Definition

We start by considering the natural analogue of the real integral $\int_{a}^{b} f (t) d t$ along the segment $[a, b]$ , defined as follows:

Let $a, b \in R$ and $f : [a, b] \to C$ be continuous. Then we set
$\int_{a}^{b} f (t) d t := \int_{a}^{b} Re f (t) d t + i \int_{a}^{b} Im f (t) d t$
Here $Re f$ and $Im f$ denote the real and imaginary parts of the complex-valued function $f$ . Hence the integrals on the right-hand side are familiar integrals of real functions.