1-    Curves, Arcs, and Contours, and a definition of integration

A curve is given by the parametric equations x=x(t), y=y(t) where x(t) and  y(t) are real-valued function on a closed interval [a,b].  An arc (or differentiable path) is a continuously differentiable curve.  A complex-valued function f(z) is said to be continuous on an arc C if f[x(t)+iy(t)] is continuous on [a,b] and in this case we define

Under this definition almost all the rules of real integrals carry over to complex ones of which we mention, for instance, the following:

, where f(z) and g(z) are continuous on an arc C and and are complex constants (exercise 2).

However, using an alternative equivalent definition in the next section makes it easier to carry on with other complex integral properties.

Example.  Let .  Let C be the circle centered at the origin of radius R. We parametrize C as Then

Consequently, if,

If , 0 since n is an integer.

A collection of n arcs is said to form a contour T if the end point of the arc C coincides with the initial point of for In this case, we define


We now state and prove the Fundamental Theorem of Calculus for complex functions:

 2-    Theorem

Let F(z) be an analytic function with continuous derivative f(z).  Let T be a contour with initial point and terminal point .  Then


Proof.  Without loss of generality, it is enough to show that

for an arc C with initial point and terminal point .

We use the chain rule together with the Fundamental Theorem of Calculus for real functions:

A simple curve is a curve that does not intersect itself. A domain is said to be simply connected if the interior of every simple closed curve in the domain is contained in that domain.

3 -   An equivalent definition of integration that is best suited to prove

As in real analysis, can be obtained as a limit of sums of the form

, where are points taken in succession along the curve C

and is on C between and .  This formulation makes it obvious, that the integral is, to a large extent, independent of the parametric representation of the curve C,  and also leads to an important inequality whenever

Since the straight line segment is the shortest distance between two points does not exceed the length of the part of the arc C joining and .  Thus if L denotes the length of C ML.  Taking limits, we get   .