Contour integral
\int_C f(z)\,dz, the integral of f along a contour C, computed via \int_a^b f(z(t))z'(t)\,dt.
The contour integral of f along a contour C parametrized by z(t), a\le t\le b, is \int_C f(z)\,dz := \int_a^b f(z(t))\,z'(t)\,dt, an ordinary integral of a complex-valued function of the real variable t. Its value depends on the contour itself, not just its endpoints, unless f has an antiderivative throughout a domain containing C; reversing C's direction of travel negates the integral, and splicing two contours end to end adds their integrals.
Examples
\int_C z\,dz along C: z(t)=t+it, 0\le t\le 1, evaluates by substitution to \int_0^1 (t+it)(1+i)\,dt = i.
Along the unit circle C: z(t)=e^{it}, 0\le t\le 2\pi, \int_C \frac1z\,dz = \int_0^{2\pi} i\,dt = 2\pi i.