Asymptotic Expansions

V 5.7.9 2019/10/08

The interest in asymptotic analysis originated from the necessity of search-ing for approximations to functions close the point(s) of interest. Suppose wehave a function f(x) of single real parameter x and we are interested in anapproximation to f(x) for x “close to” x0. The function f(x) may be undefinedat x = x0, however the limit limx→x0

f(x) = A must exists, finite, 0 or ∞, asx0 is approached from below or from above.

An approximation to f(x) for x close to x0 is a function g(x) which is closeto f(x) as x→ x0. The formal definition is:

Definition. Let D j R, x0 a condensation point of D, and f(x), g(x) and η(x)real functions defined on D \ x0, i.e., on D excluding x0. The function g(x) isan approximation to the function f(x) to order η(x) as x→ x0 in D if,1

limx→x0

f(x)− g(x)

η(x)= 0,

i.e., if for any small positive number δ there exists a punctured neighborhoodUδ(x0) of x0 such that∣∣f(x)− g(x)

∣∣ ≤ δ ∣∣η(x)∣∣ for x ∈ Uδ(x0) ∩D.

The function η(x) is called gauge function and intuitively measures how closeg(x) is to f(x) as x→ x0.

Searching for an approximation to f(x) as x→ x0 actually requires to knowthe rate at which f(x) → A as x → x0, i.e., how the function f(x) approachesasymptotically the value A as x → x0 along the chosen direction. In otherwords, the asymptotic behavior of f(x) as x→ x0.

The above example is a particular case of a single real variable function.In many problems we deal with functions f(x1, . . . , xn; ε1, . . . , εm) dependingon two sets of arguments: the set (x1, . . . , xn) called variables and the set(ε1, . . . , εm) called parameters. The distinction between the two sets is notintrinsic and depends on the context.

For functions of more that one argument the above definition must be slightlymodified. Suppose we look for an approximation to the function f(x; ε) for x

1The limit is evaluated in the domain D \ x0. This will be always understood in thefollowing.

Preprint submitted to Elsevier October 8, 2019

in some region D as the parameter ε approaches the value ε → 0+. Withoutloss of generality we assume the distinguished value of 0+ for parameters. Anapproximation to f(x; ε) is a function g(x; ε) which is uniformly close to f(x; ε)as ε→ 0+ for all x in D. Thus the above definition becomes:

Definition. Given the real functions f(x, ε) and g(x, ε) defined for x in thedomain D j R and ε in the interval I : 0 < ε ≤ ε0, and the real functionη(ε) defined for ε ∈ I. The function g(x; ε) is an approximation to the functionf(x; ε) to order η(ε) for all x ∈ D as ε→ 0+ if

limε→0+

f(x; ε)− g(x; ε)

η(ε)= 0 uniformly in D,

i.e., if for any small positive number δ there exists a punctured neighborhoodIδ : 0 < ε ≤ εδ of ε = 0 such that for all x ∈ D∣∣f(x; ε)− g(x; ε)

∣∣ ≤ δ ∣∣η(ε)∣∣ for ε ∈ Iδ ∩ I,

with δ and εδ independent of x.

These definitions not only enlighten that searching for approximation closeto points of interest naturally leads to an asymptotic analysis. What is mostimportant, they show that the asymptotic behavior of functions can be studiedby comparing them with known gauge functions. The simplest gauge functionsare powers, for example for ε→ 0+

. . . , ε−2, ε−1, 1, ε, ε2, . . .

but other choices are possible. The corollary is that fundamental to asymptoticanalysis is to determine the relative order of functions.

1. The Order Relations

The concept of order of magnitude of functions is central in asymptoticanalysis. The precise mathematical definition of the intuitive idea of sameorder of magnitude, smaller order of magnitude and equivalent is provided bythe Bachmann-Landau symbols “O”, “o” and “∼”.

