Asymptotic Expansions for Integrals · 2012-05-26 · asymptotic expansions of integrals, especially focusing on Laplace-type integrals. As a contribution to the topic, we give a

Gergo Nemes

Asymptotic Expansionsfor Integrals

M. Sc. Thesis

Lorand Eotvos University

May 23, 2012

Lorand Eotvos University

Faculty of Science

M. Sc. Thesis

Asymptotic Expansions

for Integrals

Author:Gergo NemesM. Sc. Pure Mathematics

Supervisor:Dr. Arpad Toth

associate professor

Examination Committee:

Committee Chairman: Prof. Dr. Laszlo Simon . . . . . . . . . . . . . . . . . . . . . . . .

Committee Members: Dr. Balazs Csikos . . . . . . . . . . . . . . . . . . . . . . . .

Dr. Alice Fialowski . . . . . . . . . . . . . . . . . . . . . . . .

Prof. Dr. Tibor Jordan . . . . . . . . . . . . . . . . . . . . . . . .

Dr. Tamas Mori . . . . . . . . . . . . . . . . . . . . . . . .

Dr. Peter Sziklai . . . . . . . . . . . . . . . . . . . . . . . .

Dr. Janos Toth . . . . . . . . . . . . . . . . . . . . . . . .

May 23, 2012

Preface

The asymptotic theory of integrals is an important subject of applied mathematicsand physics. Although it is an old topic, with origins dating back to Laplace, newmethods, applications and formulations continue to appear in the literature. Thepurpose of this thesis is to review some of the classical procedures for obtainingasymptotic expansions of integrals, especially focusing on Laplace-type integrals.As a contribution to the topic, we give a new method for computing the coefficientsof these asymptotic series with several illustrating examples. This method is ageneralization of the one given in my paper about the Stirling Coefficients (J.Integer Seqs. 13 (6), Article 10.6.6, pp. 5 (2010)).

Chapter 1 covers the asymptotic theory of real Laplace-type integrals. Thecomputation of the coefficients appearing in the asymptotic expansions are de-scribed completely in this chapter. In Chapter 2, we discuss two methods fromthe asymptotic theory of complex Laplace-type integrals: the Method of SteepestDescents and Perron’s Method. Each chapter contains examples to demonstratethe application of the results. In Appendix A, we collect the basic properties ofsome combinatorial quantities we use in this thesis.

Budapest, Hungary, May 23, 2012 Gergo Nemes

i

Table of Contents

Preface i

Table of Contents ii

Acknowledgements iii

1 Real Laplace-type integrals 1

1.1 Fundamental concepts and results . . . . . . . . . . . . . . . . . . . 21.1.1 Asymptotic expansions . . . . . . . . . . . . . . . . . . . . . 21.1.2 Watson’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Laplace’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.1 Asymptotic expansion of Laplace-type integrals . . . . . . . 51.2.2 Generalization of Perron’s Formula . . . . . . . . . . . . . . 71.2.3 The CFWW Formula . . . . . . . . . . . . . . . . . . . . . . 81.2.4 Another method . . . . . . . . . . . . . . . . . . . . . . . . 91.2.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Complex Laplace-type integrals 21

2.1 The Method of Steepest Descents . . . . . . . . . . . . . . . . . . . 212.2 Perron’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

A Combinatorial objects 33

A.1 Ordinary Potential Polynomials . . . . . . . . . . . . . . . . . . . . 33A.2 The r-associated Stirling Numbers . . . . . . . . . . . . . . . . . . . 34

Bibliography 35

ii

Acknowledgements

I am deeply grateful to my supervisor, Professor Arpad Toth for his detailed andconstructive comments, and for his important support throughout this work. I owemy loving thanks to Dorottya Sziraki. Without her support it would have beenimpossible for me to finish this work. My special gratitude is due to my parentsand my brother for their loving support.

iii

Chapter 1

Real Laplace-type integrals

In this chapter, we investigate the asymptotic behavior of certain parametric inte-grals. The origins of the method date back to Pierre-Simon de Laplace (1749– 1827), who studied the estimation of integrals arising in probability theory ofthe form

In =

∫ b

a

e−nf(x)g (x) dx (n → +∞) . (1.1)

Here the functions f and g are real continuous functions defined on the real (finiteor infinite) interval [a, b]. Hereinafter, we call integrals of the type (1.1), Laplace-type integrals. Laplace made the observation that the major contribution to theintegral In should come from the neighborhood of the point where f attains itssmallest value. If f has its minimum value only at the point x0 in (a, b) wheref ′ (x0) = 0 and f ′′ (x0) > 0, then Laplace’s result is

∫ b

a

e−nf(x)g (x) dx ∼ g (x0) e−nf(x0)

√

2π

nf ′′ (x0).

The sign ∼ is used to mean that the quotient of the left-hand side by the right-hand side approaches 1 as n → +∞. This formula is now known as Laplace’sapproximation. A heuristic proof of this formula may proceed as follows. First,we replace f and g by the leading terms in their Taylor series expansions aroundx = x0, and then extend the integration limits to −∞ and +∞. Hence,

∫ b

a

e−nf(x)g (x) dx ≈∫ b

a

e−n

(

f(x0)+f ′′(x0)

2(x−x0)

2

)

g (x0) dx

≈ g (x0) e−nf(x0)

∫ +∞

−∞e−n

f ′′(x0)2

(x−x0)2

dx

= g (x0) e−nf(x0)

√

2π

nf ′′ (x0).

We divided this chapter into two parts. In Section 1.1, we deal with the fundamen-tal theorem of the topic, called Watson’s Lemma. Section 1.2 presents a generaltheorem on asymptotic expansions of Laplace-type integrals, and investigates sev-eral of their properties.

1

1.1. Fundamental concepts and results

1.1 Fundamental concepts and results

This section is divided into two parts. In the first part, we collect all the basicconcepts which we will use hereinafter. In the second part we state and prove animportant theorem of the topic, known as Watson’s Lemma.

1.1.1 Asymptotic expansions

Let f and g be two continuous complex functions defined on a subset H of thecomplex plane. Let z0 be a limit point of H.

Definition 1.1.1. We write f(z) = O(g(z)) (”f is big-oh g”), as z → z0, tomean that there is a constant C > 0 and a neighborhood U of z0 such that |f(z)| ≤C |g(z)| for all z ∈ U ∩H.

Definition 1.1.2. We write f(z) = o(g(z)) (”f is little-oh g”), as z → z0, to meanthat for every ε > 0, there exists a neighborhood Uε of z0 such that |f(z)| ≤ ε |g(z)|for all z ∈ Uε ∩H.

To define the asymptotic expansion of a function we need the concept of anasymptotic sequence.

Definition 1.1.3. Let {ϕn}n≥0 be a sequence of continuous complex functionsdefined on a subset H of the complex plane. Let z0 be a limit point of H. Wesay that {ϕn}n≥0 is an asymptotic sequence as z → z0 in H if, for all n ≥ 0,ϕn+1(z) = o(ϕn(z)), as z → z0.

Definition 1.1.4. If {ϕn}n≥0 is an asymptotic sequence as z → z0, we say that

∞∑

n=1

anϕn(z),

where the an’s are constants, is an asymptotic expansion of the function f if foreach N ≥ 0

f(z) =N∑

n=1

anϕn(z) + o(ϕN(z)) as z → z0. (1.2)

If a function possesses an asymptotic expansion, we write

f(z) ∼∞∑

n=1

anϕn(z) as z → z0.

Note that (1.2) may be written as

f(z) =N−1∑

n=1

anϕn(z) +O(ϕN(z)),

which implies that the error is of the same order of magnitude as the first termomitted. Usually, we use ϕn(z) = z−ρn with 0 < Re (ρ0) < Re (ρ1) < Re (ρ2) < . . .,as z → ∞ in some sector of the complex plane.

2

1.1. Fundamental concepts and results

1.1.2 Watson’s Lemma

Consider the integral

I (λ) =

∫ +∞

0

e−λx

1 + xdx

where λ > 0. Using the well-known identity

1

1 + x=

N−1∑

k=0

(−1)k xk + (−1)NxN

1 + x(N ≥ 0) ,

term-by-term integration gives

I (λ) =N−1∑

k=0

(−1)kk!

λk+1+ (−1)N

∫ +∞

0

xN

1 + xe−λxdx,

where∣∣∣∣(−1)N

∫ +∞

0

xN

1 + xe−λxdx

∣∣∣∣=

∫ +∞

0

xN

1 + xe−λxdx ≤

∫ +∞

0

xNe−λxdx =N !

λN+1.

And hence, by definition,

I (λ) ∼∞∑

k=0

(−1)kk!

λk+1, as λ → +∞.

This is a special case of a much more general result, now known as Watson’sLemma. The theorem is due to George Neville Watson (1886 – 1965). It is asignificant result in the theory of asymptotic expansions of Laplace-type integrals.In view of its importance, the proof of the result is reproduced below.

Theorem 1.1.1 (Watson’s Lemma). Let f (x) be a complex valued function of areal variable x such that

(i) f is continuous on (0,∞);

(ii) as x → 0+,

f (x) ∼∞∑

k=0

akxρk−1

with 0 < Re (ρ0) < Re (ρ1) < Re (ρ2) < . . ., limk→+∞Re (ρk) = +∞; and

(iii) for some fixed c > 0, f (x) = O (ecx) as x → +∞.

Then we have∫ +∞

0

e−λxf (x) dx ∼∞∑

k=0

akΓ (ρk)

λρk,

as λ → +∞.

3

1.2. Laplace’s Method

Proof. By the conditions (i), (ii) and (iii); the integral

∫ +∞

0

e−λxf (x) dx

converges for λ > c. Conditions (ii) and (iii) imply that

∣∣∣∣∣f (x)−

N−1∑

k=0

akxρk−1

∣∣∣∣∣≤ KNe

cx∣∣xρN−1

∣∣ for x > 0,

for every N ≥ 0 with some constant KN . Thus we have

∣∣∣∣∣

∫ +∞

0

e−λxf (x) dx−N−1∑

k=0

ak

∫ +∞

0

e−λxxρk−1dx

∣∣∣∣∣≤ KN

∫ +∞

0

e−(λ−c)x∣∣xρN−1

∣∣ dx.

Note that we have for λ > 0∫ +∞

0

e−λxxρk−1dx =1

λρk

∫ +∞

0

e−ttρk−1dt =Γ (ρn)

λρk.

