1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

304 CHAPTER 8 • RESIDUE THEORY s s: g(t) tt/4 rrJ4 F igure 8.3 Graph of g (t) = J 1 + 3 1 0082 t dt = -Arct~ (^2 cot t). probl ...

8 .2 • TRIGONOMETRIC INTEGRALS 30 5 We solve for cos 28 and sin 28 to obtain the substitutions 1 and sin 28 = 2 i (z^2 - z-^2 ). ...

306 CHAPTER 8 • RESIDUE THEORY . 2 (} 5. fi2" Sill dO. (^0) 5 + 4cos8 . 2 (} 6. r2" Sill d(J. Jo 5 - 3cos8 r2w 1 Jo 2 dO. (5+ 3 ...

8.3 • IMPROPER INTEGRALS OF RATIONAL FUNCTIONS 307 provided the limit exists. If f is defined for all real x, then the integral ...

308 CHAPTER 8 • RESIDUE THEORY If f (x) = ~~:j, where P and Qare polynomials, then f is called a rational function. In calculus ...

8 .3 • IMPROPER. !NTEGR.ALS OF RATIONAL FUNCTIONS 309 EXAMPLE 8.15 Evaluate J::oo (:e•+i'i<o:•+w Solution We write the inte ...

310 CHAPTER 8 • RESIDUE THEORY the residues, we obtain Res[/, i] = ~i and Res [f, 2i) = 1 i 2 . Using Theorem 8.3, we conclude t ...

8.4 • IMPROPER INTEGRALS INVOLVING TRIGONOMETRIC FUNCTIONS 311 7 J"° x2da: -oo x^4 + 4 · 8 Joo x 2 dx -oo (x2 + 4 )3 · (^9) ...

312 CHAPTER 8 • RESIDUE THEORY (8-12) (8-13} The proof of Theorem 8.4 is similar to the proof of Theorem 8.3. Before tuu~ing to ...

8.4 • IMPROPER INTEGRALS INVOLVING TRIGONOMETR.!C FUNCTIONS 313 the residues with the aid of L'Hopital's rule: R 1 / 1 . 1 1 . ( ...

814 CHAPTER 8 8 RESIDUE THEORY provided R 2 R~. The parametrization of CR leads to the equation ldzl = R d(J and ie"I = e- 11 = ...

8.4 • IMPROPER I NTEGRALS INVOLVING TRIGONOMETRIC F UNCTIONS 315 -------~EXERCISES FOR SECTION 8.4 Use residues to find the Cauc ...

316 CHAPTER 8 • RESIDUE THEORY (^1) J ·oo cos 2x dx 1.. -oo x^2 + 2x + 2 (^2) J oo x^3 sin 2x dx oo x (^4) + 4. 1 3. Why do y ...

8.5 • INDENTED CONTOUR INTEGRALS 3 17 In this section we show how to use residues to evaluate the Cauchy principal value of the ...

318 CHAPTER 8 • RESIDUE THEORY has simple poles at the points t 1 = 2 on the x-axis and z 1 = -1 + iJ3 in the upper half-plane. ...

8.5 • INDENTED CONTOUR INTEGRALS 319 and the Cauchy principal limit at t = 2 as r --> 0 is lim [g(2 +r)-g(2-r)) = O. r-o+ The ...

320 CHAPTER 8 • RESIDUE THEORY where g is analytic at z = to. Using the parametrization of Gr and Equation (8-23), we get 1 1 " ...

8.5 • INDENTED CONTOUR INTEGRALS 321 -------~EXERCISES FOR SECTION 8.5 Use residues to compute J oo dx 1.P.V. _ (^00) x ( X-l )( ...

3 2 2 CHAPTER 8 • RESIDUE THEORY co x^4 dx 6. P.V. f -cox - 6 - - l. 7. P.V. f~oo sinx dx. x S. P.V. I"'° cosx dx. J-oo x2 - x c ...

8.6 • INTEGRANDS WlTH BRANCH POINTS 323 y Figure 8 .7 The contour C that encloses the nonzero poles z1, z2,... , Zk of G· ...