Question

9-19 CAUCHY'S THEOREM APPLICABLE? Integrate f(z) counterclockwise around the unit circle. Indicate whether Cauchy's integral theorem applies. Show the details. 9. f(z) = exp (-z^2) 10. f(z) = tan 1/4z 11. f(z) = 1/(2z - 1) 12. f(z) = z^3 13. f(z) = 1/(z^4 - 1.1) 14. f(z) = 1/z 15. f(z) = Im z 16. f(z) = 1/(?z - 1) 17. f(z) = 1/|z|^2 18. f(z) = 1/(4z - 3) 19. f(z) = z^3 cot z

          9-19 CAUCHY'S THEOREM APPLICABLE?
Integrate f(z) counterclockwise around the unit circle.
Indicate whether Cauchy's integral theorem applies. Show
the details.
9. f(z) = exp (-z^2)
10. f(z) = tan 1/4z
11. f(z) = 1/(2z - 1)
12. f(z) = z^3
13. f(z) = 1/(z^4 - 1.1)
14. f(z) = 1/z
15. f(z) = Im z
16. f(z) = 1/(?z - 1)
17. f(z) = 1/|z|^2
18. f(z) = 1/(4z - 3)
19. f(z) = z^3 cot z

9-19 CAUCHY'S THEOREM APPLICABLE?
Integrate f(z) counterclockwise around the unit circle.
Indicate whether Cauchy's integral theorem applies. Show
the details.
9. f(z) = exp (-z^2)
10. f(z) = tan 1/4z
11. f(z) = 1/(2z - 1)
12. f(z) = z^3
13. f(z) = 1/(z^4 - 1.1)
14. f(z) = 1/z
15. f(z) = Im z
16. f(z) = 1/(?z - 1)
17. f(z) = 1/|z|^2
18. f(z) = 1/(4z - 3)
19. f(z) = z^3 cot z

Added by Kevin R.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Adi S

09/03/2023

Step 1

For 9.fz=cxp(-z), Cauchy's integral theorem applies if the function is analytic, which it is. The integral over the unit circle is 0. Show more…

Show all steps

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Theorem 1 states that if f(z) is analytic inside and on a simple closed contour C, and if z0 is any point inside C, then the integral of f(z) around C is equal to 2̀́i times the sum of the residues of f(z) at its singular points inside C. Q12: Integrate fz counterclockwise around the unit circle. Indicate whether Cauchy's integral theorem applies. Show and conclude from the details. f(z) = exp(-z) f(z) = tan(z) f(z) = 1/2z-1 f(z) = 3 f(z) = 1/z^4-1 f(z) = 1/z f(z) = Im(z) f(z) = 1/̀́z-1 f(z) = 1/|z|^2 f(z) = 1/4z-3 f(z) = zcot(z) Q15: Determine the value of the integral of f(z) = exp(-z) over the contour C, where C is the unit circle |z| = 1. Q19: Determine the value of the integral of f(z) = 1/̀́z-1 over the annulus 1 < |z| < 2.