Cauchy Integral Theorem

Cauchy Integral Theorem:

If f is a holomorphic function on an open disc U in the complex plane, andif : [a, b] → U is a C¹curve in U with (a) = (b), then

There are a number of different ways to prove the Cauchy integral theorem. One of the most natural is by way of a complex-analytic form of Stokes’s
theorem: If is a simple, closed curve surrounding a region U in the plane then,

An important converse of Cauchy’s theorem is called Morera’s theorem: Let f be a continuous function on a connected open set U $\subseteq \!\,$ C. If

for every simple, closed curve in U, then f is holomorphic on U.

Cauchy Integral Formula:

Suppose that U is an open set in C and that f is a holomorphic function on U. Let z0 $\in \!\,$ U and let r > 0 be such that D(P, r) $\subseteq \!\,$ U. Let : [0, 1] → C be the C¹ curve (t) = P r cos (2πt) ir sin (2πt). Then, for each z $\in \!\,$ D (P, r)

One may derive this result directly from Stokes’s theorem

General Forms of the Cauchy Theorems:

Now we present the very useful general statements of the Cauchy integral theorem and formula. First we need a piece of terminology. A curve : [a, b] → C is said to be piecewise C^k if

with a = a0 < a1 < · · · am = b and [aj−1,aj] is C^k for 1 ≤ j ≤ m. In other words, is piecewise C^k if it consists of finitely many C^k curves chained end to end.