In mathematics, Watson's lemma, proved by G. N. Watson (1918, p. 133), has significant application within the theory on the asymptotic behavior of integrals.
Statement of the lemma
Let
be fixed. Assume
, where
has an infinite number of derivatives in the neighborhood of
, with
, and
.
Suppose, in addition, either that
![{\displaystyle |\varphi (t)|<Ke^{bt}\ \forall t>0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a94f0fd427548916a8db7cccf515d34bb802be2b)
where
are independent of
, or that
![{\displaystyle \int _{0}^{T}|\varphi (t)|\,\mathrm {d} t<\infty .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fbcc49004e4971fe57127b37d54afbf5a1ca6e67)
Then, it is true that for all positive
that
![{\displaystyle \left|\int _{0}^{T}e^{-xt}\varphi (t)\,\mathrm {d} t\right|<\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/36648336df5d02a34426f74bc040bbabb2c36e64)
and that the following asymptotic equivalence holds:
![{\displaystyle \int _{0}^{T}e^{-xt}\varphi (t)\,\mathrm {d} t\sim \ \sum _{n=0}^{\infty }{\frac {g^{(n)}(0)\ \Gamma (\lambda +n+1)}{n!\ x^{\lambda +n+1}}},\ \ (x>0,\ x\rightarrow \infty ).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea32a40d7d9c188d6729e6cf8f4bed0aac0343d9)
See, for instance, Watson (1918) for the original proof or Miller (2006) for a more recent development.
Proof
We will prove the version of Watson's lemma which assumes that
has at most exponential growth as
. The basic idea behind the proof is that we will approximate
by finitely many terms of its Taylor series. Since the derivatives of
are only assumed to exist in a neighborhood of the origin, we will essentially proceed by removing the tail of the integral, applying Taylor's theorem with remainder in the remaining small interval, then adding the tail back on in the end. At each step we will carefully estimate how much we are throwing away or adding on. This proof is a modification of the one found in Miller (2006).
Let
and suppose that
is a measurable function of the form
, where
and
has an infinite number of continuous derivatives in the interval
for some
, and that
for all
, where the constants
and
are independent of
.
We can show that the integral is finite for
large enough by writing
![{\displaystyle (1)\quad \int _{0}^{T}e^{-xt}\varphi (t)\,\mathrm {d} t=\int _{0}^{\delta }e^{-xt}\varphi (t)\,\mathrm {d} t+\int _{\delta }^{T}e^{-xt}\varphi (t)\,\mathrm {d} t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f32c1af85a1031b95c0a3f49b48ab1cc3333bf74)
and estimating each term.
For the first term we have
![{\displaystyle \left|\int _{0}^{\delta }e^{-xt}\varphi (t)\,\mathrm {d} t\right|\leq \int _{0}^{\delta }e^{-xt}|\varphi (t)|\,\mathrm {d} t\leq \int _{0}^{\delta }|\varphi (t)|\,\mathrm {d} t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8767d9426207f1ec64185c74f1d9553f93e7df5b)
for
, where the last integral is finite by the assumptions that
is continuous on the interval
and that
. For the second term we use the assumption that
is exponentially bounded to see that, for
,
![{\displaystyle {\begin{aligned}\left|\int _{\delta }^{T}e^{-xt}\varphi (t)\,\mathrm {d} t\right|&\leq \int _{\delta }^{T}e^{-xt}|\varphi (t)|\,\mathrm {d} t\\&\leq K\int _{\delta }^{T}e^{(b-x)t}\,\mathrm {d} t\\&\leq K\int _{\delta }^{\infty }e^{(b-x)t}\,\mathrm {d} t\\&=K\,{\frac {e^{(b-x)\delta }}{x-b}}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c04add0b5819e323cb5884e163a02dd1f580cd51)
The finiteness of the original integral then follows from applying the triangle inequality to
.
We can deduce from the above calculation that
![{\displaystyle (2)\quad \int _{0}^{T}e^{-xt}\varphi (t)\,\mathrm {d} t=\int _{0}^{\delta }e^{-xt}\varphi (t)\,\mathrm {d} t+O\left(x^{-1}e^{-\delta x}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/191621b3377f98fdc77ccee34cdcb86c79b89347)
as
.