1.1. The O and ∼ symbols

The formal definition of the O-symbol used in asymptotical analysis is:

Definition. Let D j R, x0 a condensation point of D and f, g : D \ x0 → Rreal functions defined on D \ x0. Then f = O(g) as x→ x0 in D if there existsa positive constant C and a punctured neighborhood UC(x0) of x0 such that

|f(x)| ≤ C |g(x)| for all x ∈ UC(x0) ∩D. (1)

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We say that f(x) is asymptotically bounded by g(x), or simply f(x) is “orderbig O” of g(x), as x→ x0 in D. If the ratio f/g is defined then (1) implies that|f/g| is bounded from above by C in UC(x0) ∩D, or, equivalently that

limx→x0

∣∣∣∣f(x)

g(x)

∣∣∣∣ <∞.These definitions do not inquire into the fate of f and g in x0; asymptotic

analysis concerns with the asymptotic behavior of the functions in the neigbor-hood of the point x0. In x0 the functions f and g can be defined or not. Iff = O(g) as x → x0 for all x0 in D then f is bounded by g everywhere in D.In this case we say that f = O(g) in D.

The O-symbol provides a one-side bound, the statement f = O(g) as x →x0 indeed does not necessarily imply that f and g are of the same order ofmagnitude. Consider for example the functions f = xα and g = xβ , where αand β are constants with α > β. Then xα = O(xβ) as x → 0+ because thechoice C = δα−β and UC(0) : 0 < x < δ satisfies the requirement (1). Clearly,the reverse xβ = O(xα) as x → 0+ is not true. Notice that xα/xβ → 0 whilexβ/xα → +∞ as x→ 0+.

Two functions f and g are strictly of the same order of magnitude if f = O(g)and g = O(f) as x → x0. To stress this point one sometimes introduces thenotation

f = Os(g), as x→ x0

for f strictly of order g to indicate f = O(g) and g = O(f) as x → x0. Inthis case limx→x0

|f/g| exists and it is neither zero or infinity. We shall notemphasize the distinction and use only the symbol “O”. However, the possibleone-side nature of O-symbol should always be kept in mind.

If the limx→x0(f/g) exists and is equal to 1 we say that f(x) is asymptoticallyequal to g(x) as x→ x0, or simply f(x) “goes as” g(x) as x→ x0, and write

f ∼ g, as x→ x0.

For example, as x→ 0

sinx ∼ x, sin(7x) = O(x), 1− cosx = O(x2)

log(1 + x) ∼ x, ex ∼ 1, sin(1/x) = O(1).

The assertion 1 = O(sin(1/x)) as x → 0 is not true because sin(1/x) vanishesin every neighborhood of x = 0. Similarly as x→ +∞

cosh(3x) = O(e3x), log(1 + 7x) ∼ log x,

N∑n=1

anxn = O

(xN).

In the case of functions f(x; ε), defined for x in a domain D and ε in theinterval I : 0 < ε ≤ ε0, we say that

f(x; ε) = O[g(x; ε)

]as ε→ 0+ in D,

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if for each x inD there exist a positive number Cx and a punctured neighborhoodIδ(x) : 0 < ε ≤ εδ(x) of ε = 0, such that

|f(x; ε)| ≤ Cx |g(x; ε)| for all ε ∈ Iδ(x) ∩ I. (2)

If the ratio f/g exists then from (2) follows that

limε→0+

∣∣∣∣f(x; ε)

g(x; ε)

∣∣∣∣ <∞, for all x ∈ D.

The limit may depend on x. If the positive constant Cx and εδ(x) do notdepend on x for all x ∈ D then f(x; ε) = O(g(x; ε) as ε → 0+ uniformly in D.For example

cos(x+ ε) = O(1) as ε→ 0 uniformly in R,eεx − 1 = O(ε) as ε→ 0 nonuniformly in R.

A relation not uniform in D can be uniform in a subdomain of D. For example,the second example is uniform in the subdomain [0 : 1] ⊂ R.

