1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

ANSWERS 621

Section 12.3. Vibrations in Mechanical Systems: page 535

(^00) (-1rs (^00) (-1jA(n (^2) - 1).

la. Up (t) = L:: n•+ 7 n 2 + 1 cos(nt) + L:: n•+ 1 n•+n sm(nt).

n=l n=l

Section 12.4. The Fourier Transform: page 540

1. 
   

   j (U (t )) = s~ww.

   
   


2. 
   

   ~(U (t )) = 1 -;.~~w = 2s~: 2 'f.

   
   

(^5). ) "' ( e -<>ltl ) _ - .-(a. J<>+I w1).
7.~(~)={
i sinw for

• 2 - lwl ~ ?r,
  0 for lwl > 1r.

9. j(~) = {

l -lwl for

4ir lwl ~ 1,

0 for lwl > 1.

Section 12.5. The Laplace Transform: page 548

1. Use s = u + iT and the integ ral J e-(<1+ir)t d t = .-·• 1 -<Tc(.,+r•in('T't)) +
   i e-•• 1 .,.c"!.\;~~t"•in('rt)) = u(t) + iv(t) and supply the details showing t hat
   Jim u(t) = 0 and Jim v(t) = 0. T hen C (1) = fi
   00
   e-<<T+ i.,.)tdt = 0 +Qi=
   t - + oo t-+ oo 0

• 1 I

cr+i'r = $·

3. [, (f (t)) = ~ - ce:•• - •:; •.

1 e"- •

5. [, (f (t)) = s-<> - s-a.

7. 
   

   C (3t^2 - 4t+5) = f!r-ti + ~-

   
   


8. 
   

   c ( e2t-3) = :=~.

   
   

11.J.., r (( t + 1)4) -_ ~^24 + .. 2 4 +^1 .,^2 +:;v^4 +,I.

13. c-^1 (.2!25) =! sin 5t.

15. c-^1 (^1 +>;-•

3

) = - 1 + t + ~-

11. c-^1 (.t 4 ) = 3 e-^21 + 3e^21 = 6 cosh 2t,