Integration in the complex plane

Abstract

Abstract This chapter continues the study of paths begun in Chapter 4. It contains basic material on integrals along paths which underpins the presentation of Cauchy’s theorem in Chapters 11 and 12. To qualify as a path, the function γ is required to satisfy a rather technical differentiability condition. Intuitively, we may interpret this condition as telling us that the image γ* is made up of finitely many smooth sections (as the image of a contour certainly is). Essentially, the definition of a path is set up so that we are able to apply the theory of integration of continuous functions, piecewise, to the finitely many smooth paths which join to form γ We assume familiarity with integration of real-valued functions on compact intervals in JR, at least at a fairly basic level. Specifically, we take for granted simple techniques for evaluating real integrals, elementary properties of integrals (such as linearity) and the fact that, at the least, continuous real-valued functions on compact intervals in JR are integrable. [Here ‘integrable’ may be interpreted as having the meaning it has in any treatment of basic integration theory, whether a Riemann-style approach or a more sophisticated one leading to the Lebesgue integral.]