As in the case of a single variable function, if limε→0+(f/g) = 1 for all x ∈ Dwe say that f is asymptotic equal to g in D and write

f(x; ε) ∼ g(x; ε) as ε→ 0+ for all x ∈ D.

1.2. The o-symbol

The formal definition of the o-symbol is:

Definition. Let D j R, x0 a condensation point of D and f, g : D \ x0 → Rreal functions defined on D \ x0. We say f = o(g) as x → x0 in D if for everyδ > 0 there exists a punctured neighborhood Uδ(x0) of x0 such that

|f(x)| ≤ δ |g(x)| for all x ∈ Uδ(x0) ∩D. (3)

This inequality indicates that |f | becomes arbitrarily small compared to |g|as x→ x0 in D. Thus, if the ratio f/g is defined, (3) implies that

limx→x0

∣∣∣∣f(x)

g(x)

∣∣∣∣ = 0.

Notice that f = o(g) as x→ x0 always implies f = O(g) as x→ x0 not in strictsense. The converse is not true. Asymptotic equivalence f(x) ∼ g(x) as x→ x0means that f(x) = g(x) + o

(g(x)

)as x→ x0.

If f = o(g) as x→ x0 we say that f(x) is asymptotically smaller than g(x)as x → x0, or simply f(x) is “order little o” of g(x) as x → x0. Often thealternative notation f � g as x → x0, or f “much smaller than” g as x → x0,is used in place of f = o(g).

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For example, as x→ 0

x3 = o(x), sin(7x) = o(1),

cosx = o(x−1

), log(1 + x2) = o(x),

ex − 1 = o(x−1/3

), e−1/x

2

= o(xn)

for all n ∈ N,

while as x→ +∞

x2 = o(x3), log x = o(x),

N∑n=1

anxn = o

(xN+1

).

Generalization to functions f(x, ε), defined for x in a domain D and ε in theinterval I : 0 < ε ≤ ε0, is immediate. The statement

f(x, ε) = o[g(x, ε)

]as ε→ 0+ for x ∈ D,

means that for each x in D and any given δ > 0 there exist a punctured neigh-borhood Iδ(x) : 0 < ε ≤ εδ(x) of ε = 0, such that

|f(x, ε)| ≤ δ |g(x, ε)| for all ε in Iδ(x) ∩ I.

This inequality implies that if the ratio f/g exists then

limε→0+

∣∣∣∣f(x; ε)

g(x; ε)

∣∣∣∣ = 0, for all x ∈ D.

If εδ is independent of x we say that f(x; ε) = o(g(x; ε)) as ε→ 0+ uniformlyin D. For example

cos(x+ ε) = o(ε−1) as ε→ 0+ uniformly in R,

eεx − 1 = o(ε1/2) as ε→ 0+ nonuniformly in R.

eεx − 1 = o(ε1/2) as ε→ 0+ uniformly in D : 0 ≤ x ≤ 1.

The uniform validity of the order relation is more important for the O-symbol.In asymptotic analysis the O (in strict sense) and ∼ symbols are usually morerelevant than the o-symbol because the latter hide so much information. We aremore interested to know how a function behaves close to some point, rather thanjust knowing that it is much smaller than some other functions. For example,sinx = x + o

(x2)

as x → 0 tells us that sinx − x vanishes faster than x2 as

x→ 0, while sinx = x+O(x3)

as x→ 0 (in strict sense) tells us that sinx− xvanishes as x3 as x→ 0.

2. Asymptotic Expansion

An asymptotic expansion describes the asymptotic behavior of a functionclose to a point in terms of some reference functions, the gauge functions.

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The definition was introduced by Poincare (1886) and provides a mathemat-ical framework for the use of many divergent series.

We shall call the set of functions ϕn : D\x0 → R, n = 0, 1, . . . , an asymptoticsequence as x→ x0 in D \ x0, if for each n we have

ϕn+1(x) = o[ϕn(x)

]as x→ x0 in D.

