Cauchy's Theorems.

Cauchy's Theorems.

Let be a path of integration. It is smooth at a point if is derivable at and if its first derivative is continuous at

. The path is smooth if it is smooth at every point; it is smooth by parts if it is not smooth at only a finite number of points.

Example 5.3.1   The arc of the parabola whose equation is is smooth.

Figure 2: Paths.

 

A loop is an integrating path whose origin and endpoints are identical. If it is smooth and does not intersect itself at another point, we will call it a Jordan curve. If we travel exactly once along the loop, we will call it a simple loop. The integral of a function along a simple loop will be denoted as follows:

Note the little circle on the integration symbol.

A Jordan curve determines in the plane three disjoint regions: the curve itself, the interior (= a bounded region) and the exterior (= an unbounded region).

Theorem 5.3.2 (Cauchy-Goursat)   Let be a function, analytic on an open simply connected subset of . If is a Jordan curve in , then .

Note that the converse is not true: if is not analytic on the interior of , the integral can either vanish or not. For example, compute the following integral:

Using the parametrization , we can show that . The integral vanishes, despite the fact that the function fails to be analytic at 0, which is an interior point of the unit circle (defined here by the equation ).

Corollary 5.3.4   Let be a function defined and analytic on a connected domain . Let and be two paths with the same origin and the same endpoint such that both path contain only interior points of (see Figure 3). Then .

Figure 3: Two paths

Theorem 5.3.5   Consider two Jordan curves and such that all the points of are interior to (Figure 4). Let be a function, analytic on , on and at every point of the ``annulus'' bounded by and . Then:

 

Figure 4: Loop within loop.