Hence we have∣∣∣∣∣

∫ +∞

0

e−λxf (x) dx−N−1∑

k=0

akΓ (ρn)

λρk

∣∣∣∣∣≤ KN

∫ +∞

0

e−(λ−c)x∣∣xρN−1

∣∣ dx

= KNΓ (Re (ρN))

|(λ− c)ρN | ,

that is∫ +∞

0

e−λxf (x) dx =N−1∑

k=0

akΓ (ρn)

λρk+O

(1

λρN

)

as λ → +∞, which proves Watson’s Lemma.

1.2 Laplace’s Method

In this section, we revisit the classical Laplace Method. In the first subsection, weprove the fundamental theorem on asymptotic expansion of Laplace-type integrals,an extension of the formula mentioned in the introduction of the chapter. Whenthe functions appearing in the integral are holomorphic, the coefficients of theasymptotic expansion can be given explicitly by Perron’s Formula. In the secondpart, we extend this formula to the case when the power series of the functions arenot necessarily convergent but asymptotic. In the third subsection, we reproducethe proof of the CFWW Formula which gives another explicit form of the coeffi-cients in the asymptotic expansion. The fourth part of the section is about a newand simpler explicit formula for these coefficients. Finally, in the fifth subsection,we give three examples to demonstrate the application of the results.

4


1.2.1 Asymptotic expansion of Laplace-type integrals

Laplace’s Method is one of the best-known techniques developing asymptotic ex-pansions for integrals. Here we present an extension of the formula mentioned inthe introduction of this chapter. Consider the integral of the form

I (λ) =

∫ b

a

e−λf(x)g (x) dx,

where (a, b) is a real (finite or infinite) interval, λ is a large positive parameterand the functions f and g are continuous. We observe that by subdividing therange of integration at the minima and maxima of f , and by reversing the sign ofx whenever necessary, we may assume, without loss of generality, that f has onlyone minimum in [a, b] which occurs at x = a. Next, we assume that, as x → a+,

f (x) ∼ f (a) +∞∑

k=0

ak (x− a)k+α, (1.3)

and

g (x) ∼∞∑

k=0

bk (x− a)k+β−1 (1.4)

with α > 0, Re (β) > 0; and that the expansion of f can be term wise differenti-ated, that is,

f ′ (x) ∼∞∑

k=0

ak (k + α) (x− a)k+α−1 (1.5)

as x → a+. Also, we suppose, without loss of generality, that a0 6= 0 and b0 6= 0.The following theorem is due to Arthur Erdelyi (1908 – 1977) (see, e.g., [8, p.85], [14, p. 57]).

Theorem 1.2.1. For the integral

I (λ) =

∫ b

a

e−λf(x)g (x) dx,

we assume that

(i) f(x) > f(a) for all x ∈ (a, b), and for every δ > 0 the infimum of f(x)−f(a)in [a+ δ, b) is positive;

(ii) f ′(x) and g(x) are continuous in a neighborhood of x = a, expect possibly ata;

(iii) the expansions (1.3), (1.4) and (1.5) hold; and

(iv) the integral I (λ) converges absolutely for all sufficiently large λ.

Then

I (λ) ∼ e−λf(a)

∞∑

n=0

Γ

(n+ β

α

)cn

λ(n+β)/α, (1.6)

as λ → +∞, where the coefficients cn are expressible in terms of an and bn.

5


The first three coefficients cn are given explicitly by

c0 =b0

αaβ/α0

, c1 =1

a(β+1)/α0

(b1α

− (β + 1) a1b0α2a0

)

,

and

c2 =1

a(β+2)/α0

(b2α

− (β + 2) a1b1α2a0

+((β + α + 2) a21 − 2αa0a2

) (β + 1) b02α2a20

)

.

Proof. By conditions (ii) and (iii) there exists a number c ∈ (a, b) such thatf ′(x) and g(x) are continuous in (a, c], and f ′(x) is also positive there. Let T =f(c)− f(a), and introduce the variable t as

t = f(x)− f(a).

Since f(x) is increasing in (a, c), we can write

eλf(a)∫ c

a

e−λf(x)g (x) dx =

∫ T

0

e−λth (t) dt (1.7)

with h(t) being the continuous function in (0, T ] given by

h(t) = g(x)dx

dt=

g(x)

f ′(x). (1.8)

By assumption

t ∼∞∑

k=0

ak (x− a)k+α as x → a+,

and thus, by series reversion we obtain

x− a ∼∞∑

k=1

dktk/α as t → 0+.

Substituting this into (1.8) yields

h (t) ∼∞∑

k=0

ckt(k+β)/α−1

as t → 0+, where the coefficients ck are expressible in terms of ak and bk. We nowapply Watson’s Lemma to the integral on the right-hand side of (1.7) to obtain

∫ c

a

e−λf(x)g (x) dx ∼ e−λf(a)

∞∑

n=0

Γ

(n+ β

α

)cn

λ(n+β)/α,

as λ → +∞. All that remains is to show that the integral on the remaining range(c, b) is negligible. Define

εdef= inf

c≤x<b(f (x)− f (a)) .

6


This is positive by condition (i). Let λ0 be a value of λ for which I(λ) is absolutelyconvergent. Assume that λ ≥ λ0, then

λ (f (x)− f (a)) = (λ− λ0) (f (x)− f (a)) + λ0 (f (x)− f (a))

≥ (λ− λ0) ε+ λ0 (f (x)− f (a))

and ∣∣∣∣eλf(a)

∫ b

c

e−λf(x)g (x) dx

∣∣∣∣≤ Ke−ελ,

where

K = eλ0(ε+f(a))

∫ b

c

e−λ0f(x) |g (x)| dx

is a constant.

1.2.2 Generalization of Perron’s Formula

In this subsection, we derive an explicit formula for the coefficients cn appearingin the asymptotic expansion (1.6). The notations are the same as in the previoussubsection.

Theorem 1.2.2. The coefficients cn appearing in (1.6) are given explicitly by

cn =1

αn!

[

dn

dxn

{

G (x)

((x− a)α

f (x)− f (a)

)(n+β)/α}]

x=a

=1

α

n∑

k=0

bn−k

k!

[

dk

dxk

((x− a)α

f (x)− f (a)

)(n+β)/α]

x=a

,

(1.9)

where G (x) is the (formal) power series∑∞

k=0 bk (x− a)k and f(x) should be iden-tified with its (formal) expansion (1.3).

Remark. When (f (x)− f (a)) (x− a)−α and g (x) (x− a)1−β are holomorphicfunctions, the representation (1.9) is known as Perron’s Formula [10] [14, p. 103](see also Section 2.2). But here we show that this formula holds also when theexpansions (1.3), (1.4) are merely asymptotic. An alternative proof of essentiallythe same formula has been given previously by Wojdylo [13].

Proof. Let ℓ > 0 be the index of the first nonvanishing coefficient in the expansion(1.3) of f apart from a0. Define

fℓ (x)def=

f (x)− f (a)− a0 (x− a)α

(x− a)α+ℓ∼

∞∑

k=0

ak+ℓ (x− a)k as x → a+.

Then

I (λ) =

∫ b

a

e−λf(x)g (x) dx = e−λf(a)

∫ b

a

e−a0λ(x−a)α e−λ(x−a)α+ℓfℓ(x)g (x)︸︷︷︸

h(λ,x)

dx

= e−λf(a)

∫ b

a

e−a0λ(x−a)αh (λ, x) dx

=e−λf(a)

λ1/α

∫ λ1/α(b−a)

0

e−a0tαh(λ, λ−1/αt+ a

)dt.

(1.10)

7


Set s = λ−1/αt; then the function h(λ, λ−1/αt+ a

)reads

h (λ, s+ a) = e−tαsℓfℓ(s+a)g (s+ a) .

We have

e−tαsℓfℓ(s+a) ∼∞∑

k=0

1

k!

[dk

dwke−tαwℓfℓ(w+a)

]

w=0

sk as s → 0+

and hence,

h (λ, s+ a) ∼∞∑

n=0

(n∑

k=0

bn−k

k!

[dk

dwke−tαwℓfℓ(w+a)

]

w=0

)

sn+β−1 as s → 0+.

Introducing this expansion in the last integral of (1.10) we find

I (λ) =e−λf(a)

λ1/α

∫ λ1/α(b−a)

0

e−a0tαh(λ, λ−1/αt+ a

)dt

∼ e−λf(a)

λ1/α

∞∑

n=0

(n∑

k=0

bn−k

k!

[

dk

dwk

(∫ λ1/α(b−a)

0

e−(a0+wℓfℓ(w+a))tαsn+β−1dt

)]

w=0

)

= e−λf(a)

∞∑

n=0

(n∑

k=0

bn−k

k!

[

dk

dwk

(∫ λ1/α(b−a)

0

e−(a0+wℓfℓ(w+a))tαtn+β−1dt

)]

w=0

)

1

λ(n+β)/α

∼ e−λf(a)

∞∑

n=0

(n∑

k=0

bn−k

k!

[dk

dwk

(∫ +∞

0

e−(a0+wℓfℓ(w+a))tαtn+β−1dt

)]

w=0

)

1

λ(n+β)/α

= e−λf(a)

∞∑

n=0

(

Γ(n+βα

)

α

n∑

k=0

bn−k

k!

[

dk

dwk

(

1

(a0 + wℓfℓ (w + a))(n+β)/α

)]

w=0

)

1

λ(n+β)/α

and using the definition of fℓ (w + a):

I (λ) ∼ e−λf(a)

∞∑

n=0

(

Γ(n+βα

)

α

n∑

k=0

bn−k

k!

[

dk

dwk

(wα

f (w + a)− f (a)

)(n+β)/α]

w=0

)

1

λ(n+β)/α

= e−λf(a)

∞∑

n=0

(

Γ(n+βα

)

α

n∑

k=0

bn−k

k!

[

dk

dxk

((x− a)α

f (x)− f (a)

)(n+β)/α]

x=a

)

1

λ(n+β)/α

as λ → +∞. Formula (1.9) now follows from the asymptotic expansion (1.6) andthe uniqueness theorem on asymptotic series.

1.2.3 The CFWW Formula

In the previous subsection, we introduced formula (1.9) for the coefficients cnapperaing in the asymptotic expansion of certain Laplace-type integrals. In thissubsection, we shall extract the higher order derivatives in formula (1.9) by means

8


of the Partial Ordinary Bell Polynomials (see Appendix A.1). Using the seriesexpansion (1.3) of f(x) we find

cn =1

α

n∑

k=0

bn−k

k!

[

dk

dxk

((x− a)α

f (x)− f (a)

)(n+β)/α]

x=a

=1

αa(n+β)/α0

n∑

k=0

bn−k

k!

[

dk

dxk

(a0 (x− a)α

f (x)− f (a)

)(n+β)/α]

x=a

=1

αa(n+β)/α0

n∑

k=0

bn−k

k!

