§2.3 Integrals of a Real Variable

§2.3(i) Integration by Parts

Assume that the Laplace transform

converges for all sufficiently large , and is infinitely differentiable in a neighborhood of the origin. Then

If, in addition, is infinitely differentiable on and

is finite and bounded for , then the th error term (that is, the difference between the integral and th partial sum in (2.3.2)) is bounded in absolute value by when exceeds both 0 and .

For the Fourier integral

assume and are finite, and is infinitely differentiable on . Then

Alternatively, assume , is infinitely differentiable on , and each of the integrals , , converges as uniformly for all sufficiently large . Then

In both cases the th error term is bounded in absolute value by , where the variational operator is defined by

see §1.4(v). For other examples, see Wong (1989, Chapter 1).

§2.3(ii) Watson’s Lemma

Assume again that the integral (2.3.1) converges for all sufficiently large , but now

where and are positive constants. Then the series obtained by substituting (2.3.7) into (2.3.1) and integrating formally term by term yields an asymptotic expansion:

For the function see §5.2(i).

This result is probably the most frequently used method for deriving asymptotic expansions of special functions. Since need not be continuous (as long as the integral converges), the case of a finite integration range is included.

Other types of singular behavior in the integrand can be treated in an analogous manner. For example,

provided that the integral on the left-hand side of (2.3.9) converges for all sufficiently large values of . (In other words, differentiation of (2.3.8) with respect to the parameter (or ) is legitimate.)

Another extension is to more general factors than the exponential function. In addition to (2.3.7) assume that and are piecewise continuous (§1.4(ii)) on , and

where are positive constants. Then

where is the Mellin transform of 2.5(i)).

For a more detailed treatment of the integral (2.3.12) see §§2.5, 2.6.

§2.3(iii) Laplace’s Method

When is real and is a large positive parameter, the main contribution to the integral

derives from the neighborhood of the minimum of in the integration range. Without loss of generality, we assume that this minimum is at the left endpoint . Furthermore:

1. (a)

and are continuous in a neighborhood of , save possibly at , and the minimum of in is approached only at .

2. (b)

As

and the expansion for is differentiable. Again and are positive constants. Also (consistent with (a)).

3. (c)

The integral (2.3.13) converges absolutely for all sufficiently large .

Watson’s lemma can be regarded as a special case of this result.

For error bounds for Watson’s lemma and Laplace’s method see Boyd (1993) and Olver (1997b, Chapter 3). These references and Wong (1989, Chapter 2) also contain examples.

§2.3(iv) Method of Stationary Phase

When the parameter is large the contributions from the real and imaginary parts of the integrand in

oscillate rapidly and cancel themselves over most of the range. However, cancellation does not take place near the endpoints, owing to lack of symmetry, nor in the neighborhoods of zeros of because changes relatively slowly at these stationary points.

The first result is the analog of Watson’s lemma (§2.3(ii)). Assume that again has the expansion (2.3.7) and this expansion is infinitely differentiable, is infinitely differentiable on , and each of the integrals , , converges at , uniformly for all sufficiently large . Then

where the coefficients are given by (2.3.7).

For the more general integral (2.3.19) we assume, without loss of generality, that the stationary point (if any) is at the left endpoint. Furthermore:

1. (a)

On , and are infinitely differentiable and .

2. (b)

As the asymptotic expansions (2.3.14) apply, and each is infinitely differentiable. Again , , and are positive.

3. (c)

If the limit of as is finite, then each of the functions

2.3.21,

tends to a finite limit .

4. (d)

If , then and each of the integrals

converges at uniformly for all sufficiently large .

If is finite, then both endpoints contribute:

But if (d) applies, then the second sum is absent. The coefficients are defined as in §2.3(iii).

For proofs of the results of this subsection, error bounds, and an example, see Olver (1974). For other estimates of the error term see Lyness (1971). For extensions to oscillatory integrals with logarithmic singularities see Wong and Lin (1978).

§2.3(v) Coalescing Peak and Endpoint: Bleistein’s Method

In the integral

() and are positive constants, is a variable parameter in an interval with and , and is a large positive parameter. Assume also that and are continuous in and , and for each the minimum value of in is at , at which point vanishes, but both and are nonzero. When Laplace’s method (§2.3(iii)) applies, but the form of the resulting approximation is discontinuous at . In consequence, the approximation is nonuniform with respect to and deteriorates severely as .

A uniform approximation can be constructed by quadratic change of integration variable:

2.3.25

where and are functions of chosen in such a way that corresponds to , and the stationary points and correspond. Thus

2.3.26
2.3.27

the upper or lower sign being taken according as . The relationship between and is one-to-one, and because

it is free from singularity at .

The integral (2.3.24) transforms into

where

being the value of at . We now expand in a Taylor series centered at the peak value of the exponential factor in the integrand:

2.3.31

with the coefficients continuous at . The desired uniform expansion is then obtained formally as in Watson’s lemma and Laplace’s method. We replace the limit by and integrate term-by-term:

where

For examples and proofs see Olver (1997b, Chapter 9), Bleistein (1966), Bleistein and Handelsman (1975, Chapter 9), and Wong (1989, Chapter 7).