1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

548 CHAPTER 12 • FOURlER SERIES AND THE LAPLACE TRANSFORM

Definition .C (! (t)) = F (s).

First derivative .C(f'(t)) = sF(s)-f(O).

Second derivative .C(f"(t)) = s^2 F(s)-sf(O)-f' (0).

Integral .C (J~ f (r) dr) = F ;s).

Multiplication by t C (tf (t)) = -F' (s).

Division by t .C ( f ~t)) = f 8

00

F (11) d<1.

s-axis shifting C (e"t f (t)) = F (s -a).

t-a.xis shifting .C (U., (t) f (t - a))= e-^0 • F (s), for a> O.

Convolution .C(h(t)) = F(s)G(s),

where h (t) = J~ f (t -r) g (r) dr.

Table 12 .S Properties of the Laplace Transform

• ------~EXERCISES FOR SECTION 12.5
  
  
  1. Show that .C(l) =! by using the integral definition of the Laplace transform.
     s
     Assume thats is restricted to values satisfying Re(s) > O.
  
  
  
  

2. Let U (t) = { 1, 0, otherwise. for^1 <. t < 2;

Find l. ( f (t)).

3. Let U(t) = { ~.

Find l. ( f (t)).

forO:S:t<c;

otherwise.

4. Show that .C ( t^2 ) =^23 by using t he integral definition for the Laplace transform.
   s
   Assume that s is restricted to values satisfying Re ( s) > 0.


5. Let U(t) = { e••, for 0 '.S_t < l;
   0, otherwise.

Find l. (/ (t)).

6. Let U (t) = { ~i'n (t)'

Find l. (! (t)).

for 0 :S: t :S: ?rj

otherwise.

For Exercises 7- 12, use the linearity of Laplace transform and Table 12.2.

7. Find l.(3t^2 -4t+5).