Complex Analysis/Contour integrals

Integrals of complex-valued functions on real intervalsEdit

In calculus, we learned how to integrate (say, continuous) functions   on a finite interval  . What happens now if the function   we wish to integrate has values in the complex numbers (that is,  )? The answer is straightforward. We decompose   by the formula   and define the integral as follows:

Definition 4.1:

Let   continuous and complex-valued (or, alternatively, such that both   and   are integrable). Then we set

 .

The idea behind contour integralsEdit

In this chapter, given a function  , we want to integrate   along a differentiable curve; roughly speaking, we want to determine the measure of the area under the graph which arises when flattening out the curve as indicated in the following animation:

 

We now want to figure out which formula could make sense for obtaining this measure (note that as in normal integration, we want the area where the function is negative to be subtracted from the value of the integral, instead of being added to it). The idea is to approximate the desired integral. Let a differentiable curve   be given. We choose a certain decomposition

 

where  . Then we approximate the desired integral, which we denote by  , by a finite sum as follows:

 .

This sum sums small squares which approximate the integral, just like Riemann sums. As the maximum distance between consecutive   gets smaller, we obtain better and better approximations. On the other hand,

 ,

where the latter integral converges to

 

as  . This is why we define:

Definition 4.2:

Let a   (in the real sense) curve   (which we shall also call contour) be given. Then we define the integral of   along the contour   to be

 .

In fact, even before talking about cycles (chapter 10) and related things we need a more general, but not much more difficult, definition of contour integrals, namely one which also holds for piecewise   curves.

Definition 4.3:

A function   is called a piecewise   contour if and only if there exists a decomposition  ,   such that for all   the restriction

 

is  .

Rules for contour integralsEdit

In this section, we state and prove some formulas which hold for contour integrals and which we shall extensively use throughout the subsequent chapters.

Theorem 4.4:

Assume that   has a primitive, that is a function   such that   is holomorphic and   for all  . Then for all piecewise   contours   we have

 .

Theorem 4.5:

The contour integral is linear, that is for   holomorphic,   and   piecewise   we have

 .