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§2.6 Distributional Methods

Contents

§2.6(i) Divergent Integrals

Consider the integral

2.6.1 S(x)=01(1+t)1/3(x+t)dt,

where x>0. For t>1,

2.6.2 (1+t)-1/3=s=0(-13s)t-s-(1/3).

Motivated by Watson’s lemma (§2.3(ii)), we substitute (2.6.2) in (2.6.1), and integrate term by term. This leads to integrals of the form

2.6.3 0t-s-(1/3)x+tdt,
s=1,2,3,.

Although divergent, these integrals may be interpreted in a generalized sense. For instance, we have

2.6.4 0tα-1(x+t)α+βdt=Γ(α)Γ(β)Γ(α+β)1xβ,
α>0, β>0.

But the right-hand side is meaningful for all values of α and β, other than nonpositive integers. We may therefore define the integral on the left-hand side of (2.6.4) by the value on the right-hand side, except when α,β=0,-1,-2,. With this interpretation

2.6.5 0t-s-(1/3)x+tdt=2π3(-1)sxs+(1/3),
s=0,1,2,.

Inserting (2.6.2) into (2.6.1) and integrating formally term-by-term, we obtain

2.6.6 S(x)2π3s=0(-1)s(-13s)x-s-(1/3),
x.

However this result is incorrect. The correct result is given by

2.6.7 S(x)2π3s=0(-1)s(-13s)x-s-(1/3)-s=13s(s-1)!25(3s-1)x-s;

see §2.6(ii).

The fact that expansion (2.6.6) misses all the terms in the second series in (2.6.7) raises the question: what went wrong with our process of reaching (2.6.6)? In the following subsections, we use some elementary facts of distribution theory (§1.16) to study the proper use of divergent integrals. An important asset of the distribution method is that it gives explicit expressions for the remainder terms associated with the resulting asymptotic expansions.

For an introduction to distribution theory, see Wong (1989, Chapter 5). For more advanced discussions, see Gel’fand and Shilov (1964) and Rudin (1973).

§2.6(ii) Stieltjes Transform

Let f(t) be locally integrable on [0,). The Stieltjes transform of f(t) is defined by

2.6.8 𝒮(f;z)=0f(t)t+zdt.

To derive an asymptotic expansion of 𝒮(f;z) for large values of |z|, with |phz|<π, we assume that f(t) possesses an asymptotic expansion of the form

2.6.9 f(t)s=0ast-s-α,
t+,

with 0<α1. For each n=1,2,3,, set

2.6.10 f(t)=s=0n-1ast-s-α+fn(t).

To each function in this equation, we shall assign a tempered distribution (i.e., a continuous linear functional) on the space 𝒯 of rapidly decreasing functions on . Since f(t) is locally integrable on [0,), it defines a distribution by

2.6.11 f,ϕ=0f(t)ϕ(t)dt,
ϕ𝒯.

In particular,

2.6.12 t-α,ϕ=0t-αϕ(t)dt,
ϕ𝒯,

when 0<α<1. Since the functions t-s-α, s=1,2,, are not locally integrable on [0,), we cannot assign distributions to them in a similar manner. However, they are multiples of the derivatives of t-α. Motivated by the definition of distributional derivatives, we can assign them the distributions defined by

2.6.13 t-s-α,ϕ=1(α)s0t-αϕ(s)(t)dt,
ϕ𝒯,

where (α)s=α(α+1)(α+s-1). Similarly, in the case α=1, we define

2.6.14 t-s-1,ϕ=-1s!0(lnt)ϕ(s+1)(t)dt,
ϕ𝒯.

To assign a distribution to the function fn(t), we first let fn,n(t) denote the nth repeated integral (§1.4(v)) of fn:

2.6.15 fn,n(t)=(-1)n(n-1)!t(τ-t)n-1fn(τ)dτ.

