Tham khảo tài liệu 'mathematical method in science and engineering episode 11', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | COMPLEX TECHNIQUES IN TAKING SOME DEFINITE INTEGRALS 353 dtì o a cos 0 Using Equations and we can write this integral as 1 dz 2ĩ 2ị i z2 2az l The denominator can be factorized as z - a z - Ị3 where a fl a2 1 2 3 a a2 l 5 . For a 1 we have a 1 and 3 1 thus only the root z a is present inside the unit circle. We can now use the Cauchy integral theorem to find I 2i 2tt a 3 2tt fl2 - 1 Example . Complex contour integral technique We now consider the integral 1 ỉ21ĩ Z -L sin2i0d0. 2tt Jo We can use Equations and to write z as a contour integral over the unit circle as r -1 1 H f dz l 2i 2 w z V zj We can now evaluate this integral by using the residue theorem as 354 COMPLEX INTEGRALS AND SERIES Using the binomial formula we can write 1 1 2 _1A 21 1V w z 7g 2TT Ei 2 4 where the residue we need is the coefficient of the 1 z term. This can be easily found as -l - and the result of the definite integral I becomes I - . 22i Z 2 Fig. Contour for the type II integrals II. Integrals of the type I dxR x where R x is a rational function of the form _ a0 ai 1 H-------1- anxn R x --------- - -------- ---- bo biX b2x2 H-------F bmxm a With no singular points on the real axis b l-R z goes to zero at least as in the limit as zI oo . Under these conditions I has the same value with the complex contour integral I ỷ R z dz COMPLEX TECHNIQUES IN TAKING SOME DEFINITE INTEGRALS 355 Fig. Contour for Example where c is a semicircle in the upper half of the z-plane considered in the limit as the radius goes to infinity Fig. . Proof is fairly straightforward if we write I as I jj R z dz ỉ R x dx ị R z dz Jc J oo J and note that the integral over the semicircle vanishes in the limit as the radius goes to infinity. We can now evaluate I using the residue theorem. Example . Complex contour integral technique Let US evaluate the integral Z dx . 13103 Since the conditions