[

dk

dxk

(f (x)− f (a)

a0 (x− a)α

)−(n+β)/α]

x=a

=1

αa(n+β)/α0

n∑

k=0

bn−k

k!

dk

dxk

(

1 +∞∑

k=1

aka0

(x− a)k)−(n+β)/α

x=a

.

From the definition of the Ordinary Potential Polynomials and the Partial Ordi-nary Bell Polynomials we obtain

cn =1

αa(n+β)/α0

n∑

k=0

bn−kA−(n+β)/α,k

(a1a0

,a2a0

, . . . ,aka0

)

=1

αa(n+β)/α0

n∑

k=0

bn−k

k∑

j=0

(−n+βα

j

)1

aj0Bk,j (a1, a2, . . . , ak−j+1)

=1

αΓ(n+βα

)

n∑

k=0

bn−k

k∑

j=0

(−1)j

a(n+β)/α+j0

Bk,j (a1, a2, . . . , ak−j+1)

j!Γ

(n+ β

α+ j

)

.

(1.11)

This is the Campbell–Froman–Walles–Wojdylo Formula (from now on the CFWWFormula) [1, 12, 13]. Sometimes it is possible to find the Partial Ordinary BellPolynomials explicitly, in other cases the recurrence (A.4) could be used.

1.2.4 Another method

In the previous subsection, we showed an explicit formula for the coefficients cnappearing in the asymptotic expansion of Laplace-type integrals. This formula,that we called the CFWW Formula, followed from the generalization of Perron’sFormula (1.9) using the Partial Ordinary Bell Polynomials. In this subsection,we give a new explicit formula using Lagrange Interpolation and the OrdinaryPotential Polynomials. We write formula (1.9) in the form

cn =1

αa(n+β)/α0

n∑

k=0

bn−k

k!

[

dk

dxk

(a0 (x− a)α

f (x)− f (a)

)(n+β)/α]

x=a

. (1.12)

Define the sequence of functions (Fk (z))k≥0 by the generating function

(a0 (x− a)α

f (x)− f (a)

)z

=∞∑

k=0

Fk (z) (x− a)k.

9


Now, equation (1.12) can be written as

cn =1

αa(n+β)/α0

n∑

k=0

bn−kFk

(n+ β

α

)

. (1.13)

Since

(a0 (x− a)α

f (x)− f (a)

)z

=

(

1 +∞∑

k=1

aka0

(x− a)k)−z

=∞∑

k=0

(k∑

j=0

(−z

j

)1

aj0Bk,j (a1, a2, . . . , ak−j+1)

)

(x− a)k,

the Fk is a polynomial of degree at most k. By Lagrange Interpolation

Fk (z) =Γ (z + k + 1)

k!Γ (z)

k∑

j=0

(−1)j(k

j

)Fk (−j)

z + j, (1.14)

where the Fk (−j)’s can be computed as follows

∞∑

k=0

Fk (−j) (x− a)k =

(f (x)− f (a)

a0 (x− a)α

)j

=

(

1 +∞∑

k=1

aka0

(x− a)k)j

=∞∑

k=0

Aj,k

(a1a0

,a2a0

, . . . ,aka0

)

(x− a)k,

that is

Fk (−j) = Aj,k

(a1a0

,a2a0

, . . . ,aka0

)

. (1.15)

From (1.14) and (1.15) we deduce

Fk

(n+ β

α

)

=Γ(n+βα

+ k + 1)

k!Γ(n+βα

)

k∑

j=0

(−1)j

n+βα

+ j

(k

j

)

Aj,k

(a1a0

,a2a0

, . . . ,aka0

)

.

Plugging this into (1.13) yields

Theorem 1.2.3. The coefficients cn appearing in (1.6) are given explicitly by

cn =1

αΓ(n+βα

)

n∑

k=0

Γ(n+βα

+ k + 1)bn−k

k!a(n+β)/α0

k∑

j=0

(−1)j

n+βα

+ j

(k

j

)

Aj,k

(a1a0

,a2a0

, . . . ,aka0

)

.

(1.16)

As we will see in the following subsection, it is possible, in several cases, toderive an explicit formula for the Ordinary Potential Polynomials due to the some-what simple generating function

(

1 +∞∑

k=1

aka0

(x− a)k)j

=

(f (x)− f (a)

a0 (x− a)α

)j

=∞∑

k=0

Aj,k

(a1a0

,a2a0

, . . . ,aka0

)

(x− a)k,

10


whereas in the case of the CFWW formula, the corresponding Partial OrdinaryBell Polynomials have more complicated generating functions, such as

exp

(

y∞∑

k=1

ak (x− a)k)

= exp

(

yf (x)− f (a)− a0 (x− a)α

(x− a)α

)

=∞∑

k=0

(k∑

j=0

Bk,j (a1, a2, . . . , ak−j+1)

j!yj

)

(x− a)k.

1.2.5 Examples

Example 1.2.1. As a first example we take the Gamma Function

Γ (λ+ 1) =

∫ +∞

0

e−ttλdt (1.17)

for λ > 0. If we put t = λ(1 + x), we obtain

Γ (λ+ 1) = λλ+1e−λ

∫ +∞

−1

e−λ(x−log(1+x))dx

and hence,

Γ (λ)

λλe−λ=

∫ +∞

0

e−λ(x−log(1+x))dx+

∫ 1

0

e−λ(−x−log(1−x))dx.

Using Theorem 1.2.1 with f (x) = x− log (1 + x) (and f (x) = −x− log (1− x)),g(x) ≡ 1, α = 2 and β = 1; one finds that

∫ +∞

0

e−λ(x−log(1+x))dx ∼∞∑

n=0

Γ

(n+ 1

2

)cn

λ(n+1)/2

and∫ 1

0

e−λ(−x−log(1−x))dx ∼∞∑

n=0

Γ

(n+ 1

2

)(−1)n cnλ(n+1)/2

,

as λ → +∞, where, by Theorem 1.2.2,

cn =1

2n!

[

dn

dxn

(x2

x− log (1 + x)

)(n+1)/2]

x=0

.

Finally,

Γ (λ) ∼√2πλλ−1/2e−λ

∞∑

n=0

γnλn

, (1.18)

as λ → +∞, where

γn =

√

2

πΓ

(

n+1

2

)

c2n (1.19)

=1

2nn!

[

d2n

dx2n

(1

2

x2

x− log (1 + x)

)n+1/2]

x=0

11


are the so-called Stirling Coefficients. The first few are γ0 = 1 and

γ1 =1

12, γ2 =

1

288, γ3 = − 139

51840, γ4 = − 571

2488320.

We shall give a more explicit formula for the γn’s using the CFWW Formula (1.11).Since

f(x) = x− log (1 + x) =∞∑

k=0

(−1)k

k + 2xk+2, g(x) ≡ 1,

we have to compute the Partial Ordinary Bell Polynomials Bk,j

(

−13, 14, . . . , (−1)k−j+1

k−j+3

)

.

We have the exponential generating function

exp

(

y∞∑

k=1

(−1)k

k + 2xk

)

= exp

(y

x2

(

x− x2

2− log (1 + x)

))

=∞∑

k=0

(k∑

j=0

Bk,j

(

−1

3,1

4, . . . ,

(−1)k−j+1

k − j + 3

)

yj

j!

)

xk.

From the generating functions of the Hermite Polynomials and the Stirling Num-bers of the First Kind we have

exp

(

t

(

x− x2

2

))

=∞∑

k=0

⌊k/2⌋∑

j=0

(−1)j

j! (k − 2j)!2jtk−j

xk

and

exp (−t log (1 + x)) =∞∑

k=0

(

(−1)kk∑

j=0

s (k, j) tj

)

xk

k!.

Performing the Cauchy product of the series and rearranging the terms in thecoefficients yields

exp

(

t

(

x− x2

2− log (1 + x)

))

=∞∑

k=0

⌊k/3⌋∑

j=0

k−j∑

i=0

(−1)i

2ii!

j−i∑

r=0

(−1)k+r s (j − i− r, k − 2i− r)

r! (k − 2i− r)!tj

xk.

Plugging t = yx2 gives

exp

(y

x2

(

x− x2

2− log (1 + x)

))

=∞∑

k=0

⌊k/3⌋∑

j=0

k−j∑

i=0

(−1)i

2ii!

j−i∑

r=0

(−1)k+r s (j − i− r, k − 2i− r)

r! (k − 2i− r)!

yj

x2j

xk

=∞∑

k=0

(k∑

j=0

j∑

i=0

(−1)i

2ii!

j−i∑

r=0

(−1)r s (k + 2j − 2i− r, j − i− r)

r! (k + 2j − 2i− r)!yj

)

xk,

12


that is

Bk,j

(

−1

3,1

4, . . . ,

(−1)k−j+1

k − j + 3

)

= j!

j∑

i=0

(−1)i

2ii!

j−i∑

r=0

(−1)r s (k + 2j − 2i− r, j − i− r)

r! (k + 2j − 2i− r)!.

Finally, formula (1.11) produces

cn =n∑

j=0

2(n−1)/2+jΓ(n+12

+ j)

Γ(n+12

)

j∑

i=0

1

2ii!

j−i∑

r=0

(−1)j+i+r s (n+ 2j − 2i− r, j − i− r)

r! (n+ 2j − 2i− r)!.

From this and the expression (1.19) we deduce

γn =2n∑

j=0

2n+jΓ(n+ j + 1

2

)

√π

j∑

i=0

1

2ii!

j−i∑

r=0

(−1)j+i+r s (2n+ 2j − 2i− r, j − i− r)

r! (2n+ 2j − 2i− r)!.

(1.20)This is the result of Lopez, Pagola and Sinusıa [6].

Now we show how our new method gives a much simpler result than (1.20) withless calculation. To use formula (1.16) we have to compute the Ordinary Potential

Polynomials Aj,k

(

−23, 12, . . . , (−1)k 2

k+2

)

. This time the generating function is

(

1 +∞∑

k=1

(−1)k2

k + 2xk

)j

=

(

2x− log (1 + x)

x2

)j

=∞∑

k=0

Aj,k

(

−2

3,1

2, . . . , (−1)k

2

k + 2

)

xk.

Using the generating function of the Stirling Numbers of the First Kind yields

(x− log (1 + x))j =

j∑

i=0

(j

i

)

xj−i (− log (1 + x))i

=

j∑

i=0

(j

i

)

xj−i

∞∑

k=0

(−1)k i!s (k, i)xk

k!

=∞∑

k=0

j∑

i=0

(j

i

)

(−1)k i!s (k, i)xk+j−i

k!

