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CHAPTER 14 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 available in Sec. 14.2. This order of the material seems desirable from a practical point of view. Comment The introduction to the chapter mentions two reasons for the importance of complex integration. Another practical reason is the extensive use of complex integral representations in the higher theory of special functions; see for instance, Ref. [GR10] listed in App. 1. SECTION 14.1. Line Integral in the Complex Plane, page 637 Purpose. To discuss the definition, existence, and general properties of complex line integrals. Complex integration is rich in methods, some of them very elegant. In this section we discuss the first two methods, integration by the use of path and (under suitable assumptions given in Theorem 1!) by indefinite integration. 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. 14.2, after we have obtained Cauchy’s integral theorem. We discuss this method here for two reasons: (i) to get going a little faster and, more importantly, (ii) to answer the students’ natural question suggested by calculus, 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 14.1, page 645 2. Vertical straight segment from 5 6 i to 5 6 i 4. Circle, center 1 i , radius 1, oriented clockwise, touching the axes 6. Semicircle, center 3 4 i , radius 5, passing through the origin 8. Portion of the parabola y 2( x 1) 2 from 1 8 i to 3 8 i , apex at x 1 10. z ( t ) 1 i (3 3 i ) t (0 t 1) 12. z ( t ) a ib [ c a i ( d b ) ] t (0 t 1) 254 im14.qxd 9/21/05 12:37 PM Page 254

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14. z ( t ) a cos t ib sin t (0 t ) 16. z ( t ) 2 3 i 4 e it (0 t 2 ) counterclockwise. For clockwise orientation the exponent is it . 18. Ellipse, z ( t ) 1 3 cos t ( 2 2 sin t ) i (0 t 2 ) 20. z ( t ) t it 2 (0 t 1), dz (1 2 it ) dt , Re z ( t ) t , so that 1 0 t (1 2 it ) dt 1 _ 2 2 _ 3 i .
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