The bilaleral Laplace traiisform exists if the regioii ol' corierge1m is not rmpty, i.e.
Trang 1472 Appendix A Solutions to the Exercisrs
~ o ~ u t i o ~ 4.2
Right-sitled functions have exponential mxkr, if 1L1, C and 7' can found such that:
I.x(t)\ 5 A l e r r for t 2 T
a) yes e.g with M = I, C = 1, T = 0
b) yes, e.g w i t h ill = 7 , C - 5, T = 0
c) yes e.g izritli M = I, C = 5, T - 0
d) yes, e.g with AI = 1 C = 6 , 2' =- 0
e ) no
f ) yes, e.g witli M - I , r' = 0, T = 0
~ o l ~ ~ t i o r ~ 4.3
First determine M , G, D a r i d T 50 that /.r(/)l 5 Mc'" for I 2 T anti l.r(t)l 5 Me"'
for t 5 -T The bilaleral Laplace traiisform exists if the regioii ol' corierge1m is not rmpty, i.e if D > c'
a) h f = I , C ' = a , D = O , T = O =+ R O C = { )