
punctuation:no_comma