=∞∑

k=0

(j∑

i=0

(j

i

)

(−1)k−j+i i!s (k − j + i, i)

(k − j + i)!

)

xk,

which gives

(

2x− log (1 + x)

x2

)j

=∞∑

k=0

(

2jj∑

i=0

(j

i

)

(−1)k−j+i i!s (k − j + i, i)

(k − j + i)!

)

xk−2j

=∞∑

k=0

(

2jj∑

i=0

(j

i

)

(−1)k−j+i i!s (k + j + i, i)

(k + j + i)!

)

xk,

13


that is

Aj,k

(

−2

3,1

2, . . . , (−1)k

2

k + 2

)

= 2jj∑

i=0

(j

i

)

(−1)k−j+i i!s (k + j + i, i)

(k + j + i)!.

Finally, by formula (1.16) we find

cn =Γ(3n+1

2+ 1)

Γ(n+12

)

n∑

j=0

2n/2+j−1/2

(n+12

+ j)(n− j)!

j∑

i=0

(−1)n+i s (n+ j + i, i)

(j − i)! (n+ j + i)!

=Γ(3n2+ 3

2

)

Γ(n+12

)

n∑

j=0

2n/2+j+1/2

(n+ 2j + 1) (n− j)!

j∑

i=0

(−1)n+j−i s (n+ 2j − i, j − i)

i! (n+ 2j − i)!.

From this and the expression (1.19) it follows that

γn =2n∑

j=0

2n+j+1Γ(3n+ 3

2

)

√π (2n+ 2j + 1) (2n− j)!

j∑

i=0

(−1)j−i s (2n+ 2j − i, j − i)

i! (2n+ 2j − i)!

=3n∑

j=n

(−1)n 2j+1Γ(3n+ 3

2

)

√π (2j + 1) (3n− j)!

j−n∑

i=0

(−1)j−i s (2j − i, j − n− i)

i! (2j − i)!,

which is much simpler than (1.20). ♣

Remark. The substitution x = log(t/λ) in (1.17) leads to the form

Γ (λ)

λλe−λ=

∫ +∞

0

e−λ(ex−x−1)dx+

∫ +∞

0

e−λ(e−x+x−1)dx.

Similar procedures to that described above give (1.18) with

γn =1

2nn!

[

d2n

dx2n

(1

2

x2

ex − x− 1

)n+1/2]

x=0

=2n∑

j=0

2n+jΓ(n+ j + 1

2

)

√π

j∑

i=0

1

2ii!

j−i∑

r=0

(−1)j+i+r S (2n+ 2j − 2i− r, j − i− r)

r! (2n+ 2j − 2i− r)!

=3n∑

j=n

(−1)n 2j+1Γ(3n+ 3

2

)

√π (2j + 1) (3n− j)!

j−n∑

i=0

(−1)j−i S (2j − i, j − n− i)

i! (2j − i)!,

using formula (1.9), (1.11) and (1.16) respectively. Here S(n, k) denotes the StirlingNumbers of the Second Kind.

Example 1.2.2. Our second example is the family of the Modified Bessel Func-tions of the Second Kind Kν . Suppose that ν, t > 0, then we have the integralrepresentation

Kν (νt) =1

2

∫ +∞

−∞e−ν(t cosh s−s)ds.

14


The substitution s = sinh−1 (1/t) + x gives

Kν (νt) =1

2

(√t2 + 1 + 1

t

)ν ∫ +∞

−∞e−ν(

√t2+1 coshx+sinhx−x)dx

=1

2

(√t2 + 1 + 1

t

)ν

e−ν√t2+1

∫ +∞

−∞e−ν(

√t2+1(coshx−1)+sinhx−x)dx,

and by splitting up the integral

2

(√t2 + 1 + 1

t

)−ν

eν√t2+1Kν (νt)

=

∫ +∞

0

e−ν(√t2+1(coshx−1)+sinhx−x)dx+

∫ +∞

0

e−ν(√t2+1(coshx−1)−sinhx+x)dx.

Using Theorem 1.2.1 with f (x) =√t2 + 1 (cosh x− 1) + sinh x − x (and f (x) =√

t2 + 1 (cosh x− 1)− sinh x+ x), g(x) ≡ 1, α = 2 and β = 1; one finds that

∫ +∞

0

e−ν(√t2+1(coshx−1)+sinhx−x)dx ∼

∞∑

n=0

Γ

(n+ 1

2

)cn

ν(n+1)/2

and∫ +∞

0

e−ν(√t2+1(coshx−1)−sinhx+x)dx ∼

∞∑

n=0

Γ

(n+ 1

2

)(−1)n cnν(n+1)/2

,

as ν → +∞, where, by Theorem 1.2.2,

cn =1

2n!

[

dn

dxn

(x2

√t2 + 1 (cosh x− 1)− sinh x+ x

)(n+1)/2]

x=0

.

Finally,

Kν (νt) ∼(√

t2 + 1 + 1

t

)ν

e−ν√t2+1

√π

2ν√t2 + 1

∞∑

n=0

(−1)nUn(τ)

νn, (1.21)

as ν → +∞, where

Un(τ) = (−1)n√

2

π

√t2 + 1Γ

(

n+1

2

)

c2n (1.22)

= (−1)n(t2 + 1)

1/4

22n+1/2n!

[

d2n

dx2n

(x2

√t2 + 1 (cosh x− 1)− sinh x+ x

)n+1/2]

x=0

=(−1)n

22n+1/2n!τ 1/2

[

d2n

dx2n

(x2

τ−1 (cosh x− 1)− sinh x+ x

)n+1/2]

x=0

,

and τ = (t2 + 1)−1/2. It is known that the Un(τ)’s are polynomials in τ of degree3n. The first few are given by U0(τ) = 1 and

U1 (τ) = − 5

24τ 3 +

1

8τ,

15


U2 (τ) =385

1152τ 6 − 77

192τ 4 +

9

128τ 2,

U3 (τ) = −85085

82944τ 9 +

17017

9216τ 7 − 4563

5120τ 5 +

75

1024τ 3.

Thus, the expansion (1.21) holds uniformly for t > 0. Since

√t2 + 1 (cosh x− 1)− sinh x+ x =

∞∑

k=1

1

τ (2k)!x2k −

∞∑

k=1

1

(2k + 1)!x2k+1,

we have

a2k =1

τ (2k + 2)!, a2k+1 = − 1

(2k + 3)!for k ≥ 0.

From the CFWW Formula (1.11) we obtain

cn =1

Γ(n+12

)

n∑

j=0

(−1)j (2τ)(n+1)/2+j

j!Bn,j

(

−1

6,

1

24τ, . . . , an−j+1

)

Γ

(n+ 1

2+ j

)

,

from which it follows by (1.22) that

Un (τ) =2n∑

j=0

(−1)n+j 2n+j+1τn+j

j!√π

Γ

(

n+ j +1

2

)

B2n,j .

Here Bn,j = Bn,j

(−1

6, 124τ

, . . . , an−j+1

), and from the recurrence of the Partial

Ordinary Bell Polynomials, Bn,0 = 0, Bn,1 = an and

Bn,j = −1

6Bn−1,j−1 +

1

24τBn−2,j−1 + · · ·+ an−j+1Bj−1,j−1.

In this case, it is quite complicated to give any explicit formula for the PartialOrdinary Bell Polynomials. Nevertheless, formula (1.16) enables us to derive afairly explicit expression for the polynomials Un (τ). To use formula (1.16) wehave to compute the Ordinary Potential Polynomials Aj,k

(− τ

3, 112, . . . , 2τak

). The

generating function is(

2ττ−1 (cosh x− 1)− sinh x+ x

x2

)j

=∞∑

k=0

Aj,k

(

−τ

3,1

12, . . . , 2τak

)

xk.

Algebraic manipulation and the generating function of the 1-associated StirlingNumbers of the Second Kind (see expression (A.5)) yields(τ−1 (cosh x− 1)− sinh x+ x

)j

=

(

(ex − x− 1)

(1

2τ− 1

2

)

+(e−x + x− 1

)(

1

2τ+

1

2

))j

=

j∑

i=0

(j

i

)(1

2τ− 1

2

)i(1

2τ+

1

2

)j−i

(ex − x− 1)i(e−x + x− 1

)j−i

= j!

j∑

i=0

(1

2τ− 1

2

)i(1

2τ+

1

2

)j−i ∞∑

k=0

S1 (k, i)xk

k!

∞∑

m=0

(−1)m S1 (m, j − i)xm

m!

= j!∞∑

k=0

(j∑

i=0

k−2j+2i∑

m=2i

(−1)k−m

(1

2τ− 1

2

)i(1

2τ+

1

2

)j−iS1 (m, i)S1 (k −m, j − i)

m! (k −m)!

)

xk,

16


which gives

(

2ττ−1 (cosh x− 1)− sinh x+ x

x2

)j

= j!∞∑

k=0

(j∑

i=0

k−2j+2i∑

m=2i

(−1)k−m (1− τ)i (1 + τ)j−i S1 (m, i)S1 (k −m, j − i)

m! (k −m)!

)

xk−2j

= j!∞∑

k=0

(j∑

i=0

k+2i∑

m=2i

(−1)k−m (1− τ)i (1 + τ)j−i S1 (m, i)S1 (k + 2j −m, j − i)

m! (k + 2j −m)!

)

xk,

that is

Aj,k

(

−τ

3,1

12, . . . , 2τak

)

= j!

j∑

i=0

k+2i∑

m=2i

(−1)k−m (1− τ)i (1 + τ)j−i S1 (m, i)S1 (k + 2j −m, j − i)

m! (k + 2j −m)!.

Finally, from formula (1.16), we find

cn = (2τ)(n+1)/2 Γ(3n+1

2+ 1)

2Γ(n+12

)

n∑

j=0

(−1)j(n+12

+ j)Aj,n

(− τ

3, 112, . . . , 2τan

)

j! (n− j)!

= (2τ)(n+1)/2 Γ(3n2+ 3

2

)

Γ(n+12

)

n∑

j=0

(−1)j

(n+ 2j + 1)

Aj,n

(− τ

3, 112, . . . , 2τan

)

j! (n− j)!,

and thus by (1.22),

Un (τ) = (−1)n2 (2τ)n Γ

(3n+ 3

2

)

√π

2n∑

j=0

(−1)j∑j

i=0 un (i, j) (1− τ)i (1 + τ)j−i

(2n+ 2j + 1) (2n− j)!,

where

un (i, j)def=

2n+2i∑

m=2i

(−1)mS1 (m, i)S1 (2n+ 2j −m, j − i)

m! (2n+ 2j −m)!.

