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Watson's lemma is a result in mathematics , proved by G. N. Watson (1918, p. 133), that has significant application within the theory on the asymptotic behavior of integrals .
Statement of the lemma
Assume
ϕ
(
t
)
=
t
λ
g
(
t
)
{\displaystyle \phi (t)=t^{\lambda }\ g(t)}
, where
g
(
t
)
{\displaystyle g(t)}
has an infinite number of derivatives in the neighborhood of
t
=
0
{\displaystyle t=0}
, with
g
(
0
)
≠
0
{\displaystyle g(0)\neq 0}
, and
λ
>
−
1
{\displaystyle \lambda >-1}
.
Suppose, in addition, that
∫
0
T
|
ϕ
(
t
)
|
d
t
<
∞
,
T
>
0.
{\displaystyle \int _{0}^{T}|\phi (t)|\ dt<\infty ,\ \ T>0.}
Then, it is true that
∫
0
T
e
−
x
t
ϕ
(
t
)
d
t
{\displaystyle \int _{0}^{T}e^{-xt}\,\phi (t)\ dt}
is finite
∀
x
>
0
{\displaystyle \forall x>0}
, and the following asymptotic equivalence holds:
∫
0
T
e
−
x
t
ϕ
(
t
)
d
t
∼
∑
n
=
0
∞
g
(
n
)
(
0
)
Γ
(
λ
+
n
+
1
)
n
!
x
λ
+
n
+
1
,
(
x
>
0
,
x
→
∞
)
.
{\displaystyle \int _{0}^{T}e^{-xt}\,\phi (t)\ dt\ \sim \ \sum _{n=0}^{\infty }{\frac {g^{(n)}(0)\ \Gamma (\lambda +n+1)}{n!\ x^{\lambda +n+1}}},\ \ (x>0,\ x\rightarrow \infty ).}
Proof: See, for instance, Watson (1918) for the original proof or Miller (2006) for a more recent development.
Example
Find a simple asymptotic approximation for
f
(
x
)
=
∫
0
T
e
−
x
t
t
−
1
2
d
t
{\displaystyle f(x)=\int _{0}^{T}e^{-xt}\,t^{-{\frac {1}{2}}}\ dt}
for large values of
x
{\displaystyle x}
, that is when
x
→
∞
{\displaystyle x\rightarrow \infty }
.
Solution: By direct application of Watson's lemma, with
λ
=
−
1
/
2
{\displaystyle \lambda =-1/2}
and
g
(
t
)
=
1
{\displaystyle g(t)=1}
,
so that
g
n
(
t
)
=
0
{\displaystyle g^{n}(t)=0}
for
n
≠
0
{\displaystyle n\neq 0}
. It is easy to see that:
f
(
x
)
=
∫
0
T
e
−
x
t
t
−
1
2
d
t
∼
π
/
x
(
as
x
→
∞
)
.
{\displaystyle f(x)=\int _{0}^{T}e^{-xt}\,t^{-{\frac {1}{2}}}\ dt\sim {\sqrt {\pi /x}}\ \ \ ({\mbox{as}}\ x\rightarrow \infty ).}
This development used the fact that
Γ
(
1
/
2
)
=
π
{\displaystyle \Gamma (1/2)={\sqrt {\pi }}}
, a known property of the Gamma function .
References
Miller, P.D. (2006), Applied Asymptotic Analysis , Providence, RI: American Mathematical Society, p. 467, ISBN 9780821840788 .
Watson, G. N. (1918), "The harmonic functions associated with the parabolic cylinder", Proc. London Math. Soc. , vol. 2, no. 17, pp. 116–148 .
[[Category:Lemmas] [Category:Asymptotic analysis]]