For 0<α<1, it is easily seen that fn,n(t) is bounded on [0,R] for any positive constant R, and is O(t-α) as t. For α=1, we have fn,n(t)=O(t-1) as t and fn,n(t)=O(lnt) as t0+. In either case, we define the distribution associated with fn(t) by

2.6.16 fn,ϕ=(-1)n0fn,n(t)ϕ(n)(t)dt,
ϕ𝒯,

since the nth derivative of fn,n is fn.

We have now assigned a distribution to each function in (2.6.10). A natural question is: what is the exact relation between these distributions? The answer is provided by the identities (2.6.17) and (2.6.20) given below.

For 0<α<1 and n1, we have

2.6.17 f,ϕ=s=0n-1ast-s-α,ϕ-s=1ncsδ(s-1),ϕ+fn,ϕ

for any ϕ𝒯, where

2.6.18 cs=(-1)s(s-1)!(f;s),

(f;z) being the Mellin transform of f(t) or its analytic continuation (§2.5(ii)). The Dirac delta distribution in (2.6.17) is given by

2.6.19 δ(s),ϕ=(-1)sϕ(s)(0),
s=0,1,2,;

compare §1.16(iii).

For α=1

2.6.20 f,ϕ=s=0n-1ast-s-1,ϕ-s=1ndsδ(s-1),ϕ+fn,ϕ

for any ϕ𝒯, where

2.6.21 (-1)s+1ds+1=ass!k=1s1k+1s!limzs+1((f;z)+asz-s-1),

for s=0,1,2,.

To apply the results (2.6.17) and (2.6.20) to the Stieltjes transform (2.6.8), we take a specific function ϕ𝒯. Let ε be a positive number, and

2.6.22 ϕε(t)=e-εtt+z,
t(0,).

From (2.6.13) and (2.6.14)

2.6.23 limε0t-s-α,ϕε=πsin(πα)(-1)szs+α,
2.6.24 limε0t-s-1,ϕε=(-1)s+1zs+1k=1s1k+(-1)szs+1lnz,

with s=0,1,2,. From (2.6.11) and (2.6.16), we also have

2.6.25 limε0f,ϕε=𝒮(f;z),
2.6.26 limε0fn,ϕε=n!0fn,n(t)(t+z)n+1dt.

On substituting (2.6.15) into (2.6.26) and interchanging the order of integration, the right-hand side of (2.6.26) becomes

(-1)nzn0τnfn(τ)τ+zdτ.

To summarize,

2.6.27 𝒮(f;z)=πsin(πα)s=0n-1(-1)saszs+α-s=1n(s-1)!cszs+Rn(z),

if α(0,1) in (2.6.9), or

2.6.28 𝒮(f;z)=lnzs=0n-1(-1)saszs+1+s=0n-1(-1)sd~szs+1+Rn(z),

if α=1 in (2.6.9). Here cs is given by (2.6.18),

2.6.29 d~s=limzs+1((f;z)+asz-s-1),

and

2.6.30 Rn(z)=(-1)nzn0τnfn(τ)τ+zdτ.

The expansion (2.6.7) follows immediately from (2.6.27) with z=x and f(t)=(1+t)-(1/3); its region of validity is |phx|π-δ (<π). The distribution method outlined here can be extended readily to functions f(t) having an asymptotic expansion of the form

2.6.31 f(t)eicts=0ast-s-α,
t+,

where c (0) is real, and 0<α1. For a more detailed discussion of the derivation of asymptotic expansions of Stieltjes transforms by the distribution method, see McClure and Wong (1978) and Wong (1989, Chapter 6). Corresponding results for the generalized Stieltjes transform

2.6.32 0f(t)(t+z)ρdt,
ρ>0,

can be found in Wong (1979). An application has been given by López (2000) to derive asymptotic expansions of standard symmetric elliptic integrals, complete with error bounds; see §19.27(vi).