By appealing to Taylor's theorem with remainder we know that, for each integer
,
![{\displaystyle g(t)=\sum _{n=0}^{N}{\frac {g^{(n)}(0)}{n!}}\,t^{n}+{\frac {g^{(N+1)}(t^{*})}{(N+1)!}}\,t^{N+1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53d93530fd4b338b7161c55c1755292f51c1e158)
for
, where
. Plugging this in to the first term in
we get
![{\displaystyle {\begin{aligned}(3)\quad \int _{0}^{\delta }e^{-xt}\varphi (t)\,\mathrm {d} t&=\int _{0}^{\delta }e^{-xt}t^{\lambda }g(t)\,\mathrm {d} t\\&=\sum _{n=0}^{N}{\frac {g^{(n)}(0)}{n!}}\int _{0}^{\delta }t^{\lambda +n}e^{-xt}\,\mathrm {d} t+{\frac {1}{(N+1)!}}\int _{0}^{\delta }g^{(N+1)}(t^{*})\,t^{\lambda +N+1}e^{-xt}\,\mathrm {d} t.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/261d9b486198f96bd8dd0c7f17f7c45120742e34)
To bound the term involving the remainder we use the assumption that
is continuous on the interval
, and in particular it is bounded there. As such we see that
![{\displaystyle {\begin{aligned}\left|\int _{0}^{\delta }g^{(N+1)}(t^{*})\,t^{\lambda +N+1}e^{-xt}\,\mathrm {d} t\right|&\leq \sup _{t\in [0,\delta ]}\left|g^{(N+1)}(t)\right|\int _{0}^{\delta }t^{\lambda +N+1}e^{-xt}\,\mathrm {d} t\\&<\sup _{t\in [0,\delta ]}\left|g^{(N+1)}(t)\right|\int _{0}^{\infty }t^{\lambda +N+1}e^{-xt}\,\mathrm {d} t\\&=\sup _{t\in [0,\delta ]}\left|g^{(N+1)}(t)\right|\,{\frac {\Gamma (\lambda +N+2)}{x^{\lambda +N+2}}}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/088c6d8aead67a29697d944b25ece6f6740c5714)
Here we have used the fact that
![{\displaystyle \int _{0}^{\infty }t^{a}e^{-xt}\,\mathrm {d} t={\frac {\Gamma (a+1)}{x^{a+1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/05fecad7653c77602fe5a05522d1ce81afd0a30b)
if
and
, where
is the gamma function.
From the above calculation we see from
that
![{\displaystyle (4)\quad \int _{0}^{\delta }e^{-xt}\varphi (t)\,\mathrm {d} t=\sum _{n=0}^{N}{\frac {g^{(n)}(0)}{n!}}\int _{0}^{\delta }t^{\lambda +n}e^{-xt}\,\mathrm {d} t+O\left(x^{-\lambda -N-2}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b41e1345138e317c40fba3e035c28df4c01ac7a)
as
.
We will now add the tails on to each integral in
. For each
we have
![{\displaystyle {\begin{aligned}\int _{0}^{\delta }t^{\lambda +n}e^{-xt}\,\mathrm {d} t&=\int _{0}^{\infty }t^{\lambda +n}e^{-xt}\,\mathrm {d} t-\int _{\delta }^{\infty }t^{\lambda +n}e^{-xt}\,\mathrm {d} t\\[5pt]&={\frac {\Gamma (\lambda +n+1)}{x^{\lambda +n+1}}}-\int _{\delta }^{\infty }t^{\lambda +n}e^{-xt}\,\mathrm {d} t,\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a636c582e462d650f8bb182df4a3967f0c9f59d)
and we will show that the remaining integrals are exponentially small. Indeed, if we make the change of variables
we get
![{\displaystyle {\begin{aligned}\int _{\delta }^{\infty }t^{\lambda +n}e^{-xt}\,\mathrm {d} t&=\int _{0}^{\infty }(s+\delta )^{\lambda +n}e^{-x(s+\delta )}\,\mathrm {d} s\\[5pt]&=e^{-\delta x}\int _{0}^{\infty }(s+\delta )^{\lambda +n}e^{-xs}\,\mathrm {d} s\\[5pt]&\leq e^{-\delta x}\int _{0}^{\infty }(s+\delta )^{\lambda +n}e^{-s}\,\mathrm {d} s\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff7014de6e6869b74a2c78b7ab408d78cf4ad82a)
for
, so that