♣

Remark. It can be shown that the Modified Bessel Functions of the First KindIν have a similar asymptotic expansion:

Iν (νt) ∼(

t√t2 + 1 + 1

)νeν

√t2+1

√

2πν√t2 + 1

∞∑

n=0

Un (τ)

νn,

as ν → +∞, uniformly for t > 0. The coefficients Un(τ) are the same as above. Itis known that a recurrence for the polynomials Un(τ) is given as follows

Un+1 (τ) =1

2τ 2(1− τ 2

)U ′n (τ)−

1

8

∫ τ

0

(5x2 − 1

)Un (x) dx,

with U0(τ) = 1 [8, p. 376].

17


Example 1.2.3. As a last example we take the Legendre Polynomials Pm. Sup-pose that t > 1, then we have the integral representation

Pm (t) =1

π

∫ π

0

(

t+ cosx√t2 − 1

)m

dx.

The substitution t = cosh θ (θ > 0) and a little algebraic manipulation gives

Pm (cosh θ) =emθ

π

∫ π

0

e−m(− log(1−sin2(x2 )(1−e−2θ)))dx.

Using Theorem 1.2.1 with f (x) = − log(1− sin2

(x2

) (1− e−2θ

)), g(x) ≡ 1, α = 2

and β = 1; one finds that

∫ π

0

e−m(− log(1−sin2(x2 )(1−e−2θ)))dx ∼

∞∑

n=0

Γ

(n+ 1

2

)cn

m(n+1)/2

as m → +∞, where, by Theorem 1.2.2,

cn =1

2n!

dn

dxn

(

x2

− log(1− sin2

(x2

)(1− e−2θ)

)

)(n+1)/2

x=0

.

It is not hard to see that cn = (−1)ncn, thus

Pm (cosh θ) ∼ e(m+1)θ

√

πm (e2θ − 1)

∞∑

n=0

ρn (θ)

mn, (1.23)

as m → +∞, where

ρn (θ) =1√π

√

1− e−2θΓ

(

n+1

2

)

c2n (1.24)

=

√1− e−2θ

22n+1n!

d2n

dx2n

(

x2

− log(1− sin2

(x2

)(1− e−2θ)

)

)n+1/2

x=0

.

The first few are ρ0 (θ) = 1 and

ρ1 (θ) = − e2θ − 3

8 (e2θ − 1), ρ2 (θ) =

e4θ + 10e2θ + 25

128 (e2θ − 1)2, ρ3 (θ) =

5e6θ + 35e4θ + 455e2θ + 105

1024 (e2θ − 1)3.

Since

− log(

1− sin2(x

2

) (1− e−2θ

))

=∞∑

k=1

(1− e−2θ

)k

ksin2k

(x

2

)

=∞∑

k=1

(1− e−2θ

)k

k

∞∑

j=0

(

(−1)k+j

22k−1 (2k + 2j)!

k∑

i=1

(−1)i(

2k

k − i

)

i2j+2k

)

(x2)k+j

=∞∑

k=1

(k−1∑

j=0

(−1)k(1− e−2θ

)j+1

22j+1 (j + 1) (2k)!

j+1∑

i=1

(−1)i(

2j + 2

j − i+ 1

)

i2k

)

x2k,

18


we have

a2k =k∑

j=0

(−1)k+1 (1− e−2θ)j+1

22j+1 (j + 1) (2k + 2)!

j+1∑

i=1

(−1)i(

2j + 2

j − i+ 1

)

i2k+2

and a2k+1 = 0 for k ≥ 0. From the CFWW Formula (1.11) we obtain

cn =1

2Γ(n+12

)

n∑

j=0

(−1)j 4(n+1)/2+j

j! (1− e−2θ)(n+1)/2+jΓ

(n+ 1

2+ j

)

Bn,j,

with Bn,j = Bn,j

(

0, 3e−4θ−4e−2θ+1

96, . . . , an−j+1

)

. From this and (1.24) it follows that

ρn (θ) =2n∑

j=0

(−1)j4n+j

j!√π (1− e−2θ)n+jΓ

(

n+ j +1

2

)

B2n,j .

By the recurrence of the Partial Ordinary Bell Polynomials, B2n,0 = 0, B2n,1 = a2nand

B2n,2j =3e−4θ − 4e−2θ + 1

96B2n−2,2j−1 + · · ·+ a2n−2jB2j,2j ,

B2n,2j+1 =3e−4θ − 4e−2θ + 1

96B2n−2,2j + · · ·+ a2n−2jB2j,2j

for j ≥ 0. If one thinks of the generating function of these Partial Ordinary BellPolynomials, it is clear that giving an explicit formula for them would be quitecomplicated. On the other hand, formula (1.16) enables us to give an explicitexpression for the coefficients ρn (θ) in terms of the Stirling Numbers of the FirstKind. To use formula (1.16) we have to compute the Ordinary Potential Polyno-

mials Aj,k

(

0, 1−3e−2θ

24, . . . , 4ak

1−e−2θ

)

. The generating function is

(

4− log

(1− sin2

(x2

) (1− e−2θ

))

x2 (1− e−2θ)

)j

=∞∑

k=0

Aj,k

(

0,1− 3e−2θ

24, . . . ,

4ak1− e−2θ

)

xk.

To compute the corresponding Ordinary Potential Polynomials, note that

(

− log(

1− sin2(x

2

) (1− e−2θ

)))j

=∞∑

k=1

j!s (k, j)

(1− e−2θ

)k

k!sin2k

(x

2

)

=∞∑

k=1

j!s (k, j)

(1− e−2θ

)k

k!

∞∑

m=0

(

(−1)k+m

22k−1 (2k + 2m)!

k∑

i=1

(−1)i(

2k

k − i

)

i2k+2m

)

(x2)k+m

=∞∑

k=1

(k∑

m=0

(−1)k j!s (m, j)(1− e−2θ

)m

22m−1m! (2k)!

m∑

i=1

(−1)i(

2m

m− i

)

i2k

)

x2k,

for j ≥ 1, which gives

Aj,2k

(

0,1− 3e−2θ

24, . . . ,

4a2k1− e−2θ

)

=

k+j∑

m=0

(−1)k+j 22j−2m+1j!s (m, j)

(1− e−2θ)j−mm! (2k + 2j)!

m∑

i=1

(−1)i(

2m

m− i

)

i2k+2j,

19


for k, j ≥ 1. Finally, formula (1.16) yields

cn =1

2Γ(n+12

)2n+1Γ

(n+12

+ n+ 1)

n! (1− e−2θ)(n+1)/2

n∑

j=0

(−1)j(n+12

+ j)

(n

j

)

Aj,n

=1

Γ(n+12

)2n+1Γ

(3n2+ 3

2

)

(1− e−2θ)(n+1)/2

n∑

j=0

(−1)j

(n+ 2j + 1)

Aj,n

(n− j)!j!,

for n ≥ 1 with Aj,n = Aj,n

(

0, 1−3e−2θ

24, . . . , 4an

1−e−2θ

)

. From this and the expression

(1.24) we deduce

ρn (θ) =Γ(3n+ 3

2

)

√π

2n∑

j=0

(−1)n

(2n− j)!

n+j∑

m=0

22n+2j−2m+2s (m, j)

(1− e−2θ)n+j−mm! (2n+ 2j + 1)!

m∑

i=1

(−1)i(

2m

m− i

)

i2n+2j

=Γ(3n+ 3

2

)

√π

3n∑

j=n

(−1)n

(3n− j)!

j∑

m=0

4j−m+1s (m, j − n)

(1− e−2θ)j−m m! (2j + 1)!

m∑

i=1

(−1)i(

2m

m− i

)

i2j ,

(1.25)

for n ≥ 1, with ρ0(θ) = 1. ♣

Remark. Another possible way to derive the asymptotic expansion (1.23) is touse the generating function

1√1− 2x cosh θ + x2

=∞∑

m=0

Pm (cosh θ) xm, θ > 0.

A simple algebraic manipulation yields

1√1− z

eθ√e2θ − z

=∞∑

m=0

e−mθPm (cosh θ) zm.

This generating function has an algebraic singularity at z = 1, the full asymptoticexpansion can now be carried out by Darboux’s Method (see, e.g., [8, p. 309], [14, p.116]). The coefficients ρn (θ) this time take the form

ρn (θ) =n∑

k=0

(−1)k (2n− 4k + 1)

24n−4k (2n− 2k + 1) (e2θ − 1)n−k

(2n− 2k

n− k

)2(n− k + 1/2

k

)

B(n−k+1/2)k ,

where B(µ)n denote the Generalized Bernoulli Numbers. Using the explicit formula

[5]

B(µ)k =

k∑

j=0

(k+µk−j

)(k−µk+j

)

(k−µk

) S (k + j, j),

we get the following expression involving the Stirling Numbers of the Second Kind:

ρn (θ) =n∑

k=0

(−1)k (2n− 4k + 1)

24n−4k (e2θ − 1)n−k

(2n− 2k

n− k

)2(n+ 1/2

2k

) k∑

j=0

(2k

k − j

)(−1)j S (k + j, j)

2n− 2k + 2j + 1,

which is even simpler than (1.25).

20

Chapter 2

Complex Laplace-type integrals

In this chapter, we investigate the asymptotic behavior of Laplace-type integralswith complex parameter. The integrals under consideration are of the form

I (λ) =

∫

C

eλf(z)g (z) dz, (2.1)

where the path of integration C is a contour in the complex plane and λ is a largereal parameter. The functions f and g are independent of λ and holomorphic in adomain containing the path of integration.

Chapter 2 is organized as follows. In the first section, we revisit the Method ofSteepest Descents, a well-known procedure in the asymptotic theory of complexLaplace-type integrals. In the second part, we present Perron’s Method that avoidsthe computation of the path of steepest descent. Finally, in the third section, wegive three examples to demonstrate the application of these methods.

2.1 The Method of Steepest Descents

Consider the integral (2.1) with the assumptions we made. The basic idea of themethod is to deform the contour of integration C into a new path of integrationC ′ so that the following conditions hold:

(i) C ′ passes through one or more zeros of f ′;

(ii) the function Im(f) is constant on C ′.

The choice of a path with Im(f) = constant has two major advantages. It removesthe rapid oscillations of the integrand and on such paths Re(f) changes the mostrapidly (see, e.g., [9, p. 5]). Thus, the dominant contribution will arise from aneighborhood of the point where Re(f) is the greatest.

In order to obtain a geometrical interpretation of the method and the new pathof integration C ′, we introduce the notations

f(z) = u(x, y) + iv(x, y)

with z = x + iy, and the functions u and v are real. Suppose that z0 = x0 + iy0is a zero of f ′. It follows easily that (x0, y0) is a critical point of u, and since u isharmonic, it must be a saddle point of u. For this reason, we call z0 a saddle pointof f .

