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§2.3 Integrals of a Real Variable

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§2.3(i) Integration by Parts

Assume that the Laplace transform

2.3.1 0-xtq(t)t

converges for all sufficiently large x, and q(t) is infinitely differentiable in a neighborhood of the origin. Then

2.3.2 0-xtq(t)ts=0q(s)(0)xs+1,
x+.

If, in addition, q(t) is infinitely differentiable on [0,) and

2.3.3 σn=sup(0,)(t-1ln|q(n)(t)/q(n)(0)|)

is finite and bounded for n=0,1,2,, then the nth error term (that is, the difference between the integral and nth partial sum in (2.3.2)) is bounded in absolute value by |q(n)(0)/(xn(x-σn))| when x exceeds both 0 and σn.

For the Fourier integral

abxtq(t)t

assume a and b are finite, and q(t) is infinitely differentiable on [a,b]. Then

2.3.4 abxtq(t)taxs=0q(s)(a)(x)s+1-bxs=0q(s)(b)(x)s+1,
x+.

Alternatively, assume b=, q(t) is infinitely differentiable on [a,), and each of the integrals xtq(s)(t)t, s=0,1,2,, converges as t uniformly for all sufficiently large x. Then

2.3.5 axtq(t)taxs=0q(s)(a)(x)s+1,
x+.

In both cases the nth error term is bounded in absolute value by x-n𝒱a,b(q(n-1)(t)), where the variational operator 𝒱a,b is defined by

2.3.6 𝒱a,b(f(t))=ab|f(t)t|;

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 x, but now

2.3.7 q(t)s=0ast(s+λ-μ)/μ,
t0+,

where λ and μ 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:

2.3.8 0-xtq(t)ts=0Γ(s+λμ)asx(s+λ)/μ,
x+.

For the function Γ see §5.2(i).

This result is probably the most frequently used method for deriving asymptotic expansions of special functions. Since q(t) 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,

2.3.9 0-xtq(t)lntts=0Γ(s+λμ)asx(s+λ)/μ-(lnx)s=0Γ(s+λμ)asx(s+λ)/μ,

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

Another extension is to more general factors than the exponential function. In addition to (2.3.7) assume that f(t) and q(t) are piecewise continuous (§1.4(ii)) on (0,), and

2.3.10 |f(t)|Aexp(-atκ),
0t<,
2.3.11 q(t)=O(exp(btκ)),
t+,

where A,a,b,κ are positive constants. Then

2.3.12 0f(xt)q(t)ts=0(f;s+λμ)asx(s+λ)/μ,
x+,

where (f;α) is the Mellin transform of f(t)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 p(t) is real and x is a large positive parameter, the main contribution to the integral

2.3.13 I(x)=ab-xp(t)q(t)t

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

  1. (a)

    p(t) and q(t) are continuous in a neighborhood of a, save possibly at a, and the minimum of p(t) in [a,b) is approached only at a.

  2. (b)

    As ta+

    2.3.14 p(t) p(a)+s=0ps(t-a)s+μ,
    q(t) s=0qs(t-a)s+λ-1,

    and the expansion for p(t) is differentiable. Again λ and μ are positive constants. Also p0>0 (consistent with (a)).

  3. (c)

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

Then

2.3.15 ab-xp(t)q(t)t-xp(a)s=0Γ(s+λμ)bsx(s+λ)/μ,
x+,

where the coefficients bs are defined by the expansion

2.3.16 q(t)p(t)s=0bsv(s+λ-μ)/μ,
v0+,

in which v=p(t)-p(a). For example,

2.3.17 b0 =q0μp0λ/μ,
b1 =(q1μ-(λ+1)p1q0μ2p0)1p0(λ+1)/μ,
b2 =(q2μ-(λ+2)(p1q1+p2q0)μ2p0+(λ+2)(λ+μ+2)p12q02μ3p02)1p0(λ+2)/μ.