Some examples of asymptotic sequence are:

Example 2.1. The functions ϕn(x) = (x− x0)n form an asymptotic sequenceas x→ x0.

Example 2.2. The functions ϕn(x) = xn/2, ϕn(x) = (log x)−n and ϕn(x) =(sinx)n are examples of asymptotic sequences as x → 0+, while the functionsϕn(x) = x−n form an asymptotic sequence as x→ +∞.

Example 2.3. The sequence {log x, x2, x3, . . . } form an asymptotic sequenceas x→ 0+.

Example 2.4. The sequence {. . . , x−2, x−1, 1, x, x2, . . . } form an asymptoticsequence as x → 0+, while the sequence {. . . , x2, x, 1, x−1, x−2, . . . } form anasymptotic sequence as x→ +∞.

The asymptotic expansion of a function f(x) is defined as:

Definition. Let the set of functions ϕn(x), n = 0, 1, 2, . . . , defined on thedomain D \ x0 form an asymptotic sequence as x→ x0 in D and an a sequenceof numbers. The formal series

∑∞n=0 anϕn(x) is an asymptotic expansion of

f(x) as x→ x0 in D with gauge functions ϕn(x) if, and only if, for any N :

f(x)−N∑n=0

an ϕn(x) = o[ϕN (x)

]as x→ x0 in D,

or, equivalently,

f(x)−N−1∑n=0

an ϕn(x) = O[ϕN (x)

]as x→ x0 in D. (4)

In this case we shall write

f(x) ∼∞∑n=0

an ϕn(x), as x→ x0 in D.

Asymptotic expansion of this form are also called asymptotic series.The definition of the asymptotic expansion of a single variable function is

not general enough to handle functions of more than one variable. For them weneed the more general definition:

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Definition. Let the functions ϕn(ε), n = 0, 1, 2, . . . , with ε ∈ I : 0 < ε ≤ ε0form an asymptotic sequence as ε→ 0+. We say that the sequence of functionsgn(x; ε) form an uniform asymptotic sequence of approximations to f(x; ε) inthe domain x ∈ D with gauge functions ϕn(ε) as ε→ 0+ if

limε→0+

f(x; ε)− gn(x; ε)

ϕn(ε)= 0, for all n, (5)

uniformly in D.

The boundary of the domain D where the approximation is uniformly validmay depend on ε, and also on n. Domains with ε-dependent boundaries areessential in asymptotic analysis because the gauge functions are not unique.

The definition (5) is not an asymptotic expansion of f(x; ε); to find an asymp-totic expansion of f(x; ε) one constructs a sequence of functions an(x; ε) suchthat

gn(x; ε) =

n∑m=0

am(x; ε), (6)

and (5) holds. The formal series∑∞n=0 an(x; ε) is a generalized uniform asymp-

totic expansion of f(x; ε) and we write

f(x; ε) ∼∞∑n=0

an(x; ε) as ε→ 0+, (7)

uniformly in D. The formal expression (7) is just the restatement of the defini-tion (5) with the particular choice (6). The leading term a0(x; ε) of the seriesis called asymptotic representation of f(x; ε) as ε → 0+. Notice that nothingis said about the convergence of the series. The series (7) is just formal andgenerally it does not converge; if it converges it may not converge to f(x; ε).

In most cases the functions an(x; ε) do not have a simple expression in termsof ϕn(ε), and finding them may be rather difficult. To overcome this problem,Poincare introduced a definition in terms of the functions ϕn(ε).

Definition. The uniform asymptotic expansion of the function f(x; ε) in thedomain x ∈ D and ε in the interval I : 0 < ε ≤ ε0, as ε → 0+ in the sense ofPoincare, or simply of Poincare type, is an asymptotic expansion of the form

f(x; ε) ∼∞∑n=0

an(x)ϕn(ε), as ε→ 0+ uniformly for x ∈ D. (8)

The functions an(x) are independent of ε and play the role of coefficients of theasymptotic expansion.

