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§2.10 Sums and Sequences

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§2.10(i) Euler–Maclaurin Formula

As in §24.2, let Bn and Bn(x) denote the nth Bernoulli number and polynomial, respectively, and B~n(x) the nth Bernoulli periodic function Bn(x-x).

Assume that a,m, and n are integers such that n>a, m>0, and f(2m)(x) is absolutely integrable over [a,n]. Then

2.10.1 j=anf(j)=anf(x)x+12f(a)+12f(n)+s=1m-1B2s(2s)!(f(2s-1)(n)-f(2s-1)(a))+anB2m-B~2m(x)(2m)!f(2m)(x)x.

This is the Euler–Maclaurin formula. Another version is the Abel–Plana formula:

2.10.2 j=anf(j)=anf(x)x+12f(a)+12f(n)-20(f(a+y))2πy-1y+s=1mB2s(2s)!f(2s-1)(n)+2(-1)m(2m)!0(f(2m)(n+ϑny))y2my2πy-1,

ϑn being some number in the interval (0,1). Sufficient conditions for the validity of this second result are:

  1. (a)

    On the strip azn, f(z) is analytic in its interior, f(2m)(z) is continuous on its closure, and f(z)=o(2π|z|) as z±, uniformly with respect to z[a,n].

  2. (b)

    f(z) is real when azn.

  3. (c)

    The first infinite integral in (2.10.2) converges.

Example

2.10.3 S(n)=j=1njlnj

for large n. From (2.10.1)

2.10.4 S(n)=12n2lnn-14n2+12nlnn+112lnn+C+s=2m-1(-B2s)2s(2s-1)(2s-2)1n2s-2+Rm(n),

where m (2) is arbitrary, C is a constant, and

2.10.5 Rm(n)=nB~2m(x)-B2m2m(2m-1)x2m-1x.

From §24.12(i), (24.2.2), and (24.4.27), B~2m(x)-B2m is of constant sign (-1)m. Thus Rm(n) and Rm+1(n) are of opposite signs, and since their difference is the term corresponding to s=m in (2.10.4), Rm(n) is bounded in absolute value by this term and has the same sign.

Formula (2.10.2) is useful for evaluating the constant term in expansions obtained from (2.10.1). In the present example it leads to

2.10.6 C=γ+ln(2π)12-ζ(2)2π2=112-ζ(-1),

where γ is Euler’s constant (§5.2(ii)) and ζ is the derivative of the Riemann zeta function (§25.2(i)). C is sometimes called Glaisher’s constant. For further information on C see §5.17.

Other examples that can be verified in a similar way are:

2.10.7 j=1n-1jαζ(-α)+nα+1α+1s=0(α+1s)Bsns,
n,

where α (-1) is a real constant, and

2.10.8 j=1n-11jlnn+γ-12n-s=1B2s2s1n2s,
n.

In both expansions the remainder term is bounded in absolute value by the first neglected term in the sum, and has the same sign, provided that in the case of (2.10.7), truncation takes place at s=2m-1, where m is any positive integer satisfying m12(α+1).

For extensions of the Euler–Maclaurin formula to functions f(x) with singularities at x=a or x=n (or both) see Sidi (2004). See also Weniger (2007).

For an extension to integrals with Cauchy principal values see Elliott (1998).

§2.10(ii) Summation by Parts

The formula for summation by parts is

2.10.9 j=1n-1ujvj=Un-1vn+j=1n-1Uj(vj-vj+1),

where

2.10.10 Uj=u1+u2++uj.

This identity can be used to find asymptotic approximations for large n when the factor vj changes slowly with j, and uj is oscillatory; compare the approximation of Fourier integrals by integration by parts in §2.3(i).

Example

2.10.11 S(α,β,n)=j=1n-1jβjα,

where α and β are real constants with β1.

As a first estimate for large n

2.10.12 |S(α,β,n)|j=1n-1jα=O(1), O(lnn), or O(nα+1),

according as α<-1, α=-1, or α>-1; see (2.10.7), (2.10.8). With uj=jβ, vj=jα,

2.10.13 Uj=β(jβ-1)/(β-1),

and

2.10.14 S(α,β,n)=ββ-1((n-1)βnα-1+j=1n-1jβ(jα-(j+1)α)).

Since

2.10.15 jα-(j+1)α=-αjα-1+α(α-1)O(jα-2)

for any real constant α and the set of all positive integers j, we derive

2.10.16 S(α,β,n)=ββ-1((n-1)βnα-αS(α-1,β,n)+O(nα-1)+O(1)).

From this result and (2.10.12)

2.10.17 S(α,β,n)=O(nα)+O(1).

Then replacing α by α-1 and resubstituting in (2.10.16), we have

2.10.18 S(α,β,n)=nββ-1nα+O(nα-1)+O(1),
n,

which is a useful approximation when α>0.

For extensions to α0, higher terms, and other examples, see Olver (1997b, Chapter 8).

§2.10(iii) Asymptotic Expansions of Entire Functions

The asymptotic behavior of entire functions defined by Maclaurin series can be approached by converting the sum into a contour integral by use of the residue theorem and applying the methods of §§2.4 and 2.5.

