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Suppose that, as

t ↓ 0

f (t) \sim \sum_{n = 0}^{\infty} a_{n} t^{λ_{n} - 1}, 0 < λ_{0} < λ_{1} < \dots,

and that the corresponding Laplace integral has suitable integrability and growth control away from zero. Watson lemma gives the endpoint expansion

\int_{0}^{b} e^{- x t} f (t) d t \sim \sum_{n = 0}^{\infty} a_{n} Γ (λ_{n}) x^{- λ_{n}}

x \to + \infty

. Each finite truncation of the local amplitude expansion produces the corresponding finite truncation and remainder of the integral expansion.

Watson's lemma