![{\displaystyle \int _{0}^{\delta }t^{\lambda +n}e^{-xt}\,\mathrm {d} t={\frac {\Gamma (\lambda +n+1)}{x^{\lambda +n+1}}}+O\left(e^{-\delta x}\right){\text{ as }}x\to \infty .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0a8ccbd528eca1adc0b15b833d2ec9105c36201)
If we substitute this last result into
we find that
![{\displaystyle {\begin{aligned}\int _{0}^{\delta }e^{-xt}\varphi (t)\,\mathrm {d} t&=\sum _{n=0}^{N}{\frac {g^{(n)}(0)\ \Gamma (\lambda +n+1)}{n!\ x^{\lambda +n+1}}}+O\left(e^{-\delta x}\right)+O\left(x^{-\lambda -N-2}\right)\\&=\sum _{n=0}^{N}{\frac {g^{(n)}(0)\ \Gamma (\lambda +n+1)}{n!\ x^{\lambda +n+1}}}+O\left(x^{-\lambda -N-2}\right)\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef63996924d324d04bf81a540a13c7acdef35b45)
as
. Finally, substituting this into
we conclude that
![{\displaystyle {\begin{aligned}\int _{0}^{T}e^{-xt}\varphi (t)\,\mathrm {d} t&=\sum _{n=0}^{N}{\frac {g^{(n)}(0)\ \Gamma (\lambda +n+1)}{n!\ x^{\lambda +n+1}}}+O\left(x^{-\lambda -N-2}\right)+O\left(x^{-1}e^{-\delta x}\right)\\&=\sum _{n=0}^{N}{\frac {g^{(n)}(0)\ \Gamma (\lambda +n+1)}{n!\ x^{\lambda +n+1}}}+O\left(x^{-\lambda -N-2}\right)\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54c24a2a1535c717b1d66f1f3244afd19b6d75d1)
as
.
Since this last expression is true for each integer
we have thus shown that
![{\displaystyle \int _{0}^{T}e^{-xt}\varphi (t)\,\mathrm {d} t\sim \sum _{n=0}^{\infty }{\frac {g^{(n)}(0)\ \Gamma (\lambda +n+1)}{n!\ x^{\lambda +n+1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ce231e956e0a306d29907abbd09f48427e01237)
as
, where the infinite series is interpreted as an asymptotic expansion of the integral in question.
Example
When
, the confluent hypergeometric function of the first kind has the integral representation
![{\displaystyle {}_{1}F_{1}(a,b,x)={\frac {\Gamma (b)}{\Gamma (a)\Gamma (b-a)}}\int _{0}^{1}e^{xt}t^{a-1}(1-t)^{b-a-1}\,\mathrm {d} t,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c8ba92f9e9c9dfcf9c4b1ad5e7bd71e9dd11755)
where
is the gamma function. The change of variables
puts this into the form
![{\displaystyle {}_{1}F_{1}(a,b,x)={\frac {\Gamma (b)}{\Gamma (a)\Gamma (b-a)}}\,e^{x}\int _{0}^{1}e^{-xs}(1-s)^{a-1}s^{b-a-1}\,ds,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92a223a1166478d8dee9b1b3ae4f2319fee44ea3)
which is now amenable to the use of Watson's lemma. Taking
and
, Watson's lemma tells us that
![{\displaystyle \int _{0}^{1}e^{-xs}(1-s)^{a-1}s^{b-a-1}\,ds\sim \Gamma (b-a)x^{a-b}\quad {\text{as }}x\to \infty {\text{ with }}x>0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/249202e3aa949411aecd670d47f7d83f08679ecf)
which allows us to conclude that
![{\displaystyle {}_{1}F_{1}(a,b,x)\sim {\frac {\Gamma (b)}{\Gamma (a)}}\,x^{a-b}e^{x}\quad {\text{as }}x\to \infty {\text{ with }}x>0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc119a19c1dbeb028fa9fa2ed02c4ac96b384ab1)
References
- Miller, P.D. (2006), Applied Asymptotic Analysis, Providence, RI: American Mathematical Society, p. 467, ISBN 978-0-8218-4078-8.
- Watson, G. N. (1918), "The harmonic functions associated with the parabolic cylinder", Proceedings of the London Mathematical Society, vol. 2, no. 17, pp. 116–148, doi:10.1112/plms/s2-17.1.116.
- Ablowitz, M. J., Fokas, A. S. (2003). Complex variables: introduction and applications. Cambridge University Press.