Poincare type asymptotic expansions have useful properties not shared bygeneralized asymptotic expansion, for example they can be added or integratedterm by term, hence they are largely used in asymptotic analysis. Their formis, however, not general enough to handle all problems that can be studied with

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perturbation techniques. Nevertheless, they can generally be used as startingpoint to construct more suitable generalized asymptotic expansions.

Since Poincare type expansions simply extend the definition of asymptoticexpansion of functions f(x) to functions f(x; ε), they are usually introducedusing the following equivalent definition:

Definition. Let the functions ϕn(ε), n = 0, 1, 2, . . . , with ε ∈ I : 0 < ε ≤ ε0form an asymptotic sequence as ε→ 0+ and an(x) a sequence of functions inde-pendent of ε defined on the domain D. Then the formal series

∑∞n=0 an(x)ϕn(ε)

is a uniform asymptotic expansion of Poincare type of f(x; ε) in the domainx ∈ D with gauge functions ϕn(ε) as ε→ 0+ if, and only if, for any N :

f(x; ε)−N∑n=0

an(x)ϕn(ε) = o[ϕN (ε)

], as ε→ 0+,

or, equivalently,

f(x; ε)−N−1∑n=0

an(x)ϕn(ε) = O[ϕN (ε)

], as ε→ 0+,

uniformly for x ∈ D.

2.1. Uniqueness of Poincare type Expansions and Subdominance

The asymptotic expansion of a function is not unique because there existsan infinite number of asymptotic sequences that can be used in the expansion.However, given an asymptotic sequence of gauge functions ϕn the Poincare typeexpansion in terms of ϕn is unique. Consider for simplicity a function f(x) ofa single variable, the result holds for the Poincare type expansions of functionsf(x; ε) as well. From the definition (4) it follows that

limx→x0

f(x)−N−1∑n=0

anϕn(x)

ϕN (x)= aN .

Thus the coefficients an are uniquely determined by the gauge functions ϕn; ifthe function f(x) admits an asymptotic expansion with respect to the sequenceof gauge functions ϕn(x) then the expansion is unique.

The converse is not true. For any asymptotic sequence ϕn(x) as x → x0there always exists a function ψ(x) which is transcendentally small with respectto the sequence, i.e.,

ψ = o(ϕn), ∀n, as x→ x0.

One may then form a sequence of transcendentally small terms, by say positivepowers of ψ. As a consequence an asymptotic expansion does not determinesthe function f uniquely. Different functions may have the same asymptoticexpansion with respect to the given asymptotic sequence of gauge functions ϕn.

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Example 2.5. For any constant c 6= 0, the functions

f(x) =1

1− xand g(x) =

1

1− x+ ce−1/x,

have the same asymptotic expansion as x→ 0+ in terms of the gauge functionsϕn(x) = xn:

f(x) ∼ g(x) ∼ 1 + x+ x2 + . . . as x→ 0+,

because e−1/x is transcendentally small with respect to xn: e−1/x = o(xn) asx→ 0+ for every n ∈ N.

An asymptotic expansion establishes an equivalence relation among func-tions: if for all n

f(x)− g(x) = o[ϕn(x)

]as x→ x0,

then the functions f and g are asymptotically equivalent as x→ x0 with respectto the gauge functions ϕn. The function h(x) = f(x) − g(x) is said subdom-inant to the asymptotic expansion with gauge functions ϕn as x → x0. Anasymptotic expansion is, thus, asymptotic to a class of functions that differ oneanother by subdominant functions transcendentally small with respect to thegauge function ϕn as x→ x0.

Notice that the concept of transcendentally smallness is an asymptotic one.Transcendentally small terms might be numerically important for values of xnot too close to x0, even for relatively small non-zero values of x− x0.

