Let $f$ be a complex-valued function defined on the image of a curve $\alpha$ : $[ a, b ] \rightarrow \mathbb{C}$ , let $P = \{ a_{0}, ..., a_{n} \}$ be a partition of $[ a, b ]$ . We will restrict our attention to contours, i.e. curves for which the parametric equations consist of a finite number of continuously differentiable arcs. If the sum

$$\sum_{i = 1}^{n} f(z_{i}) (\alpha(a_{i}) - \alpha(a_{i - 1})),$$

where $z_{i}$ is some point $\alpha(t_{i})$ such that $a_{i - 1} \leqslant t_{i} \leqslant a_{i}$ , converges as $n$ tends to infinity and the greatest of the numbers $a_{i} - a_{i - 1}$ tends to zero, then we define the contour integral of $f$ along $\alpha$ to be the integral

$$\int_{\alpha} f(z) dz:=\int_a^b f(\alpha(t))d\alpha(t)$$

Notes

(ii) An alternative definition, making use of the Riemann-Stieltjes integral, is based on the fact that the definition of this can be extended without any other changes in the wording to cover the cases where $f$ and $\alpha$ are complex-valued functions.

Now let $\alpha$ be any curve $[a, b] \rightarrow \mathbb{R}^{2}$ . Then $\alpha$ can be expressed in terms of the components $(\alpha_{1}, \alpha_{2})$ and can be associated with the complex-valued function

$$z(t) = \alpha_{1}(t) + i \alpha_{2}(t).$$

Given any complex-valued function of a complex variable, $f$ say, defined on $\Im(\alpha)$ we define the contour integral of $f$ along $\alpha$ , denoted by

$$\int_{\alpha} f(z) dz$$

by

$$\int_{\alpha} f(z) dz = \int_{a}^{b} f(z(t)) dz(t)$$ whenever the complex Riemann-Stieltjes integral on the right exists.

(iii) Reversing the direction of the curve changes the sign of the integral.

(iv) The contour integral always exists if $\alpha$ is rectifiable and $f$ is continuous.

(v) If $\alpha$ is piecewise smooth and the contour integral of $f$ along $\alpha$ exists, then

$$\int_{\alpha} f dz = \int_{a}^{b} f(z(t)) z'(t) dt.$$