1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

600 ANSWERS

19. The absolute value of t he integrand is .Jx^2 + (1 - x^2 ) cos^2 rl, which simpli-

fies to .J x^2 sin^2 8 + cos^2 e" (show the details for t his assertion). The maxi-

mum of this expression occurs when x = 1. Now simplify and apply the ML

inequality.

Section 6.3. The Cauchy-Goursat Theorem: page 227

la. Analytic everywhere except z = ±v'2i, so fc{(O) f (z) dz = 0, since ±v'2i

both lie outside t he circle C1 (0).

le. Analytic everywhere except z = (n + 4) 11', where n is an integer, so

fct(o) f (z) dz= 0, since all nonanalytic points lie outside the circle C1 (0).

s. By the quadratic formula (see Theorem 1.5), 4z^2 - 4z + 5 = 0 when z =

~ ± i (verify). Since both these points lie outside C 1 (0), the function

(4z^2 -4z+5)-

1

is analytic inside C1 (0), so fct(o) (4z^2 -4z +5)-

1

dz=0

by the Cauchy-Goursat theorem.

5a. 411'i.

5b. 211'i.

(^7) a. 1ri 4 ·
7 b. -JI/.
7c. 0.

9. -3di ·


10. 
    
    
    
    0. 
       
    
    
    
    

lSa. 411'i.

lSb. 0.

Section 6.4. The Fundamental Theorems of Integration: page 233

1. ~ + 3i.

S. - e^2 +i.

5. 
   
   
   
   • 1 + i "!^2.
   
   
   
   

(^7). - 5 +^7 t2. · l

9. 
   
   
   
   • 1 - sinhl + cosh 1.