For continuous functions in the complex plane, the contourintegral can be defined in analogy to the line integral by first defining the integral along a directed smooth curve in terms of an integral over a real valued parameter.
Contourintegrals are very useful tools to evaluate integrals. For example, there are many functions whose indefinite integrals can’t be written in terms of elementary functions, but their definite integrals (often from −∞ to ∞) are known. They can often be derived using contourintegrals.
Whereas for definite real integrals we only needed to specify the end-points, for a contourintegral it is not enough to just state the end-points—we need to specify the entire contour that starts and ends at those points!
In this section, we define and evaluate integrals of the form , ∫ c f (z) d z, where f is complex-valued and C is a contour in the plane (so that z is complex, with z ∈ C). Theorem 6.2.11 provides our main result, which shows how to transform the latter type of integral into the kind we investigated in Section 6.1.
To evaluate this type of integrals under conditions where the curve partially lies at infinity, one makes use of complex variable methods and in particular the Cauchy Integral Theorem.
Example 2.1. Calculate where 2. ContourIntegrationContourintegration is a method to compute de nite integrals by integrating a function with isolated singularities on a region around the boun. ary of the region and applying the residue theorem. Typically the contour we integrate ove. gets larger and we have to identify various ter. s. There is
We illustrate these steps for a set of five types of definite integral. By “trig function” we mean a function of cos θ and sin θ. The obvious way to turn this into a contourintegral is to choose the unit circle as the contour, in other words to write z = exp iθ, and integrate with respect to θ.
Evaluating integrals over a contour c. The function consists of polynomial, and no poles lies on real axis. - Download as a PDF or view online for free.
