On the general asymptotic expansions of k_υ-transforms

Theorems are established to obtain the general asymptotic expansions of the K_υ-transforms. It is shown that, under certain circumstances, the asymptotic expansion of K_υ-transform F(s) as s → 0⁺ or s → ∞ is obtained by integrating term by term the asymptotic expansion of the function at t → ∞ or t → 0⁺. In the case that the function has a power expansion, the treatment of this problem can be found in the existing literature (e.g. [4],[5]). In this paper we assume that the function has a general expansion form. Three examples, including the case that the function has an exponentially small behaviour at the origin, are discussed.