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 \[\int_{C} (u\, dx – v\, dy) + i\int_{C} (v\, dx + u\, dy).\]
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