Guide Extensions of asymptotic fields via meromorphic functions

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Explanations (6)

Then from Table 1. We now apply 2.

This is allowable in view of the asymptotic formula. Then as in 2. From 2. Also, let.

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To ensure that the integral 2. Following Handelsman and Lew , we now give an extension of this method in which none of these conditions is required. With these definitions and the conditions 2. By Table 2. Hence we can extend the definition of the Mellin transform of f by setting. First we note that.

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Next from Table 2. From Table 2. For further discussion of this method and examples, see Wong , Chapter 3 , Paris and Kaminski , Chapter 5 , and Bleistein and Handelsman , Chapters 4 and 6. The first reference also contains explicit expressions for the error terms, as do Soni and Carlson and Gustafson The Mellin transform method can also be extended to derive asymptotic expansions of multidimensional integrals having algebraic or logarithmic singularities, or both; see Wong , Chapter 3 , Paris and Kaminski , Chapter 7 , and McClure and Wong On substituting 2.