21

2.1. The Method of Steepest Descents

Consider the surface F in the (x, y, u) space defined by u = u(x, y). The shapeof the surface F can also be represented on the (x, y) plane by drawing the levelcurves on which u is constant. From the Cauchy–Riemann equations it followsthat the families of curves corresponding to constant values of u(x, y) and v(x, y)are orthogonal at all their points of intersection [9, p. 6]. The regions whereu(x, y) > u(x0, y0) are called hills and those where u(x, y) < u(x0, y0) are calledvalleys. The level curve through the saddle, u(x, y) = u(x0, y0), separates theneighborhood of the saddle point (x0, y0) into a series of hills and valleys.

Suppose that z0 is a saddle point of order m− 1, m ≥ 2, i.e.,

f ′ (z0) = f ′′ (z0) = · · · = f (m−1) (z0) = 0,

with f (m) (z0) = aeiϕ, a > 0. Then, if z = z0 + reiθ, r > 0, we have

f (z) = f (z0) +rm

m!aei(mθ+ϕ) + · · ·

and hence, near the saddle point z0 the level curves and steepest paths are roughlythe same as

u (x, y) = u (x0, y0) +rm

m!a cos (mθ + ϕ) ,

v (x, y) = v (x0, y0) +rm

m!a sin (mθ + ϕ) .

The directions of the level curves where u is constant, are given by the solutionsof the equation cos (mθ + ϕ) = 0, i.e.,

θ = − ϕ

m+

(

k +1

2

)π

m, k = 0, 1, . . . , 2m− 1.

Similarly, the directions of the steepest paths satisfy sin (mθ + ϕ) = 0, i.e.,

θ = − ϕ

m+ k

π

m, k = 0, 1, . . . , 2m− 1.

Therefore, there are 2m equally spaced steepest directions from z0: m directionsof steepest descent and m directions of steepest ascent. In the neighborhood of z0,the level curves u = u(x0, y0) form the boundaries of m valleys surrounding thesaddle point, in which cos (mθ + ϕ) < 0, and m hills on which cos (mθ + ϕ) > 0.The valleys and hills are situated respectively entirely below and above the saddlepoint, and each has angular width equal to π/m.

Now suppose that the path of integration C in (2.1) is a steepest path throughthe saddle point z0 of order m− 1. On this path we have

f (z) = f (z0)− τ,

where τ is non-negative and monotonically increasing as one progresses down thepath of steepest descent. Then (2.1) becomes

I (λ) = eλf(z0)∫ T

z0

e−λτg (z) dz = eλf(z0)∫ T ′

0

e−λτg (z)dz

dτdτ , (2.2)

where T denotes some point on the steepest descent path and T ′ > 0 is the mapof T in the τ -plane. Since, for large positive λ, the factor e−λτ decays rapidly, the

22

2.1. The Method of Steepest Descents

main contribution comes from the neighborhood of τ = 0. Thus, we can applyWatson’s Lemma to the integral in the right-hand side of (2.2). To do this, wefirst require a series expansion for g (z) dz

dτin ascending powers of τ . Near z0,

f (z) = f (z0)−∞∑

k=0

ak (z − z0)m+k, a0 6= 0,

that is

τ =∞∑

k=0

ak (z − z0)m+k

as z → z0. If we put τ = υm, then

υ = a1/m0 (z − z0)ϕ (z) ,

where ϕ is analytic around z0 with ϕ(z0) 6= 0 and a1/m0 takes its principal value.

Hence, υ is a single-valued analytic function of z in a neighborhood of z0 and thatυ′(z0) 6= 0. Therefore, by the inverse function theorem (see [2, p. 121])

z − z0 =∞∑

k=1

αkυk =

∞∑

k=1

αkτk/m,

where

α1 =1

a1/m0

, α2 = − a1

ma1+2/m0

, α3 =(m+ 3) a21 − 2ma0a2

2m2a2+3/m0

, . . . .

Furthermore, for small τ , g (z) dzdτ

has a convergent expansion of the form

g (z)dz

dτ=

∞∑

n=0

cnτ(n−m+1)/m. (2.3)

The first three coefficients cn are given explicitly by

c0 =b0

ma1/m0

, c1 =1

a2/m0

(b1m

− 2a1b0m2a0

)

,

and

c2 =1

a3/m0

(b2m

− 3a1b1m2a0

+((m+ 3) a21 − 2ma0a2

) b0m2a20

)

.

Here n!bn = g(n) (z0). We are now in a position to derive the asymptotic expansionof the integral (2.2) taken from the saddle point z0 of order m − 1 down one ofthe m valleys. Depending on the path C , we choose one of the m valleys, labelledby l, say. We replace τ by τe2πil in the expansion (2.3) and substitute it into theintegral (2.2). The asymptotic expansion of I(λ) can now be obtained directlyfrom Watson’s Lemma:

I (λ) ∼ eλf(z0)∞∑

n=0

Γ

(n+ 1

m

)cne

2πil(n+1)/m

λ(n+1)/m, (2.4)

23

2.2. Perron’s Method

as λ → +∞. To obtain an explicit formula for the coefficients cn we observe thatwith τ = υm, so that dz

dτ= υ1−m

mdzdυ, we find from (2.3)

g (z)dz

dυ= m

∞∑

n=0

cnυn.

Cauchy’s formula then shows that

cn =1

2πim

∫

γ

g (z)dz

dυ

dυ

υn+1=

1

2πim

∫

γ′

g (z)

(f (z0)− f (z))(n+1)/mdz,

with an appropriate branch of (f (z0)− f (z))1/m. Here γ and γ′ are simple closedcontours with positive orientation that enclose the points υ = 0 and z = z0,respectively. This is Dingle’s Formula [3, p. 119]. It follows that

cn =1

mn!

[

dn

dzn

{

g (z)

((z − z0)

m

f (z0)− f (z)

)(n+1)/m}]

z=z0

=1

m

n∑

k=0

bn−k

k!

[

dk

dzk

((z − z0)

m

f (z0)− f (z)

)(n+1)/m]

z=z0

,

(2.5)

which is Perron’s Formula. In the next subsection we derive this formula in moregeneral conditions.

Remark. It can be shown that the asymptotic expansion (2.4) is also valid when

|arg λ+ arg a0 +mω − 2πl| ≤ π

2− δ <

π

2

and |λ| → +∞, where 0 < δ ≤ π2is fixed and

ω = limC∋z→a

arg (z − a)

is the angle of slope of C at a.

2.2 Perron’s Method

In the previous subsection, we presented a method, namely, the Method of SteepestDescents, to derive asymptotic expansions for integrals of the form

I (λ) =

∫

C

eλf(z)g (z) dz. (2.6)

However, in many specific cases, the construction of steepest descent path can beextremely complicated. In this subsection, we shall describe a method due to Per-ron that avoids the computation of the path of steepest descent. Our presentationof Perron’s Method follows closely that given by Wong [14, p. 103].

In this subsection we allow λ to be complex. Let a be the starting point, and bbe the endpoint of the continuous curve C (b can by finite or infinite). We imposethe following four conditions:

24


(i) The path C lies in the sector

|arg(λ(f(a)− f(z)))| ≤ π

2− δ,

where δ is a fixed positive number.

(ii) For each point c 6= a in C , there exists a number η = η(c), such that|f(a)− f(z)| ≥ η > 0 for all z on the portion of C joining c to b.

(iii) In a neighborhood of a,

f (a)− f (z) =∞∑

k=0

ak (z − a)k+α, (2.7)

where a0 6= 0 and α > 0. Since α is not necessarily a positive integer, f isneed not be analytic at a. We make (z − a)α single-valued by introducing acut in the z-plane from a to infinity along a convenient radial line.

(iv) The contour C must not cross this cut. Furthermore, suppose that thereexists a point c′ 6= a on C such that for any c on C with |ac| < |ac′|, theportion of C from a to c can be deformed into the straight line arg(z − a) =arg(c− a).

Letω = lim

C∋z→aarg (z − a)

be the angle of slope of C at a. In order to condition (i) holds, the set throughwhich |λ| → +∞ must be contained in the sector

(

2l − 1

2

)

π + δ ≤ arg λ+ arg a0 + αω ≤(

2l +1

2

)

π − δ (2.8)

for some fixed δ in 0 < δ ≤ π2, where l is a fixed integer.

Theorem 2.2.1. Consider the integral (2.6), and assume that it exists absolutelyfor every fixed λ satisfying the inequalities in (2.8). Furthermore, assume that ina neighborhood of a,

g (z) =∞∑

k=0

bk (z − a)k+β−1, b0 6= 0,

with some fixed complex number β, Re(β) > 0. Then, under the conditions (i) to(iv), we have

I (λ) ∼ eλf(a)∞∑

n=0

Γ

(n+ β

α

)cne

2πli(n+β)/α

λ(n+β)/α,

as |λ| → +∞, uniformly with respect to arg λ confined to the sector (2.8). Thecoefficients cn are given by

cn =1

αa(n+β)/α0 n!

[

dn

dzn

{

G (z)

(a0 (z − a)α

f (a)− f (z)

)(n+β)/α}]

z=a

=1

αa(n+β)/α0

n∑

k=0

bn−k

k!

[

dk

dzk

(a0 (z − a)α

f (a)− f (z)

)(n+β)/α]

z=a

,

(2.9)

where G (z)def=∑∞

k=0 bk (z − a)k.

25


Proof. Let F be the function

F (z)def= −

∞∑

k=1

aka0

(z − a)k

which is analytic at z = a and satisfies

f (z) = f (a)− a0 (z − a)α (1− F (z)) (2.10)

in a neighborhood of a. Since the function G (z) exp (wF (z)) is, for each fixed w,analytic at z = a, we have

G (z) exp (wF (z)) =∞∑

k=0

Pk (w) (z − a)k, |z − a| < ρ,

for sufficiently small ρ, where Pk (w) is a polynomial in w whose degree does notexceed k, and

Pk (w) =1

k!

[dk

dzk(G (z) exp (wF (z)))

]

z=a

. (2.11)

Using (2.10), (2.11) can be written as

e−wPk (w) =1

k!

