Cauchy's Integral Theorem
Also known as: Cauchy-Goursat theorem · Cauchy's theorem
If a complex function is holomorphic (smooth in the complex sense) throughout a region, then integrating it around any closed loop gives exactly zero. Equivalently, the integral between two points doesn't care which path you take. This is the complex cousin of a conservative force field having zero circulation. It's the foundation stone: from it flow Cauchy's integral formula, the residue theorem, and the whole contour-integration toolkit physicists use to crack hard integrals.
Accumulating ∮f dz around a holomorphic loop returns to zero, illustrating path independence.
Equivalent forms
Holomorphic circulation is always zero — the single fact from which almost all of complex analysis is deduced.
Where it holds
Dimensional analysis
Cauchy stated the theorem in 1825 assuming continuous derivatives; Édouard Goursat (1900) removed that assumption, proving it from mere complex differentiability — hence 'Cauchy–Goursat.' It made complex analysis a rigorous edifice rather than a formal calculus.
- Justifying path deformation in contour integration
- 2D ideal-fluid and electrostatics potential theory
- Foundations of the residue calculus used across physics
- Fails if the contour encloses a singularity — then the integral counts residues
- Requires the domain be simply connected (no holes)
- 'Holomorphic' is stronger than real-differentiable; it needs the Cauchy–Riemann conditions
What if…
1/z is not holomorphic at 0; the loop encloses a singularity and the integral is , not zero — the seed of the residue theorem.
Integrals around different loops need not agree; the difference measures what's trapped inside (winding residue).
Path independence
- entire (holomorphic on all of ℂ)
- Any closed contour lies in a simply-connected domain
- Cauchy's theorem