1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

5. ~·


6. 1r.


7. ~·


8. 7r (1-~).

13. 1f (1 - cos 1).

15. 'Ir.

Section 8.6. Integrands with Branch Points: page 326

(^1). v'321'.

5. nln2.

(^7) · (l$i;1ra·

9. 'Ir •
   ..;-;


10. 2./2.

ANSWERS 609

Section 8. 7. The Argument Principle and Rouche's Theorem: page

335

la. 1.

le. 5.

3a. Let f (z) = 15. Then I/ (z) + g (z)I = lz^5 + 4zl < 6 < I/ (z)I. As f has no

roots in D 1 (0), neither does g by Rouche's theorem.

Sa. Let f (z) = -6z^2. Then If (z) + g (z)I = lz^5 + 2z + 11. It is easy to show

that I/ (z) + g (z)I < I/ (z)I for z E C1 (0). Complete the details.

7a. Let f (z) = 7. Then If (z) + g (z)I ~ 6 < If (z)I. Show the details and

explain why this gives the conclusion you want.

9. Let f (z) = zn. T hen I/ (z) + g (z)I = lh (z)I < 1 = I/ (z)I. Complete the

argument.

Section 9.1. The z-transform: page 355

la. X (z) = I:~=o (!)"' z-n = I:~=o ( 2 ~)" = 1 /(1- 21 ,) = .~

4

.