§2.6(iii) Fractional Integrals

The Riemann–Liouville fractional integral of order μ is defined by

2.6.33 Iμf(x)=1Γ(μ)0x(x-t)μ-1f(t)dt,
μ>0;

see §1.15(vi). We again assume f(t) is locally integrable on [0,) and satisfies (2.6.9). We now derive an asymptotic expansion of Iμf(x) for large positive values of x.

In terms of the convolution product

2.6.34 (fg)(x)=0xf(x-t)g(t)dt

of two locally integrable functions on [0,), (2.6.33) can be written

2.6.35 Iμf(x)=1Γ(μ)(tμ-1f)(x).

The replacement of f(t) by its asymptotic expansion (2.6.9), followed by term-by-term integration leads to convolution integrals of the form

2.6.36 (tμ-1t-s-α)(x)=0x(x-t)μ-1t-s-αdt,
s=0,1,2,.

Of course, except when s=0 and 0<α<1, none of these integrals exists in the usual sense. However, the left-hand side can be considered as the convolution of the two distributions associated with the functions tμ-1 and t-s-α, given by (2.6.12) and (2.6.13).

To define convolutions of distributions, we first introduce the space K+ of all distributions of the form Dnf, where n is a nonnegative integer, f is a locally integrable function on which vanishes on (-,0], and Dnf denotes the nth derivative of the distribution associated with f. For F=Dnf and G=Dmg in K+, we define

2.6.37 FG=Dn+m(fg).

It is easily seen that K+ forms a commutative, associative linear algebra. Furthermore, K+ contains the distributions H, δ, and tλ, t>0, for any real (or complex) number λ, where H is the distribution associated with the Heaviside function H(t)1.16(iv)), and tλ is the distribution defined by (2.6.12)–(2.6.14), depending on the value of λ. Since δ=DH, it follows that for μ1,2,,

2.6.38 tμ-1δ(s-1)=Γ(μ)Γ(μ+1-s)tμ-s,
t>0.

Using (5.12.1), we can also show that when μ1,2, and μ-α is not a nonnegative integer,

2.6.39 tμ-1t-s-α=Γ(μ)Γ(1-s-α)Γ(μ+1-s-α)tμ-s-α,
t>0,

and

2.6.40 tμ-1t-s-1=(-1)sμs!Ds+1(tμ(lnt-γ-ψ(μ+1))),
t>0,

where γ is Euler’s constant (§5.2(ii)).

To derive the asymptotic expansion of Iμf(x), we recall equations (2.6.17) and (2.6.20). In the sense of distributions, they can be written

2.6.41 f=s=0n-1ast-s-α-s=1ncsδ(s-1)+fn,

and

2.6.42 f=s=0n-1ast-s-1-s=1ndsδ(s-1)+fn.

Substituting into (2.6.35) and using (2.6.38)–(2.6.40), we obtain

2.6.43 tμ-1f=s=0n-1asΓ(μ)Γ(1-s-α)Γ(μ+1-s-α)tμ-s-α-s=1ncsΓ(μ)Γ(μ-s+1)tμ-s+tμ-1fn

when 0<α<1, or

2.6.44 tμ-1f=s=0n-1(-1)sasμs!Ds+1(tμ(lnt-γ-ψ(μ+1)))-s=1ndsΓ(μ)Γ(μ-s+1)tμ-s+tμ-1fn

when α=1. These equations again hold only in the sense of distributions. Since the function tμ(lnt-γ-ψ(μ+1)) and all its derivatives are locally absolutely continuous in (0,), the distributional derivatives in the first sum in (2.6.44) can be replaced by the corresponding ordinary derivatives. Furthermore, since fn,n(n)(t)=fn(t), it follows from (2.6.37) that the remainder terms tμ-1fn in the last two equations can be associated with a locally integrable function in (0,). On replacing the distributions by their corresponding functions, (2.6.43) and (2.6.44) give

2.6.45 Iμf(x)=s=0n-1asΓ(1-s-α)Γ(μ+1-s-α)xμ-s-α-s=1ncsΓ(μ+1-s)xμ-s+1xnδn(x),

when 0<α<1, or

2.6.46 Iμf(x)=s=0n-1(-1)sass!Γ(μ+1)ds+1dxs+1(xμ(lnx-γ-ψ(μ+1)))-s=1ndsΓ(μ-s+1)xμ-s+1xnδn(x),

when α=1, where

2.6.47 δn(x)=j=0n(nj)Γ(μ+1)Γ(μ+1-j)Iμ(tn-jfn,j)(x),

fn,j(t) being the jth repeated integral of fn; compare (2.6.15).

