Contour integrals and Cauchy's theorem

Let γ be a piecewise continuously differentiable curve with a specified orientation, and let f be continuous on its image. Substitute $z=\gamma(t)$ and $dz=\gamma'(t)dt$. The complex product retains both components of the tangent. Reversing orientation changes the sign, and subdividing or making an orientation-preserving piecewise smooth reparameterization leaves the integral unchanged. Integrating $|f(\gamma(t))|\,|\gamma'(t)|$ instead produces a nonnegative length-weighted magnitude that generally is a different quantity.

The triangle inequality yields the useful length bound: the absolute value of the contour integral is at most the curve length times the maximum of |f| on the curve. This estimate can prove that an auxiliary arc disappears in a limiting argument without computing it exactly. Before using the maximum, check that f is continuous on the compact curve. Poles on the path invalidate an ordinary contour integral and require an explicitly different interpretation, such as an indentation or principal value.

One convenient Cauchy theorem says: if f is holomorphic on a simply connected open domain D, its integral around every closed piecewise smooth curve in D is zero. A more local formulation uses a simple closed piecewise smooth contour and holomorphicity on an open neighborhood of its curve and enclosed region. These are sufficient hypotheses, not an invitation to ignore the enclosed set. The Cauchy–Goursat theorem does not require continuity of the derivative as an additional assumption.

The geometry behind the theorem is deformation through a region containing no obstruction to holomorphicity. Integrals of a holomorphic function are unchanged under a fixed-endpoint homotopy of paths within its domain. A closed curve contractible there can therefore shrink to a point. On a punctured domain a loop may fail to contract without passing through a missing point. The integrand 1/z is holomorphic on the punctured plane, but the unit circle encloses the excluded origin and its integral is 2πi.

If a holomorphic function F satisfies $F'=f$ on a domain, integrating the chain rule along each smooth piece gives an endpoint difference. Thus any two paths with the same endpoints have the same integral. Conversely, a continuous function with path-independent integrals on a domain defines a primitive by integrating from a fixed base point; a small-segment calculation verifies its derivative. In particular, a holomorphic function on a simply connected domain has a primitive. The domain must support the full definition, not merely the two paths drawn.

Simple connectedness is sufficient for every holomorphic integrand to have a primitive, but a particular integrand can have a primitive on a domain with holes. For example, z is the derivative of z²/2 on the punctured plane too. By contrast, a global logarithm would be a primitive of 1/z, and its nonzero circle integral rules one out there. This distinction keeps a domain property separate from the behavior of one chosen function.