1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

540 CHAPTER 12 • FOURI ER SERIES AND THE LAPLACE TRANSFORM

Solution Using the result of Example 12.5 and the symmetry property, we

obtain

3" Cr (1 ~ t2)) = 2~ e-1-wl = 2~ e-lwl.

We use the linearity property and multiply each term by 7r to obtain

3" (i ~ tZ) = ~e-lwl,

establishing t he result.

-------~EXERCISES FOR SECTION 1 2. 4

1. Let U(t) = { ~:

Find ;J"(U(t)).

2. Let U(t) = { ~~t,

for itl < l;

for ltl > l.

for ltl $ 7r;

for ltl > 7r.

isin11"W

Show that ;J" (U (t)) = 7r (l _ wZ).

3. Let U (t) = { 01 ,-ltl, for itl $ l ;
   for ltl > l.
   Find ;J" (U (t)).
   2 1 -w^2
   · 4. Let U(t) = e-• 12. Show that ;J"(U(t)) = t=e-,-. Hint: Use the integral
   v27r
   definition and combine the terms in the exponent; then complete the square and
   _,2
   use the fact that J~ 00 e-,-dt = ,,/2i.


4. Use the time scaling property and Example 12. 5 in the text to show that

J (e-af<I) = lal.

7r(a2 +wz)

6. Use t he symmetry and linearity properties and the result of Exercise 1 to show
   that

for lwl < l;

for lwl > l.

1. Use the symmetry and linearity properties and the result of Exercise 2 to show

that

(

i sin 7rt) = { isin w

J 1 - t2 0, 2

for lwl $ 7r;

for lwl > 1f.