[dk

dzk

(

G (z) exp

(

wf (z)− f (a)

a0 (z − a)α

))]

z=a

. (2.12)

By Cauchy’s inequality

|Pk (w)| ≤1

rkmax

|z−a|=r|G (z) exp (wF (z))| ,

for 0 < r < ρ. Since F vanishes at a, for every 0 < r < ρ,

|Pk (w)| ≤1

rkM1 exp (M2 |w| r) ,

where M1 and M2 are fixed numbers. Thus, for any fixed N > 0,

G (z) exp (wF (z)) =N∑

k=0

Pk (w) (z − a)k +RN , (2.13)

where, for |z − a| ≤ r < ρ,

|RN | ≤ M3 |z − a|N+1 exp (M2 |w| r) ,

where M3 is a fixed number depending on N . Now, we put w = λa0(z − a)α in(2.13). It follows that

g (z) exp (λf (z)− λf (a)) =N∑

k=0

e−wPk (w) (z − a)k+β−1 + e−w (z − a)β−1 RN

(2.14)and, for |z − a| ≤ r < ρ,

∣∣∣e−w (z − a)β−1RN

∣∣∣ ≤ M3

∣∣∣(z − a)N+β

∣∣∣ exp (−Re (w) +M2 |w| r) . (2.15)

26


We write the integral (2.6) in the form

I (λ) =

∫ c

a

eλf(z)g (z) dz +

∫ b

c

eλf(z)g (z) dz, (2.16)

where c is a point on C such that the portion of C form a to c can be deformedinto the radial line arg(z − a) = arg(c − a) (see condition (iv)). Furthermore, werequire |c− a| < r so that the expansion (2.14)-(2.15) holds for all z on the newpath of integration from a to c. Inserting (2.14) in the first integral on the rightof (2.16) yields

∫ c

a

eλf(z)g (z) dz = eλf(a)

[N∑

k=0

Ik (λ) + EN (λ)

]

, (2.17)

where

Ik (λ)def=

∫ c

a

e−wPk (w) (z − a)k+β−1 dz (2.18)

and

EN (λ)def=

∫ c

a

e−w (z − a)β−1 RNdz.

By choosing c closer to a if necessary, we also have from (2.8)

(

2l − 1

2

)

π + δ0 ≤ arg λ+ arg a0 + α arg (c− a) ≤(

2l +1

2

)

π − δ0

for some 0 < δ0 < δ. This implies that

(

2l − 1

2

)

π + δ0 ≤ argw ≤(

2l +1

2

)

π − δ0, (2.19)

and thus Re(w) ≥ |w| sin δ0. Thus we obtain from (2.15)

∣∣∣e−w (z − a)β−1 RN

∣∣∣ ≤ M3

∣∣∣(z − a)N+β

∣∣∣ exp (− |w| sin δ0 +M2 |w| r)

for the points on the path of integration. If we choose r such that r < sin δ0/M2,then there will exist a constant M4 > 0 such that for all z with arg(z − a) =arg(c− a) and |z − a| ≤ r < ρ,

∣∣∣e−w (z − a)β−1 RN

∣∣∣ ≤ M3

∣∣∣(z − a)N+β

∣∣∣ exp (−M4 |w|) ,

and therefore,

|EN (λ)| ≤ M3

∫ c

a

∣∣∣(z − a)N+β

∣∣∣ exp (−M4 |λa0 (z − a)α|) |dz|.

If we let θ = arg(c− a) and z − a = ρeiθ, then

|EN (λ)| ≤ M3

∫ +∞

0

∣∣ρβ+N

∣∣ exp (−M4 |λa0| ρα) dρ = O

(λ−(β+N+1)/α

). (2.20)

27


We now turn our attention to the integrals Ik(λ) in (2.18). It is not hard toshow that

∫ a+∞eiθ

c

e−wPk (w) (z − a)k+β−1 dz = O(e−ε|λ|)

for some ε > 0. Next we note, from (2.19), that w = λa0(z − a)α gives

z − a = w1/αe2πli/αa−1/α0 λ−1/α

and(z − a)k+β−1 = w(k+β−1)/αe2πli(k+β−1)/αa

−(k+β−1)/α0 λ−(k+β−1)/α.

Hence,

Ik (λ) =1

αa−(k+β)/α0 λ−(k+β)/αe2πli(k+β)/α

∫ ∞eiθ′

0

e−wPk (w)w(k+β)/α−1dw+O

(e−ε|λ|) ,

where, for fixed λ, |θ′| < π2. Now, we deform the path of integration into the

positive real axis. Therefore

Ik (λ) =1

αa−(k+β)/α0 λ−(k+β)/αe2πli(k+β)/α

∫ +∞

0

e−wPk (w)w(k+β)/α−1dw+O

(e−ε|λ|) .

We now insert (2.12) into the integral, and then interchange the order of integrationand differentiation. This leads to

Ik (λ) = Γ

(k + β

α

)cke

2πli(k+β)/α

λ(k+β)/α+O

(e−ε|λ|) .

Using this and (2.20) in (2.17) yields

∫ c

a

eλf(z)g (z) dz = eλf(a)

(N∑

k=0

Γ

(k + β

α

)cke

2πli(k+β)/α

λ(k+β)/α+O

(1

λ(N+β+1)/α

))

.

(2.21)What remains is to consider the second integral on the right-hand side of (2.16).

We choose a point λ0 that satisfies the inequalities in (2.8) and write

∫ b

c

eλf(z)g (z) dz =

∫ b

c

e(λ−λ0)f(z)eλ0f(z)g (z) dz,

whence ∣∣∣∣

∫ b

c

eλf(z)g (z) dz

∣∣∣∣≤ max

∣∣e(λ−λ0)f(z)

∣∣

∫ b

c

∣∣eλ0f(z)g (z)

∣∣ |dz|,

where the maximum is taken over all points on the portion of C joining c to b. Byassumption, the integral on the right exist, thus

∣∣∣∣

∫ b

c

eλf(z)g (z) dz

∣∣∣∣≤ Kmax

∣∣e(λ−λ0)f(z)

∣∣

for some constant K > 0. It follows that∣∣∣∣

∫ b

c

eλf(z)g (z) dz

∣∣∣∣≤(Ke−λ0f(a)

)eλf(a) max

∣∣e−(λ−λ0)(f(a)−f(z))

∣∣ . (2.22)

28

2.3. Examples

Then

Re ((λ− λ0) (f (a)− f (z)))

= |f (a)− f (z)| {|λ| cos [arg (λ (f (a)− f (z)))]− |λ0| cos [arg (λ0 (f (a)− f (z)))]}= |λ| |f (a)− f (z)| {cos [arg (λ (f (a)− f (z)))] + o (1)} ,

as |λ| → +∞. Thus, by conditions (i) and (ii),

Re ((λ− λ0) (f (a)− f (z))) ≥ |λ| η (sin δ + o (1)) ,

as |λ| → +∞ in the sector (2.8), and consequently,

Re ((λ− λ0) (f (a)− f (z))) ≥ ε1 |λ|

for some ε1 > 0, uniformly in arg λ for all λ satisfying (2.8) and uniformly in z forall z on C from c to b. This together with (2.22) implies that

∣∣∣∣

∫ b

c

eλf(z)g (z) dz

∣∣∣∣= O

(eλf(a)−ε1|λ|) . (2.23)

The result now follows from (2.16), (2.21) and (2.23).

Remark. From (2.7) and (2.9) it can be seen that the representations (1.11) and(1.16) hold in the complex case too.

2.3 Examples

Example 2.3.1. Our first example is the Reciprocal Gamma Function

1

Γ (λ)=

1

2πi

∫

H

ett−λdt,

where the path of integration starts at ∞e−πi, goes round the origin once and endsat ∞eπi. Suppose that λ is real an positive, then the substitution t = λz gives

1

Γ (λ)=

1

2πiλλ−1

∫

H

eλ(z−log z)dz,

with the same path as before. This formula holds also when Re(λ) > 0, providedthat λλ means eλ log λ where log λ has its principal value. Using the notations ofSection 2.1, f (z) = z − log z and g(z) ≡ 1. The function f has only one saddlepoint, namely z0 = 1. The steepest paths through z0 are described by the equation

y − tan−1(y

x

)

= 0.

The equation y = 0 represents the path of steepest ascent whereas the equationx = y cot y (or rather the branch through (1, 0)) gives the path of steepest descent.

Then H can be deformed into Cdef= {(x, y) : x = y cot y, |y| < π}. On the path

of steepest descent, we have

f (z)− f (1) = −1 + z − log z = −τ. (2.24)

29

2.3. Examples

As z → 1,

f (z)− f (1) =(z − 1)2

2− (z − 1)3

3+ · · · ,

thus

z± = 1 +∞∑

n=0

(±1)n αnτn/2, (2.25)

dz±dτ

=∞∑

n=0

(±1)n+1 cnτ(n−1)/2, cn =

n+ 1

2αn+1, (2.26)

where z± denotes the upper and lower path of C through the saddle point z0 = 1.Equation (2.24) defines a many valued function z(τ) of a complex variable τ withbranch points at τ = 2πin with n ∈ Z \ {0}. Hence, the expansions (2.25)-(2.26)are converget in the disc |τ | < 2π. Since

dz±dτ

=z±

1− z±

is bounded when τ > 0, Watson’s Lemma is applicable and gives

1

Γ (λ)=

eλ

2πiλλ−1

∫ +∞

0

e−λτ

(dz+dτ

− dz−dτ

)

dτ

∼ eλλ−λ

√

λ

2π

∞∑

n=0

√

2

πΓ

(

n+1

2

) −ic2nλn

,

as λ → +∞. From Perron’s Formula (2.5) we find

cn =1

2n!

dn

dzn

(

(z − 1)2

1− z + log z

)(n+1)/2

z=1

=in+1

2n!

[

dn

dzn

(z2

z − log (1 + z)

)(n+1)/2]

z=0

,

and hence

√

2

πΓ

(

n+1

2

)

(−ic2n) =

√

2

πΓ

(

n+1

2

)(−1)n

2 (2n)!

[

d2n

dz2n

(z2

z − log (1 + z)

)n+1/2]

z=0

= (−1)n γn.

Here γn denotes the Stirling Coefficients given in Example 1.2.1. Therefore,

1

Γ (λ)∼ eλλ−λ

√

λ

2π

∞∑

n=0

(−1)n γnλn

,

as λ → +∞. This expansion is also valid when |arg λ| ≤ π2− δ < π

2, with fixed

0 < δ ≤ π2. ♣

Example 2.3.2. As a second example we take the Airy Function

Ai(µ2)=

1

2πi

∫

L

eµ2t− t3

3 dt,

30

2.3. Examples

where µ > 0 and the path L consists of two rays, one with end points ∞e43πi and

0, and the other with end points 0 and ∞e23πi. The substitution t = µz yields

Ai(µ2)=

µ

2πi

∫

L

eµ3

(

z− z3

3

)

dz,

where L can be taken as the same as in the original integral. Using the notationsof Section 2.1, f (z) = z − z3

3, g(z) ≡ 1 and λ = µ3. The saddle points of f are

z0 = −1 and z1 = 1, we choose the former one. The steepest paths through z0 aredescribed by the equation

y(y2 − 3x2 + 3

)= 0.

