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 0 < T ≤ ∞ {\displaystyle 0<T\leq \infty } be fixed. Assume φ ( t ) = t λ g ( t ) {\displaystyle \varphi (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, either that
| φ ( t ) | < K e b t ∀ t > 0 , {\displaystyle |\varphi (t)|<Ke^{bt}\ \forall t>0,}
where K , b {\displaystyle K,b} are independent of t {\displaystyle t} , or that
∫ 0 T | φ ( t ) | d t < ∞ . {\displaystyle \int _{0}^{T}|\varphi (t)|\,\mathrm {d} t<\infty .}
Then, it is true that for all positive x {\displaystyle x} that
| ∫ 0 T e − x t φ ( t ) d t | < ∞ {\displaystyle \left|\int _{0}^{T}e^{-xt}\varphi (t)\,\mathrm {d} t\right|<\infty }
and that 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}\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 ).}
See, for instance, Watson (1918) for the original proof or Miller (2006) for a more recent development.
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