Let $$AB$$ be an arc $$C$$ of a curve defined by the equations $x = \phi(t),\quad y = \psi(t),$ where $$\phi$$ and $$\psi$$ are functions of $$t$$ with continuous differential coefficients $$\phi’$$ and $$\psi’$$; and suppose that, as $$t$$ varies from $$t_{0}$$ to $$t_{1}$$, the point $$(x, y)$$ moves along the curve, in the same direction, from $$A$$ to $$B$$.

Then we define the curvilinear integral $\begin{equation*} \int_{C} \{g(x, y)\, dx + h(x, y)\, dy\}, \tag{1} \end{equation*}$ where $$g$$ and $$h$$ are continuous functions of $$x$$ and $$y$$, as being equivalent to the ordinary integral obtained by effecting the formal substitutions $$x = \phi(t)$$, $$y = \psi(t)$$,  to $\int_{t_{0}}^{t_{1}} \{g(\phi, \psi) \phi’ + h(\phi, \psi) \psi’\}\, dt.$ We call $$C$$ the path of integration.

Let us suppose now that $z = x + iy = \phi(t) + i\psi(t),$ so that $$z$$ describes the curve $$C$$ in Argand’s diagram as $$t$$ varies. Further let us suppose that $f(z) = u + iv$ is a polynomial in $$z$$ or rational function of $$z$$.

Then we define $\begin{equation*} \int_{C} f(z)\, dz \tag{2} \end{equation*}$ as meaning $\int_{C} (u + iv) (dx + i\, dy),$ which is itself defined as meaning $\int_{C} (u\, dx – v\, dy) + i\int_{C} (v\, dx + u\, dy).$