In general

2.3.18 bs=1μrest=a[q(t)(p(t)-p(a))(λ+s)/μ],
s=0,1,2,.

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

2.3.19 I(x)=abxp(t)q(t)t

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 p(t) because p(t) changes relatively slowly at these stationary points.

The first result is the analog of Watson’s lemma (§2.3(ii)). Assume that q(t) again has the expansion (2.3.7) and this expansion is infinitely differentiable, q(t) is infinitely differentiable on (0,), and each of the integrals xtq(s)(t)t, s=0,1,2,, converges at t=, uniformly for all sufficiently large x. Then

2.3.20 0xtq(t)ts=0exp((s+λ)π2μ)Γ(s+λμ)asx(s+λ)/μ,
x+,

where the coefficients as 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 (a,b), p(t) and q(t) are infinitely differentiable and p(t)>0.

  2. (b)

    As ta+ the asymptotic expansions (2.3.14) apply, and each is infinitely differentiable. Again λ, μ, and p0 are positive.

  3. (c)

    If the limit p(b) of p(t) as tb- is finite, then each of the functions

    2.3.21 Ps(t)=(1p(t)t)sq(t)p(t),
    s=0,1,2,,

    tends to a finite limit Ps(b).

  4. (d)

    If p(b)=, then P0(b)=0 and each of the integrals

    2.3.22 xp(t)Ps(t)p(t)t,
    s=0,1,2,,

    converges at t=b uniformly for all sufficiently large x.

If p(b) is finite, then both endpoints contribute:

2.3.23 abxp(t)q(t)txp(a)s=0exp((s+λ)π2μ)Γ(s+λμ)bsx(s+λ)/μ-xp(b)s=0Ps(b)(x)s+1,
x+.

But if (d) applies, then the second sum is absent. The coefficients bs 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

2.3.24 I(α,x)=0k-xp(α,t)q(α,t)tλ-1t

k () and λ are positive constants, α is a variable parameter in an interval α1αα2 with α10 and 0<α2k, and x is a large positive parameter. Assume also that 2p(α,t)/t2 and q(α,t) are continuous in α and t, and for each α the minimum value of p(α,t) in [0,k) is at t=α, at which point p(α,t)/t vanishes, but both 2p(α,t)/t2 and q(α,t) are nonzero. When x+ Laplace’s method (§2.3(iii)) applies, but the form of the resulting approximation is discontinuous at α=0. In consequence, the approximation is nonuniform with respect to α and deteriorates severely as α0.

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

2.3.25 p(α,t)=12w2-aw+b,

where a and b are functions of α chosen in such a way that t=0 corresponds to w=0, and the stationary points t=α and w=a correspond. Thus

2.3.26 a =(2p(α,0)-2p(α,α))1/2,
b =p(α,0),
2.3.27 w=(2p(α,0)-2p(α,α))1/2±(2p(α,t)-2p(α,α))1/2,

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

2.3.28 wt=±1(2p(α,t)-2p(α,α))1/2p(α,t)t

it is free from singularity at t=α.

The integral (2.3.24) transforms into

2.3.29 I(α,x)=-xp(α,0)0κexp(-x(12w2-aw))f(α,w)wλ-1w,

where

2.3.30 f(α,w)=q(α,t)(tw)λ-1tw,

κ=κ(α) being the value of w at t=k. We now expand f(α,w) in a Taylor series centered at the peak value w=a of the exponential factor in the integrand:

2.3.31 f(α,w)=s=0ϕs(α)(w-a)s,

with the coefficients ϕs(α) continuous at α=0. The desired uniform expansion is then obtained formally as in Watson’s lemma and Laplace’s method. We replace the limit κ by and integrate term-by-term:

2.3.32 I(α,x)-xp(α,0)xλ/2s=0ϕs(α)Fs(ax)xs/2,
x,

where

2.3.33 Fs(y)=0exp(-12τ2+yτ)(τ-y)sτλ-1τ.

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