Watson's lemma

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

Assume $\phi (t)=t^{\lambda }\,g(t)$, where $g(t)$ has an infinite number of derivatives in the neighborhood of $t=0$, with $g(0)\neq 0$, and $\lambda >-1$.

Suppose, in addition, that

$\int _{0}^{T}|\phi (t)|\,dt<\infty ,\ \ T>0.$

Then, it is true that

$\int _{0}^{T}e^{-xt}\,\phi (t)\,dt$

is finite $\forall x>0$, and the following asymptotic equivalence holds:

$\int _{0}^{T}e^{-xt}\,\phi (t)\ dt\ \sim \ \sum _{n=0}^{\infty }{\frac {g^{(n)}(0)\ \Gamma (\lambda +n+1)}{n!\ x^{\lambda +n+1}}},\ \ (x>0,\ x\rightarrow \infty ).$

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

$f(x)=\int _{0}^{\infty }e^{-xt}\,t^{-{\frac {1}{2}}}\ dt$

for large values of $x$, that is when $x\rightarrow \infty$.

Solution: By direct application of Watson's lemma, with $\lambda =-1/2$ and $g(t)=1$, so that $g^{(n)}(t)=0$ for $n\neq 0$. It is easy to see that:

$f(x)=\int _{0}^{\infty }e^{-xt}\,t^{-1/2}\,dt\sim {\sqrt {\frac {\pi }{x}}}{\text{ as }}x\rightarrow \infty .$

This development used the fact that $\Gamma (1/2)={\sqrt {\pi }}$, 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", Procedings of the London Mathematical Society, vol. 2, no. 17, pp. 116–148.