the contour integral

In ordinary calculus you integrate a function over an interval of the real line. In complex analysis the domain is the plane, so there is no single 'interval' to integrate over — you have to choose a route. A contour integral adds up the values of a complex function f(z) as the point z travels along a chosen path through the plane, weighting each tiny step by the complex displacement dz. It answers the question: as I walk this particular curve and watch f, what do I accumulate?

Here is how it actually works. Pick a path C and describe it by a parametrization z(t) for t from a to b, so that z(a) is the start and z(b) is the end. Then the contour integral of f over C is the ordinary integral from a to b of f(z(t)) times z'(t) dt. The factor z'(t) dt is just dz written in terms of the parameter — it carries both how fast you move and in which direction. Concretely, to integrate f(z) = 1/z once counterclockwise around the unit circle, set z(t) = e^(i t) for t from 0 to 2 pi; then dz = i e^(i t) dt, the integrand becomes (1/e^(i t)) times i e^(i t) dt = i dt, and the integral is 2 pi i.

The contour integral is the single object the whole subject is built on. Cauchy's theorem, the integral formula, residues — all of them are statements about what such an integral equals. A key surprise lurking ahead: for a holomorphic f the answer often does not depend on which path you take between two points, only on the endpoints, and around a closed loop it is frequently exactly zero. That near-magical insensitivity to the route is what makes complex analysis so powerful.