Complex Analysis/Contour integrals

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Integrals of complex-valued functions on real intervals[edit | edit source]

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 integrals[edit | edit source]

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:

Line integral of scalar field.gif

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 integrals[edit | edit source]

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

.