Let be a locally integrable function on , that is, exists for all and satisfying . The Mellin transform of is defined by
when this integral converges. The domain of analyticity of is usually an infinite strip parallel to the imaginary axis. The inversion formula is given by
One of the two convolution integrals associated with the Mellin transform is of the form
If and have a common strip of analyticity , then
where . When , this identity is a Parseval-type formula; compare §1.14(iv).
If and can be continued analytically to meromorphic functions in a left half-plane, and if the contour can be translated to with , then
The sum in (2.5.6) is taken over all poles of in the strip , and it provides the asymptotic expansion of for small values of . Similarly, if and can be continued analytically to meromorphic functions in a right half-plane, and if the vertical line of integration can be translated to the right, then we obtain an asymptotic expansion for for large values of .
where denotes the Bessel function (§10.2(ii)), and is a large positive parameter. Let and . Then from Table 1.14.5 and Watson (1944, p. 403)
In the half-plane , the product has a pole of order two at each positive integer, and
and is the logarithmic derivative of the gamma function (§5.2(i)).
We now apply (2.5.5) with , and then translate the integration contour to the right. This is allowable in view of the asymptotic formula
as , uniformly for bounded ; see (5.11.9). Then as in (2.5.6) and (2.5.7), with , we obtain
From (2.5.12) and (2.5.13), it is seen that when is even. Hence
where , .
Let and be locally integrable on and
where for , and as . Also, let
where is real, , for , and as . To ensure that the integral (2.5.3) converges we assume that
with , and
with . To apply the Mellin transform method outlined in §2.5(i), we require the transforms and to have a common strip of analyticity. This, in turn, requires , , and either or . Following Handelsman and Lew (1970, 1971) we now give an extension of this method in which none of these conditions is required.
First, we introduce the truncated functions and defined by
With these definitions and the conditions (2.5.17)–(2.5.20) the Mellin transforms converge absolutely and define analytic functions in the half-planes shown in Table 2.5.1.
|Transform||Domain of Convergence|
Furthermore, can be continued analytically to a meromorphic function on the entire -plane, whose singularities are simple poles at , , with principal part
By Table 2.5.1, is an analytic function in the half-plane . Hence we can extend the definition of the Mellin transform of by setting
for . The extended transform has the same properties as in the half-plane .
Similarly, if in (2.5.18), then can be continued analytically to a meromorphic function on the entire -plane with simple poles at , , with principal part
Alternatively, if in (2.5.18), then can be continued analytically to an entire function.
Since is analytic for by Table 2.5.1, the analytically-continued allows us to extend the Mellin transform of via
in the same half-plane. From (2.5.26) and (2.5.28), it follows that both and are defined in the half-plane .
We are now ready to derive the asymptotic expansion of the integral in (2.5.3) as . First we note that
By direct computation
Next from Table 2.5.1 we observe that the integrals for the transform pair and are absolutely convergent in the domain specified in Table 2.5.2, and these domains are nonempty as a consequence of (2.5.19) and (2.5.20).
For simplicity, write
From Table 2.5.2, we see that each is analytic in the domain . Furthermore, each has an analytic or meromorphic extension to a half-plane containing . Now suppose that there is a real number in such that the Parseval formula (2.5.5) applies and
If, in addition, there exists a number such that
as . (The last order estimate follows from the Riemann–Lebesgue lemma, §1.8(i).) The asymptotic expansion of is then obtained from (2.5.29).
For further discussion of this method and examples, see Wong (1989, Chapter 3), Paris and Kaminski (2001, Chapter 5), and Bleistein and Handelsman (1975, Chapters 4 and 6). The first reference also contains explicit expressions for the error terms, as do Soni (1980) and Carlson and Gustafson (1985).
The Mellin transform method can also be extended to derive asymptotic expansions of multidimensional integrals having algebraic or logarithmic singularities, or both; see Wong (1989, Chapter 3), Paris and Kaminski (2001, Chapter 7), and McClure and Wong (1987). See also Brüning (1984) for a different approach.
Let satisfy (2.5.18) and (2.5.20) with , and consider the Laplace transform
Put and break the integration range at , as in (2.5.23) and (2.5.24). Then
Since , by the Parseval formula (2.5.5), there are real numbers and such that , , and
Since is analytic for , by (2.5.14),
for any satisfying . Similarly, since can be continued analytically to a meromorphic function (when ) or to an entire function (when ), we can choose so that has no poles in . Thus
On substituting (2.5.41) and (2.5.42) into (2.5.38), we obtain
where () is an arbitrary integer and is an arbitrary small positive constant. The last term is clearly as .
If in (2.5.18) and in (2.5.20), and if none of the exponents in (2.5.18) are positive integers, then the expansion (2.5.43) gives the following useful result:
With , we have for . In the notation of (2.5.18) and (2.5.20), , , and . Straightforward calculation gives
where . From (2.5.28)
where is Euler’s constant (§5.2(ii)). Insertion of these results into (2.5.43) yields
To verify (2.5.48) we may use