Let us show that (2) 1 " cos 2x . _ 5Jr ) ' d.\ - 4. (.r-

Chapter 0, Problem 7.13

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Let us show that (2) 1 " cos 2x . _ 5Jr ) ' d.\ - 4. (.r- + 4)-' 32e We introduce the fu ncl ion I {C:.) = ) 4 ) . (:.- + )- CllAP. 7 and observe char che produce f(:.ki 2; is analycic evc1)'where on and above che real axis excepl at che poinl:. = 2i. This singularicy lies in che interior of lhe semicircular region whose boundary consists of lhc segment - R :=:: x :=:: R of che real axis and che upper half CR of che circle 1:.1 = R ( R > 2) from :. = R co :. = -R (Fig. I 02). Integralion of f(:.)'; 2z around chat oriemed boundary yields the equation ( 4) where -R Since I R 'i2x l . 2 ' 2 d .r = 2Jr i B - I (:. )l'j > d:. . R (.\ + 4) C11 \' 0 _, ')' f(:.) = B =Res lfl:.)ei 2 z].