Complex integration

Definition 1 (Integral of a Complex‐valued Function of a Real Variable).
Let $h : [a, b] \to C$ be a continuous function, and write

h (t) = u (t) + i v (t),

where $u, v : [a, b] \to R$ are its real and imaginary parts, respectively. The integral of $h$ over $[a, b]$ is defined by

\int_{a}^{b} h (t) d t = \int_{a}^{b} u (t) d t + i \int_{a}^{b} v (t) d t,

where each integral on the right–hand side is the usual real Riemann integral.

Definition 2 (Contour Integral of a Complex Function).
Let $γ : [a, b] \to C$ be a contour. Let $f : A \to C$ be continuous on an open set $A \subset C$ containing the image $γ ([a, b])$ . The contour integral of $f$ along $γ$ is then defined by

\int_{γ} f (z) d z = \sum_{k = 0}^{n - 1} \int_{t_{k}}^{t_{k + 1}} f (γ (t)) γ^{'} (t) d t,

each term being a standard integral of a continuous function on $[t_{k}, t_{k + 1}]$ .

Remark
In particular, if $γ (t) = t$ for $t \in [a, b]$ , then $\int_{γ} f (z) d z = \int_{a}^{b} f (t) d t$ .

Visualization

According to @Needham1997Visual, one may equivalently view $\int_{γ} f (z) d z$ as the limit of Riemann–Stieltjes sums

\int_{γ} f (z) d z \approx \sum_{j} f (γ (τ_{j})) [γ (t_{j + 1}) - γ (t_{j})],

where ${a = t_{0} < t_{1} < \dots < t_{m} = b}$ is a partition of $[a, b]$ and $τ_{j} \in [t_{j}, t_{j + 1}]$ .
Then, we can visualize the integral as the result of adding the segments $[γ (t_{j + 1}) - γ (t_{j})]$ previously amplitwisted by $f (γ (τ_{j}))$ . It can be visualized at this webpage.