# §2.4 Contour Integrals

## §2.4(i) Watson’s Lemma

The result in §2.3(ii) carries over to a complex parameter . Except that is now permitted to be complex, with , we assume the same conditions on and also that the Laplace transform in (2.3.8) converges for all sufficiently large values of . Then

as in the sector (), with assigned its principal value.

If is analytic in a sector containing , then the region of validity may be increased by rotation of the integration paths. We assume that in any closed sector with vertex and properly interior to , the expansion (2.3.7) holds as , and as , where is a constant. Then (2.4.1) is valid in any closed sector with vertex and properly interior to . (The branches of and are extended by continuity.)

For examples and extensions (including uniformity and loop integrals) see Olver (1997b, Chapter 4), Wong (1989, Chapter 1), and Temme (1985).

## §2.4(ii) Inverse Laplace Transforms

On the interval let be differentiable and be absolutely integrable, where is a real constant. Then the Laplace transform

is continuous in and analytic in , and by inversion (§1.14(iii))

where () is a constant.

Now assume that and we are given a function that is both analytic and has the expansion

in the half-plane . Here , , and has its principal value. Assume also (2.4.4) is differentiable. Then by integration by parts the integral

is seen to converge absolutely at each limit, and be independent of . Furthermore, as , has the expansion (2.3.7).

For large , the asymptotic expansion of may be obtained from (2.4.3) by Haar’s method. This depends on the availability of a comparison function for that has an inverse transform

with known asymptotic behavior as . By subtraction from (2.4.3)

If this integral converges uniformly at each limit for all sufficiently large , then by the Riemann–Lebesgue lemma (§1.8(i))

If, in addition, the corresponding integrals with and replaced by their derivatives and , , converge uniformly, then by repeated integrations by parts

The most successful results are obtained on moving the integration contour as far to the left as possible. For examples see Olver (1997b, pp. 315–320).

## §2.4(iii) Laplace’s Method

Let denote the path for the contour integral

in which is finite, is finite or infinite, and is the angle of slope of at , that is, as along . Assume that and are analytic on an open domain that contains , with the possible exceptions of and . Other assumptions are:

1. (a)

In a neighborhood of

with , , , and the branches of and continuous and constructed with as along .

2. (b)

ranges along a ray or over an annular sector , , where , , and . converges at absolutely and uniformly with respect to .

3. (c)

Excluding , is positive when , and is bounded away from zero uniformly with respect to as along .

Then

as in the sector . The coefficients are determined as in §2.3(iii), the branch of being chosen to satisfy

For examples see Olver (1997b, Chapter 4). For error bounds see Boyd (1993).

Now suppose that in (2.4.10) the minimum of on occurs at an interior point . Temporarily assume that is fixed, so that is independent of . We may subdivide

and apply the result of §2.4(iii) to each integral on the right-hand side, the role of the series (2.4.11) being played by the Taylor series of and at . If , then , is a positive integer, and the two resulting asymptotic expansions are identical. Thus the right-hand side of (2.4.14) reduces to the error terms. However, if , then and different branches of some of the fractional powers of are used for the coefficients ; again see §2.3(iii). In consequence, the asymptotic expansion obtained from (2.4.14) is no longer null.

Zeros of are called saddle points (or cols) owing to the shape of the surface , , in their vicinity. Cases in which are usually handled by deforming the integration path in such a way that the minimum of is attained at a saddle point or at an endpoint. Additionally, it may be advantageous to arrange that is constant on the path: this will usually lead to greater regions of validity and sharper error bounds. Paths on which is constant are also the ones on which decreases most rapidly. For this reason the name method of steepest descents is often used. However, for the purpose of simply deriving the asymptotic expansions the use of steepest descent paths is not essential.

In the commonest case the interior minimum of is a simple zero of . The final expansion then has the form

in which

with and their derivatives evaluated at . The branch of is the one satisfying , where is the limiting value of as from .

Higher coefficients in (2.4.15) can be found from (2.3.18) with , , and replaced by . For integral representations of the and their asymptotic behavior as see Boyd (1995). The last reference also includes examples, as do Olver (1997b, Chapter 4), Wong (1989, Chapter 2), and Bleistein and Handelsman (1975, Chapter 7).

## §2.4(v) Coalescing Saddle Points: Chester, Friedman, and Ursell’s Method

Consider the integral

in which is a large real or complex parameter, and are analytic functions of and continuous in and a second parameter . Suppose that on the integration path there are two simple zeros of that coincide for a certain value of . The problem of obtaining an asymptotic approximation to that is uniform with respect to in a region containing is similar to the problem of a coalescing endpoint and saddle point outlined in §2.3(v).

The change of integration variable is given by

with and chosen so that the zeros of correspond to the zeros , say, of the quadratic . Then

where is the -map of , and

The function is analytic at and when , and at the confluence of these points when . For large , is approximated uniformly by the integral that corresponds to (2.4.19) when is replaced by a constant. By making a further change of variable

2.4.21

and assigning an appropriate value to to modify the contour, the approximating integral is reducible to an Airy function or a Scorer function (§§9.2, 9.12).

For examples, proofs, and extensions see Olver (1997b, Chapter 9), Wong (1989, Chapter 7), Olde Daalhuis and Temme (1994), Chester et al. (1957), and Bleistein and Handelsman (1975, Chapter 9).

For a symbolic method for evaluating the coefficients in the asymptotic expansions see Vidūnas and Temme (2002).

## §2.4(vi) Other Coalescing Critical Points

The problems sketched in §§2.3(v) and 2.4(v) involve only two of many possibilities for the coalescence of endpoints, saddle points, and singularities in integrals associated with the special functions. For a coalescing saddle point and a pole see Wong (1989, Chapter 7) and van der Waerden (1951); in this case the uniform approximants are complementary error functions. For a coalescing saddle point and endpoint see Olver (1997b, Chapter 9) and Wong (1989, Chapter 7); if the endpoint is an algebraic singularity then the uniform approximants are parabolic cylinder functions with fixed parameter, and if the endpoint is not a singularity then the uniform approximants are complementary error functions.

For two coalescing saddle points and an endpoint see Leubner and Ritsch (1986). For two coalescing saddle points and an algebraic singularity see Temme (1986), Jin and Wong (1998). For a coalescing saddle point, a pole, and a branch point see Ciarkowski (1989). For many coalescing saddle points see §36.12. For double integrals with two coalescing stationary points see Qiu and Wong (2000).