the fundamental theorem for contour integrals
The fundamental theorem of calculus on the real line says that integrating a derivative gives back the original function's net change: the integral of F' from a to b is F(b) - F(a). The complex analogue says exactly this for contour integrals, and it is the lever that turns many integrals into a single subtraction.
The statement: let F be holomorphic on a domain D with continuous derivative f = F', and let C be any contour in D running from a point z1 to a point z2. Then the contour integral of f over C equals F(z2) - F(z1). The proof is short and illuminating — parametrize C by z(t), apply the chain rule so that f(z(t)) z'(t) is the derivative of F(z(t)) in t, and the real fundamental theorem in the parameter t finishes the job, leaving F(z(b)) - F(z(a)). All the complex content has been routed back through a familiar one-variable integral.
Two consequences are immediate and central. First, if f has a primitive on D, its integral depends only on the endpoints, not the path. Second, taking a closed contour (z1 = z2) gives an integral of exactly zero. This is the bridge from 'f has an antiderivative' to 'f's loops vanish', and reading it the other way around will, in Cauchy's theorem, let us conclude that holomorphic functions on nice domains always have primitives.
Since e^z is its own primitive (the derivative of e^z is e^z), the integral of e^z along any contour from 0 to i pi equals e^(i pi) - e^0 = -1 - 1 = -2, no matter how the contour wanders between the two points.
An integral evaluated by the complex fundamental theorem: just plug the endpoints into a primitive.
The theorem assumes f has a primitive throughout the domain; it does not apply when no single-valued primitive exists — that is precisely how 1/z escapes it and yields 2 pi i around the origin instead of 0.