In mathematics, the Cauchy integral theorem (also known as the Cauchy–Goursat theorem) in complex analysis, named after Augustin-Louis Cauchy (and Édouard Goursat), is an important statement about line integrals for holomorphic functions in the complex plane.
is holomorphic in a simply connected domain Ω, then for any simply closed contour
If f(z) is a holomorphic function on an open region U, and
Also, when f(z) has a single-valued antiderivative in an open region U, then the path integral
be a simply connected open set, and let
has no "holes", or in other words, that the fundamental group of
(Recall that a curve is homotopic to a constant curve if there exists a smooth homotopy (within
Intuitively, this means that one can shrink the curve into a point without exiting the space.)
The first version is a special case of this because on a simply connected set, every closed curve is homotopic to a constant curve.
In both cases, it is important to remember that the curve
does not surround any "holes" in the domain, or else the theorem does not apply.
The Cauchy integral theorem does not apply here since
cannot be shrunk to a point without exiting the space.
As Édouard Goursat showed, Cauchy's integral theorem can be proven assuming only that the complex derivative
This is significant because one can then prove Cauchy's integral formula for these functions, and from that deduce these functions are infinitely differentiable.
has no "holes" or, in homotopy terms, that the fundamental group of
is trivial; for instance, every open disk
which traces out the unit circle, and then the path integral
is nonzero; the Cauchy integral theorem does not apply here since
One important consequence of the theorem is that path integrals of holomorphic functions on simply connected domains can be computed in a manner familiar from the fundamental theorem of calculus: let
be a simply connected open subset of
be a piecewise continuously differentiable path in
The Cauchy integral theorem is valid with a weaker hypothesis than given above, e.g. given
, a simply connected open subset of
[1] The Cauchy integral theorem leads to Cauchy's integral formula and the residue theorem.
If one assumes that the partial derivatives of a holomorphic function are continuous, the Cauchy integral theorem can be proven as a direct consequence of Green's theorem and the fact that the real and imaginary parts of
must satisfy the Cauchy–Riemann equations in the region bounded by
Cauchy provided this proof, but it was later proven by Goursat without requiring techniques from vector calculus, or the continuity of partial derivatives.
By Green's theorem, we may then replace the integrals around the closed contour
But as the real and imaginary parts of a function holomorphic in the domain