Complex Analysis: Theory and Applications · Problem 15

The key is to recognize that integrating a constant function over a path in the complex plane works just like in calculus—you parametrize the curve and convert to a real integral. Parametrize the line segment as $z(t) = z_1 + t(z_2 - z_1)$ for $t \in [0,1]$, so $dz = (z_2 - z_1)\,dt$. Then the integral becomes a simple real integral that telescopes to your endpoint minus your starting point.

The key is to recognize that integrating a constant function over a path in the complex plane works just like in calculus—you parametrize the curve and convert to a real integral. Parametrize the line segment as $z(t) = z_1 + t(z_2 - z_1)$ for $t \in [0,1]$, so $dz = (z_2 - z_1)\,dt$. Then the integral becomes a simple real integral that telescopes to your endpoint minus your starting point.