5. Use Cauchy's residue theorem to evaluate the following integrals along the circle 121 = 4: C 22 5. Use Cauchy's residue theorem to evaluate the following integrals along the circle 12...
4. (a) Use Cauchy's residue Theorem to provide alternative proofs for Cauchy's integral formula and its extension. (b) Evaluate z+ 4z + 5 dz, son z2 + z where C is the positively oriented circle of radius 2 centered at the origin.
2. More integrals! Evaluate each integral, using either a Cauchy Integral Formula or Cauchy's Residue Theorem. Take C to be the circle [2] = 3, oriented counter-clockwise. 1) Sota-1jad: 6) Se TH h) Sorºcos(1/2)da
solve a and c only 1. Use the residue theorem to evaluate the following integrals: cos θ (2+sin θ) 2n 0 0 1+sin2 θ 2x-0820, de (C) Jo 5-4cos
use residue theorem to evaluate the following integrals 16) cosa 30 de 5- 4 cos 20 17) COSI det (x + 1)? sin 3x dx 18) sin x dx (x² + 4x+5 19)
using Cauchy's Residue Theorem and the so-alled pacman inte 6. (10 pts) Evaluate gration contour. using Cauchy's Residue Theorem and the so-alled pacman inte 6. (10 pts) Evaluate gration contour.
Provide an example and use Cauchy's theorem to evaluate integrals over the boundary. Also know the results of cauchy estimates.
4. Evaluate the following integrals using the Residue Theorem. Justify your calculations, show the work. (10 points each) a) 12 cos a + 13 2 da b) (x2 +6x + 10)2 x sin 2x 24 13 da: c) 4. Evaluate the following integrals using the Residue Theorem. Justify your calculations, show the work. (10 points each) a) 12 cos a + 13 2 da b) (x2 +6x + 10)2 x sin 2x 24 13 da: c)
use residue theorem to evaluate the following integrals sin z 21) 20) Cosx dx (r? + 1) X 22) sin mx dx 2(x² + a²² (a > 0, b>0) 23) cos ex - cos bx -dx x?
please 2 only, thanks Exercises dA (1) Use Cauchy's residue theorem to compute Jo 2+sin (2) Repeat the preceding exercise for 8" 131. (3) Let a be a complex number such that lal < 1. Prove that (2 27 Jo 1 - 2a cos 0 + a2d6 = 1 - 22 (4) What is the value of the integral in the preceding exercise when |al > 1? (Hint: Let b= 1.)
2.6 Exercise. [Paul Use Theorem 9.35 and then Remark 9.34 to evaluate the following integrals (for some a, b e R such that a < b) 1. ( cos (r - log (x)dr 3. s 2 (3 - 10 dr 5. S sin (1-) (2-cs( ))dr 6. cos (3r) (si(3) 3-tan(4) a cos(4) 7 a (5r2+4) 14 15 2.6 Exercise. [Paul Use Theorem 9.35 and then Remark 9.34 to evaluate the following integrals (for some a, b e R such...