cal (1, 1≤1<∞ 0 ≤ t < 1 1≤1<2 2≤1<∞ In each of Problems 19 through 21, determine whether the given integral converges or diverges. to no tuldo. 0, ce b 18. f(t)=2-1, wort weroo 19. (1²+1)-¹d²301 Ste 5.- t-2e'dt. 22. Suppose that f and f' are continuous for t≥ 0 and of exponential order as too. Use integration by parts to show that if F(s) = L{f(t)), then lim F(s) = 0. The result is actually true 818 20. 21. te 'dt under less restrictive conditions, such as those of Theorem 6.1.2. PETAN 23. The Gamma Function. The gamma function is denoted by
cal (1, 1≤1<∞ 0 ≤ t < 1 1≤1<2 2≤1<∞ In each of Problems 19 through 21, determine whether the given integral converges or diverges. to no tuldo. 0, ce b 18. f(t)=2-1, wort weroo 19. (1²+1)-¹d²301 Ste 5.- t-2e'dt. 22. Suppose that f and f' are continuous for t≥ 0 and of exponential order as too. Use integration by parts to show that if F(s) = L{f(t)), then lim F(s) = 0. The result is actually true 818 20. 21. te 'dt under less restrictive conditions, such as those of Theorem 6.1.2. PETAN 23. The Gamma Function. The gamma function is denoted by
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Question
17
Transcribed Image Text:1.
equation (6), we write
Then, from Examples 5 and 7, we obtain
L{f(t)}
2. f(t) =
6-t,
(1²,
Problems
In each of Problems 1 through 3, sketch the graph of the given
function. In each case determine whether f is continuous, piecewise
continuous, or neither on the interval 0 ≤ t ≤ 3.
1²,
f(t)=2+t,
(t-1)-¹,
1,
t²,
1,
=
L{ f(t)} = 5£{e-²¹} - 3£{sin(41)}.
0 ≤ t ≤ 1
1< t <2
2<t≤3
cosh(bt) = (+
Va
0 ≤t≤1
1< t ≤2
2 < t ≤3
=
0≤t≤1
1<t≤2
2<t≤3
5
S+2
12. f(t) = teat
13. f(t) = t sin(at)
14. f(t) = teat
15. f(t) = 1² sin(at)
-
ul6V
12
s² + 16'
s > 0.
a. f(t) = t
b. f(t) = 12
c. f(t) = t", where n is a positive integer
5. Find the Laplace transform of f(t) = cos(at), where a is a real
constant.
Recall that
In each of Problems 16 through 18, find the Laplace transform of the
given function.
In each of Problems 6 through 7, use the linearity of the Laplace lov
transform to find the Laplace transform of the given function; a and b
are real constants.
6. f(t) = cosh(bt)
7. f(t) = sinh(bt)
Recall that
1
cos(bt)
gibt + e-ibt) and sin(bt) = (eibt - e-ibt).
žleibe
1
2i
In each of Problems 8 through 11, use the linearity of the Laplace
transform to find the Laplace transform of the given function; a and b
are real constants. Assume that the necessary elementary integration
formulas extend to this case.
16. f(t)=
Isitial to go
1≤t<∞0
0 ≤ t < 1
11 < 2
2≤t<∞0
to noituldo,
In each of Problems 19 through 21, determine whether the given
integral converges or diverges.
3. f(t) =
3-t,
7011 weroo
4. Find the Laplace transform of each of the following functions: no: 19. S (t² + 1)-¹dt
17. f(t) =
8. f(t) = sin(bt)
9. f(t) = cos(bt)
10. f(t) = eat sin(bt)
11. f(t) = eat cos(bt)
In each of Problems 12 through 15, use integration by parts to find the
Laplace transform of the given function; n is a positive integer and a
is a real constant.
18. f(t) =
worl
=
20.
21.
22. Suppose that f and f' are continuous for t≥ 0 and of
exponential order as t→∞. Use integration by parts to show that
1
(eb¹ + e−br) and sinh(bt) = -(e¹¹ - e-bt). be an if F(s) = £[ƒ(1)), then lim F(s) = 0. The result is actually true
L{f(t)},
Sintienos
no 12:
818
under less restrictive conditions, such as those of Theorem 6.1.2.
1, 0≤t<T
10, T ≤t<∞
[t, 0≤t<1
1,
t,
2-t,
8
So t
vd betonsb
te tdt
[₁
t-2 e' dt
23. The Gamma Function. The gamma function is denoted by
T(p) and is defined by the integral
(7)
The integral converges as x→ ∞ for all p. For p < 0 it is also
adı 15blanos sw improper at x = 0, because the integrand becomes unbounded as
x → 0. However, the integral can be shown to converge at x = 0
for p > -1.
a.
Show that, for p > 0,
88
= √²²
T(p+1) =
b. Show that IT (1) = 1.
c.
T(p+1) = p (p).
e-xxPdx.
If p is a positive integer n, show that
T(n + 1) = n!.
possible to show that I'
Since I'(p) is also defined when p is not an integer, this function
provides an extension of the factorial function to nonintegral
values of the independent variable. Note that it is also consistent
to define 0! = 1.
d. Show that, for p > 0,
p(p+1)(p+2)
(p+n-1) =
Thus I (p) can be determined for all positive values of p if T (p)
is known in a single interval of unit length-say, 0 < p ≤ 1. It is
T(p+n)
T(p)
3
(1) = √T. Find I (²)
2
and I
11
(1)
2
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