Example

Let f(t)=t1-α/(1+t), 0<α<1. Then

2.6.48 Iμf(x)=1Γ(μ)0x(x-t)μ-1t1-α(1+t)-1dt,

where μ>0. For 0<t<

2.6.49 f(t)=s=0n-1(-1)st-s-α+(-1)nt1-n-α1+t.

In the notation of (2.6.10), as=(-1)s and

2.6.50 fn(t)=(-1)nt1-n-α1+t.

Since

2.6.51 (f;s)=(-1)sπ/sin(πα),

from (2.6.45) it follows that

2.6.52 Iμf(x)=s=0n-1(-1)sΓ(1-s-α)Γ(μ+1-s-α)xμ-s-α-πsin(πα)s=1n1Γ(μ+1-s)xμ-s(s-1)!+1xnδn(x).

Moreover,

2.6.53 |δn(x)|Γ(μ+1)Γ(1-α)Γ(μ+1-α)Γ(n+α)j=0n(nj)Γ(n+α-j)|Γ(μ+1-j)|xμ-α

for x>0.

It may be noted that the integral (2.6.48) can be expressed in terms of the hypergeometric function F12(1,2-α;2-α+μ;-x); see §15.2(i).

For proofs and other examples, see McClure and Wong (1979) and Wong (1989, Chapter 6). If both f and g in (2.6.34) have asymptotic expansions of the form (2.6.9), then the distribution method can also be used to derive an asymptotic expansion of the convolution fg; see Li and Wong (1994).

§2.6(iv) Regularization

The method of distributions can be further extended to derive asymptotic expansions for convolution integrals:

2.6.54 I(x)=0f(t)h(xt)dt.

We assume that for each n=1,2,3,,

2.6.55 f(t)=s=0n-1asts+α-1+fn(t),

where 0<α1 and fn(t)=O(tn+α-1) as t0+. Also,

2.6.56 h(t)=s=0n-1bst-s-β+hn(t),

where 0<β1, and hn(t)=O(t-n-β) as t. Multiplication of these expansions leads to

2.6.57 f(t)h(xt)=j=0n-1k=0n-1ajbktj+α-1-k-βx-k-β+j=0n-1ajtj+α-1hn(xt)+k=0n-1bkx-k-βt-k-βfn(t)+fn(t)hn(xt).

On inserting this identity into (2.6.54), we immediately encounter divergent integrals of the form

2.6.58 0tλdt,
λ.

However, in the theory of generalized functions (distributions), there is a method, known as “regularization”, by which these integrals can be interpreted in a meaningful manner. In this sense

2.6.59 0tλdt=0,
λ.

From (2.6.55) and (2.6.59)

2.6.60 (f;z)=(fn;z),

where (f;z) is the Mellin transform of f or its analytic continuation. Also, when αβ,

2.6.61 (hx;j+α)=x-j-α(h;j+α),

where hx(t)=h(xt). Inserting (2.6.57) into (2.6.54), we obtain from (2.6.59)–(2.6.61)

2.6.62 I(x)=j=0n-1aj(h;j+α)x-j-α+k=0n-1bk(f;1-k-β)x-k-β+δn(x)

when αβ, where

δn(x)=0fn(t)hn(xt)dt.

There is a similar expansion, involving logarithmic terms, when α=β. For rigorous derivations of these results and also order estimates for δn(x), see Wong (1979) and Wong (1989, Chapter 6).