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4. f(z) = (2 + 3)2 sin (= ) ; Res(f(z), -3) 5. f(z) = e-2/2; Res(f(=), 0) 6. f(=) = 7 (z - 2)2; Res(f(=), 2) In Problems 7-16, use (1), (2), or (4) to find the residue at each pole of the given function. z 8. f(z) = 4z + 8 7. f ( = ) = = 22 + 16 2z - 1 9. f(=) = 4+23 - 222 10. f(z) = (2 2 - 2z + 2) 2 522 - 4z + 3 22 - 1 11. f(z) = 7 (z + 1 ) (2 + 2)(=+3) 12. f(2) = ( 2 - 1)4(2 + 3) 13. f(z) = cos z 2:2 ( 2 - 7) 3 14. f( 2) = - 15. f(=) = sec z 16. f(2) = z sin z In Problems 17-20, use Cauchy's residue theorem, where appropriate, to evaluate the given integral along the indicated contours. 1 17. P ( 2 - 1)(2 + 2)2 dz (a) 121 = } (b) 121 = 2 (c) |21 = 3 18. z +1 Jo z2(2 -21) dz (a) |= = 1 (b) |z - 2i| =1 (c) |z - 2i] = 4 19. $ 3e-1/23 dz (a) |= =5 (b) |= til =2 (c) |z -31 =1 20. 1 dz Jo z sin z (a) |z - 2i| =1 (b) |z - 2i| =3 (c) |z| =5 In Problems 21-34, use Cauchy's residue theorem to evaluate the given integral along the indicated contour. 21. $ 2 2 + 42 + 13 dz, C: |z - 3i) = 3 22. Po 23(2 - 1)4 dz, C: 12 - 21 = } 23. $ 2 dz, C: 121 = 2 24. 2 Jo (z + 1)(22 + 1) dz, C: 16x2 +y? = 4 25. Po 22 _ ] da, C: |= = 2 26. Jo 23 + 222 adz, C: |z| = 3 27. tan & dz, C: |z - 1| = 2 2 28. 22 cot " Z dz, C: 1=1 =