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CHAPTER 13 Complex Integration Change ‘We now discuss the two main integration methods (indefinite integration and integration by the use of the representation of the path) directly after the definition of the integral, postponing the proof of the first of these methods until Cauchy’s integral formula is avail- i able in Sec. 13.2. This compactification of the material seems desirable from a practical i point of view. Comment The introduction to the chapter mentions two reasons for the importance of complex in- tegration. Another practical reason is the extensive use of complex integral representa- tions in the higher theory of special functions; see Ref. [11] listed in Appendix 1. SECTION 13.1. Line Integral in the Complex Plane, Page 704 Purpose. To discuss the definition, existence, and general properties of complex line in- tegrals. Complex integration is rich in methods, some of them very elegant, In this sec- tion we discuss the first tvo methods, integration by the use of path and (under suitable | assumptions given in Theorem 1!) by indefinite integration. i Main Content, Important Concepts : Definition of the complex line integral Existence Basic properties Indefinite integration (Theorem 1) Integration by the use of path (Theorem 2) Integral of 1/z around the unit circle (basic!) ML-inequality (13) (needed often in our work) Comment on Content Indefinite integration will be justified in Sec. 13.2, after we have obtained Cauchy's in- tegral theorem. We discuss this method here for two reasons: (i) to get going a little faster and, more importantly, (ii) te answer the students’ natural question suggested by calcu- lus, that is, whether the method works and under what condition—that it does not work unconditionally can be seen from Example 7! SOLUTIONS TO PROBLEM SET 13.1, page 711 i | 2.44+3i-@+ 4) O@St=1) : 4, 3cost + 2isint (0 = ¢ = 2m). Here (and elsewhere) one should emphasize the ad- i vantage of parametric representations, that one gets the entire curve, whereas y=y) would give only the upper half (or the lower half), and y'@ > was xs +3, GQrtiP®(-28123) 8. 2cosh? + isinht (—~ <1 < a) 10. Upper semicircle (radius V3, center 5i) 187 LT | i docsity.com 