The equation y = 0 represents the path of steepest ascent whereas the equationy2− 3x2+3 = 0 (left branch of a hyperbola) gives the path of steepest descent. It

is clear that L can be deformed into Cdef= {(x, y) : y2 − 3x2 + 3 = 0, x < 0}. We

put

f (z)− f (−1) = z − z3

3+

2

3= −τ.

We have

f (z)− f (−1) = (z + 1)2 − (z + 1)3

3

as z → −1, thus

z± = −1 +∞∑

n=0

(±1)n αnτn/2, (2.27)

dz±dτ

=∞∑

n=0

(±1)n+1 cnτ(n−1)/2, cn =

n+ 1

2αn+1, (2.28)

where z± denotes the upper and lower path of C through the saddle point z0 = −1.Since

dz±dτ

=1

z2± − 1,

and τ (1) = −43, the expansions (2.27)-(2.28) are converget in the disc |τ | < 4

3.

Watson’s Lemma then gives

Ai(µ2)=

µ

2πi

∫ +∞

0

e−23µ3−µ3τ

(dz+dτ

− dz−dτ

)

dτ

∼ e−23µ3

2√πµ

∞∑

n=0

Γ

(

n+1

2

) −2ic2n√πµ3n

,

as µ → +∞. From Perron’s Formula (2.5) we find

cn =1

2n!

[

dn

dzn

(3

z − 2

)(n+1)/2]

z=−1

=(−i)n+1

2n!

[dn

dzn

(

1− z

3

)−(n+1)/2]

z=0

=(−i)n+1

2

Γ(3n+1

2

)

3nn!Γ(n+12

) ,

31

2.3. Examples

and thus

Ai(µ2)∼ e−

23µ3

2√πµ

∞∑

n=0

(−1)nΓ(3n+ 1

2

)

√π9n (2n)!

1

µ3nas µ → +∞.

This expansion is also valid when |arg µ3| ≤ π2− δ < π

2, with fixed 0 < δ ≤ π

2. The

substitution z = µ2 leads to the final form

Ai (z) ∼ e−23z3/2

2√πz1/4

∞∑

n=0

(−1)nΓ(3n+ 1

2

)

√π9n (2n)!

1

z3n/2,

as |z| → +∞ in the sector |arg z| ≤ π3− ε < π

3, with fixed 0 < ε ≤ π

3. ♣

Example 2.3.3. Our last example is the integral

I (λ, ν) =1√π

∫ +∞

0

eλ(2z−z2)zνdz, Re(ν) > −1.

We shall use Perron’s Method to obtain the asymptotic expansion of I (λ, ν) forfixed ν and |λ| → +∞ in an appropriate sector of the complex plane. We split upthe integral into two parts as follows

I (λ, ν) =1√π

∫ +∞

0

eλ(1−z2) (1 + z)ν dz +1√π

∫ 1

0

eλ(1−z2) (1− z)ν dz.

Using the notations of Section 2.2, f (z) = 1 − z2, g+(z) = (1 + z)ν , g−(z) =(1− z)ν . The only saddle point of f is z0 = 0. We have f (0)− f (z) = z2 and

g± (z) =∞∑

k=0

(±1)k(ν

k

)

zk,

as z → 0. We can apply Theorem 2.2.1 with α = 2, β = 1:

∫ +∞

0

eλ(1−z2) (1 + z)ν dz ∼ eλ∞∑

n=0

Γ

(n+ 1

2

)cn

λ(n+1)/2,

∫ 1

0

eλ(1−z2) (1− z)ν dz ∼ eλ∞∑

n=0

Γ

(n+ 1

2

)(−1)n cnλ(n+1)/2

,

as |λ| → +∞ in the sector |arg λ| ≤ π2− δ < π


2. Here

cn =1

2n!

[dn

dzn(1 + z)ν

]

z=0

=1

2

(ν

n

)

.

After simplification, the final result is

I (λ, ν) ∼ eλ√λ

∞∑

n=0

(ν

2n

)(2n)!

22nn!

1

λn,

as |λ| → +∞ and |arg λ| ≤ π2− δ < π


2. ♣

32

Appendix A

Combinatorial objects

A.1 Ordinary Potential Polynomials

Let F (x) = 1 +∑∞

n=1 fnxn be a formal power series. For any complex number ρ,

we define the Ordinary Potential Polynomial Aρ,n (f1, f2, . . . , fn) (associated to F )by the generating function

(F (x))ρ =

(

1 +∞∑

n=1

fnxn

)ρ

=∞∑

n=0

Aρ,n (f1, f2, . . . , fn) xn.

The first few are Aρ,0 = 1, Aρ,1 = ρf1, Aρ,2 = ρf2 +(ρ2

)f 21 , and in general

Aρ,n (f1, f2, . . . , fn) =∑

(ρ

k

)k!

k1!k2! · · · kn!fk11 fk2

2 · · · fknn , (A.1)

where the sum extending over all sequences k1, k2, . . . , kn of non-negative integerssuch that k1 + 2k2 + · · ·+ nkn = n and k1 + k2 + · · ·+ kn = k. We write

Aρ,n (f1, f2, . . . , fn) =n∑

k=1

(ρ

k

)

Bn,k (f1, f2, . . . , fn−k+1),

where the Bn,k (f1, f2, . . . , fn−k+1)’s are called the Partial Ordinary Bell Polyno-mials. From (A.1) it follows that

Bn,k (f1, f2, . . . , fn−k+1) =∑ k!

k1!k2! · · · kn−k+1!fk11 fk2

2 · · · fkn−k+1

n−k+1 . (A.2)

Here the sum runs over all sequences k1, k2, . . . , kn of non-negative integers suchthat k1 + 2k2 + · · ·+ (n− k + 1) kn−k+1 = n and k1 + k2 + · · ·+ kn−k+1 = k.

Since (F (x))ρ = (F (x))ρ−1 F (x), we have the recurrence

Aρ,n (f1, f2, . . . , fn) = Aρ−1,n (f1, f2, . . . , fn−k) +n∑

k=1

fkAρ−1,n−k (f1, f2, . . . , fn−k).

Taking into account formula (A.2), one finds that

(F (x)− 1)k =

( ∞∑

n=1

fnxn

)k

=∞∑

n=k

Bn,k (f1, f2, . . . , fn−k+1) xn, (A.3)

33

A.2. The r-associated Stirling Numbers

and therefore

Bn,k+1 (f1, f2, . . . , fn−k) =n−k∑

k=1

fkBn−k,k (f1, f2, . . . , fn−2k+1). (A.4)

If G (x) =∑∞

n=0 gnxn is a formal power series, then by (A.3), we have

G (y (F (x)− 1)) =∞∑

n=0

(n∑

k=0

gkBn,k (f1, f2, . . . , fn−k+1) yk

)

xn.

Specially,

exp

(

y∞∑

n=1

fnxn

)

=∞∑

n=0

(n∑

k=0

Bn,k (f1, f2, . . . , fn−k+1)

k!yk

)

xn.

For more details see, e.g., Riordan’s book [11, p. 189].

A.2 The r-associated Stirling Numbers

For every nonnegative integers r and k we define the r-associated Stirling Numbersof the First and Second Kind by the generating functions

1

k!

(

− log (1− x)−r∑

m=1

xm

m

)k

=∞∑

n=(r+1)k

sr (n, k)xn

n!,

1

k!

(

ex −r∑

m=0

xm

m!

)k

=∞∑

n=(r+1)k

Sr (n, k)xn

n!. (A.5)

If r = 0 then s (n, k)def= s0 (n, k) and S (n, k)

def= S0 (n, k) are the Stirling Numbers

of the First and Second Kind, respectively.It follows that

exp

(

y

(

− log (1− x)−r∑

m=1

xm

m

))

=∞∑

n=0

⌊n/r+1⌋∑

j=0

sr (n, j) yj

xn

n!,

exp

(

y

(

ex −r∑

m=0

xm

m!

))

=∞∑

n=0

⌊n/r+1⌋∑

j=0

Sr (n, j) yj

xn

n!.

It is known (see [4]) that

sr (n, k) = n!k∑

j=0

(−1)j sr−1 (n− rj, k − j)

rjj! (n− rj)!,

Sr (n, k) = n!k∑

j=0

(−1)j Sr−1 (n− rj, k − j)

(r!)j j! (n− rj)!.

For further identities see, e.g., Howard’s paper [4].

34

Bibliography

[1] J. A. Campbell, P. O. Froman, E. Walles, Explicit series formulae forthe evaluation of integrals by the method of steepest descent, Studies in AppliedMathematics 77 (2) (1987), 151–172.

[2] E. T. Copson, Theory of Functions of a Complex Variable, Oxford UniversityPress, London, 1935.

[3] R. B. Dingle, Asymptotic Expansions: Their Derivation and Interpretation,Academic Press, New York, 1973.

[4] F. T. Howard, Associated Stirling Numbers, Fibonacci Quart. 18 (1980),303–315.

[5] F. T. Howard, A theorem relating potential and Bell polynomials, DiscreteMathematics 39 (2) (1982), 129–143.

[6] J. L. Lopez, P. Pagola, E. P. Sinusıa, A simplification of Laplace’smethod: Applications to the Gamma function and Gauss hypergeometric func-tion, Journal of Approximation Theory 161 (1) (2009), 280–291.

[7] G. Nemes, On the coefficients of the asymptotic expansion of n!. J. IntegerSeqs. 13 (6) (2010), Article 10.6.6, pp. 5.

[8] F. W. J. Olver, Asymptotics and Special Functions, Academic Press, NewYork, 1974.

[9] R. B. Paris, Hadamard Expansions and Hyperasymptotic Evaluation: AnExtension of the Method of Steepest Descents, Cambridge University Press,2011.

[10] O. Perron, Uber die naherungsweise Berechnung von Funktionen großerZahlen, Sitzungsberichte der Koniglich Bayerischen Akademie der Wis-senschaften (1917), 191–219.

[11] J. Riordan, Combinatorial Identities. Wiley, New York, 1979.

[12] J. Wojdylo, Computing the coefficients in Laplace’s method, SIAM Review48 (1) (2006), 76–96.

[13] J. Wojdylo, On the coefficients that arise from Laplace’s method, Journalof Computational and Applied Mathematics 196 (1) (2006), 241–266.

[14] R. Wong, Asymptotic Approximations of Integrals, Society for Industrial andApplied Mathematics, 2001.

35

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