Example

From §§16.2(i)16.2(ii)

2.10.19 F20(-;1,1;x)=j=0xj(j!)3.

We seek the behavior as x+. From (1.10.8)

2.10.20 j=0n-1xj(j!)3=12𝒞xt(Γ(t+1))3cot(πt)t,

where 𝒞 comprises the two semicircles and two parts of the imaginary axis depicted in Figure 2.10.1.

See accompanying text
Figure 2.10.1: t-plane. Contour 𝒞. Magnify

From the identities

2.10.21 cot(πt)2=-12-1-2πt-1=12+12πt-1,

and Cauchy’s theorem, we have

2.10.22 j=0n-1xj(j!)3=-1/2n-(1/2)xt(Γ(t+1))3t-𝒞1xt(Γ(t+1))3t-2πt-1+𝒞2xt(Γ(t+1))3t2πt-1,

where 𝒞1,𝒞2 denote respectively the upper and lower halves of 𝒞. (5.11.7) shows that the integrals around the large quarter circles vanish as n. Hence

2.10.23 F20(-;1,1;x)=-1/2xt(Γ(t+1))3t+2-1/2xt(Γ(t+1))3t-2πt-1=0xt(Γ(t+1))3t+O(1),
x+,

the last step following from |xt|1 when t is on the interval [-12,0], the imaginary axis, or the small semicircle. By application of Laplace’s method (§2.3(iii)) and use again of (5.11.7), we obtain

2.10.24 F20(-;1,1;x)exp(3x1/3)2π31/2x1/3,
x+.

For generalizations and other examples see Olver (1997b, Chapter 8), Ford (1960), and Berndt and Evans (1984). See also Paris and Kaminski (2001, Chapter 5) and §§16.11(i)16.11(ii).

§2.10(iv) Taylor and Laurent Coefficients: Darboux’s Method

Let f(z) be analytic on the annulus 0<|z|<r, with Laurent expansion

2.10.25 f(z)=n=-fnzn,
0<|z|<r.

What is the asymptotic behavior of fn as n or n-? More specially, what is the behavior of the higher coefficients in a Taylor-series expansion?

These problems can be brought within the scope of §2.4 by means of Cauchy’s integral formula

2.10.26 fn=12π𝒞f(z)zn+1z,

where 𝒞 is a simple closed contour in the annulus that encloses z=0. For examples see Olver (1997b, Chapters 8, 9).

However, if r is finite and f(z) has algebraic or logarithmic singularities on |z|=r, then Darboux’s method is usually easier to apply. We need a “comparison function” g(z) with the properties:

  1. (a)

    g(z) is analytic on 0<|z|<r.

  2. (b)

    f(z)-g(z) is continuous on 0<|z|r.

  3. (c)

    The coefficients in the Laurent expansion

    2.10.27 g(z)=n=-gnzn,
    0<|z|<r,

    have known asymptotic behavior as n±.

By allowing the contour in Cauchy’s formula to expand, we find that

2.10.28 fn-gn=12π|z|=rf(z)-g(z)zn+1z=12πrn02π(f(rθ)-g(rθ))-nθθ.

Hence by the Riemann–Lebesgue lemma (§1.8(i))

2.10.29 fn=gn+o(r-n),
n±.

This result is refinable in two important ways. First, the conditions can be weakened. It is unnecessary for f(z)-g(z) to be continuous on |z|=r: it suffices that the integrals in (2.10.28) converge uniformly. For example, Condition (b) can be replaced by:

  1. (b´)

    On the circle |z|=r, the function f(z)-g(z) has a finite number of singularities, and at each singularity zj, say,

    2.10.30 f(z)-g(z)=O((z-zj)σj-1),
    zzj,

    where σj is a positive constant.

Secondly, when f(z)-g(z) is m times continuously differentiable on |z|=r the result (2.10.29) can be strengthened. In these circumstances the integrals in (2.10.28) are integrable by parts m times, yielding

2.10.31 fn=gn+o(r-n|n|-m),
n±.

Furthermore, (2.10.31) remains valid with the weaker condition

2.10.32 f(m)(z)-g(m)(z)=O((z-zj)σj-1),

in the neighborhood of each singularity zj, again with σj>0.

Example

Let α be a constant in (0,2π) and Pn denote the Legendre polynomial of degree n. From §14.7(iv)

2.10.33 f(z)1(1-2zcosα+z2)1/2=n=0Pn(cosα)zn,
|z|<1.

The singularities of f(z) on the unit circle are branch points at z=±α. To match the limiting behavior of f(z) at these points we set

2.10.34 g(z)=-π/4(2sinα)-1/2(-α-z)-1/2+π/4(2sinα)-1/2(α-z)-1/2.

Here the branch of (-α-z)-1/2 is continuous in the z-plane cut along the outward-drawn ray through z=-α and equals α/2 at z=0. Similarly for (α-z)-1/2. In Condition (c) we have

2.10.35 gn=(2πsinα)1/2Γ(n+12)n!cos(nα+12α-14π),

and in the supplementary conditions we may set m=1. Then from (2.10.31) and (5.11.7)

2.10.36 Pn(cosα)=(2πnsinα)1/2cos(nα+12α-14π)+o(n-1).

For higher terms see §18.15(iii).

For uniform expansions when two singularities coalesce on the circle of convergence see Wong and Zhao (2005).

For other examples and extensions see Olver (1997b, Chapter 8), Olver (1970), Wong (1989, Chapter 2), and Wong and Wyman (1974). See also Flajolet and Odlyzko (1990).