Chapter 7
Improper Integrals and
Indeterminate Forms
7.1Integrals over Unbounded Intervals
Definition 7.1.1 Suppose that a function f is continuous on (−∞, ∞). Then we define the following improper integrals when the limits exist
Z ∞ f(x)dx = lim |
Z b f(x)dx |
(1) |
ab→∞ a
b |
a→−∞ |
b |
|
Z−∞ |
Za |
(2) |
f(x)dx = |
lim |
f(x)dx |
∞ |
c |
|
|
Z−∞ f(x)dx = |
Z−∞ f(x)dx + Zc ∞ f(x)dx |
(3) |
provided the integrals on the right hand side exist for some c. If these improper integrals exist, we say that they are convergent; otherwise they are said to be divergent.
Definition 7.1.2 Suppose that a function f is continuous on [0, ∞). Then the Laplace transform of f, written L(f) or F (s), is defined by
Z ∞
L(f) = F (s) = e−stf(t)dt.
0
296CHAPTER 7. IMPROPER INTEGRALS AND INDETERMINATE FORMS
provided s > a.
(iii) L(cosh at) = Z0 |
∞ |
|
eat + e |
at |
e−stdt |
|
− |
|
|
2 |
|
= 12[L(eat) + L(e−at)]
s
= s2 − a2 , s > |a|.
(iv) L(sinh at) = Z ∞ 1 (eat − e−at)e−stdt
0 2
|
1 |
1 |
|
1 |
|
= |
|
|
|
|
− |
|
, s > |a| |
2 |
|
s − a |
s + a |
a
= s2 − a2 , s > |a|.
(v) |
L(cos ωt) = Z0 |
∞ cos ωte−stdt |
|
= |
|
|
|
1 |
|
e−st(−s cos ωt + ω sin ωt) 0∞ |
|
|
|
|
|
|
ω2 |
+ s2 |
|
|
|
|
|
s |
|
= |
|
|
|
. |
|
ω2 |
+ s2 |
(vi) |
L(sin ωt) = Z0 |
∞ sin ωte−stdt |
|
= |
|
|
|
1 |
|
e−st(−s sin ωt − ω cos ωt) 0∞ |
|
|
|
|
|
ω2 + s2 |
|
|
|
|
|
ω |
|
= |
|
|
. |
|
|
ω2 + s2 |
7.1. INTEGRALS OVER UNBOUNDED INTERVALS |
297 |
(vii) L(t) = Z0 |
∞ te−stdt; |
|
(u = t, dv = e−stdt) |
|
|
|
te st |
∞ |
|
|
∞ e−st |
|
= |
|
|
− |
|
|
0 |
+ |
|
|
|
dt |
|
|
|
s |
|
|
0 |
s |
|
|
|
− |
|
|
|
Z |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
= |
|
e−st |
|
|
∞ |
|
|
|
|
|
|
|
− |
s2 |
0 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
= |
1 |
. |
|
|
|
|
|
|
|
|
s2 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
This completes the proof of Theorem 7.1.1.
Theorem 7.1.2 Suppose that f f(x) ≤ g(x) on [a, ∞).
Z∞
(i)If g(x)dx converges, then
and g are continuous on [a, ∞) and 0 ≤
Z ∞
f(x)dx converges.
a |
a |
(ii) If Za∞ f(x)dx diverges, then |
Za∞ g(x)dx diverges. |
Proof. The proof of this follows from the order properties of the integral and is omitted.
Definition 7.1.3 For each x > 0, the Gamma function, denoted (x), is defined by
(x) = Z0 |
∞ tx−1e−tdt. |
|
Theorem 7.1.3 The Gamma function has the following properties: |
|
(1) = 1 |
|
(11) |
(x + 1) = x (x) |
(12) |
(n + 1) = n!, n = natural number |
(13) |
298CHAPTER 7. IMPROPER INTEGRALS AND INDETERMINATE FORMS
Proof.
Z ∞
(1) = e−tdt
0
= −e−t ∞
0
=1
Z ∞
(x + 1) = txe−tdt; (u = tx, dv = e−tdt)
|
0 |
+ x Z0 |
∞ |
|
= −txe−t 0∞ |
|
tx−1e−tdt |
|
|
|
|
= x (x), x > 0(2) = 1 (1) = 1(3) = 2 (2) = 1 · 2 = 2!
If (k) = (k − 1)!, then(k + 1) = k (k)
=k((k − 1)!)
=k!.
By the principle of mathematical induction,
(n + 1) = n!
for all natural numbers n. This completes the proof of this theorem.
Theorem 7.1.4 Let f be the normal probability distribution function defined by
f(x) = σ√2π |
e− √2σ |
|
2 |
1 |
x−µ |
|
|
|
|
|
|
where µ is the constant mean of the distribution and σ is the constant standard deviation of the distribution. Then the improper integral
Z ∞
f(x)dx = 1.
−∞
Let F be the normal distribution function defined by
Z x
F (x) = f(x)dx.