2.2. Uniformly vs Nonuniformly valid Asymptotic Expansions

In perturbation calculations one often encounters situations where the quan-tity to be expanded is a function f(x; ε), where ε is a (small) perturbation pa-rameter and x is a variable independent of ε varying in a domain D. In thesecases one generally constructs an asymptotic expansion of the Poincare type(8) with some gauge functions ϕn(ε). The functions ϕn(ε) are not uniquelydetermined by the problem. Rather, these are usually found in the course ofthe solution of the problem. Due to this arbitrariness different asymptotic ap-proximations can be constructed. Only asymptotic expansions uniform in thedomain D, however, lead to useful approximations to f(x; ε).

An asymptotic expansion of f(x; ε), e.g.∑∞n=0 an(x)ϕn(ε), is only formal.

However the partial sum of the first N terms is a well defined object for anyfinite N and gives an approximation to the function f(x; ε). Thus, we can write

f(x; ε)−N−1∑n=0

an(x)ϕn(ε) = RN (x; ε)

where RN (x; ε) is the reminder or error of the approximation. The asymptoticproperties of the gauge functions ϕn(ε) ensures that for any fixed N and x ∈ D:

RN (x; ε) = O[ϕN (ε)

], as ε→ 0. (9)

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The asymptotic expansion gives both an approximation to f(x; ε) as ε → 0and the estimation of the error in the approximation. The error RN (x; ε) is ingeneral a function of both ε and x. For a useful approximation, however, theerror must not depend upon the value of x, hence (9) must be valid uniformlyin D. Useful asymptotic expansions of f(x; ε) are only those which are uniformin D.

Notice that since ϕn(ε) = o[ϕn−1(ε)

]as ε → 0 the requirement of uniform

validity implies that each coefficient an(x) of a Poincare type expansion cannotbe more singular than an−1(x) for all x in the domain D. In other words, asε → 0 each term an(x)ϕn(ε) must be a small correction to the preceding termirrespective of the value of x.

Example 2.6. The expansion

sin(x+ ε

)=

(1− ε2

2+ . . .

)sinx+

(ε− ε3

3!+ . . .

)cosx

= sinx+ ε cosx− ε2

2sinx− ε3

3!cosx+ . . . , as ε→ 0,

is a uniform asymptotic expansion in R. The coefficients of the power εn arebounded for all x ∈ R, thus an(x) is not more singular than an−1(x) and theexpansion is uniform in R.

Example 2.7. Consider the function

f(x; ε) =1

x+ ε,

defined for all x ≥ 0 and ε > 0. If x > 0, then

f(x; ε) ∼ 1

x

[1− ε

x+ε2

x2+ . . .

], as ε→ 0+. (10)

Each term of this expansion is singular at x = 0, and more singular than thepreceding one. The expansion is not uniform in the domain D : x ≥ 0. It is,nevertheless, uniform in the domain D1 : 0 < x0 ≤ x where x0 is a positivenumber independent of ε. Uniformity breaks down as we get too close to x = 0,signaling that some change must occurs near x = 0. Indeed for x = 0 theexpansion has a completely different form:

f(0; ε) ∼ 1

ε, as ε→ 0+. (11)

Generally speaking, the uniform validity of the expansion (10) breaks when twosuccessive terms become of the same order, i.e.,

ε

x= O(1) ⇒ x = O(ε) as ε→ 0+.

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If we introduce the variable ξ = x/ε and rewrite f(x; ε) in terms of ξ the differentasymptotic expansion emerges:

f(εξ; ε) ∼ 1

ε

1

ξ + 1, as ε→ 0+. (12)

The expansion (12) is uniform in the domain D2 : 0 ≤ ξ ≤ ξ0 < 1/ε, with ξ0a constant independent of ε. Notice that D2 differs from D1. Nevertheless theexpansion (12) matches with (11) in the limit ξ → 0+ and with (10) as ξ � 1.We shall give the precise meaning of the vague word “matches” later, discussingasymptotic matching.

2.3. Asymptotic versus Convergent series

Convergence of a series is an intrinsic property of the coefficients of theseries, one can indeed prove the convergence of a series without knowing thefunction to which it converges. Asymptotic is, instead, a relative concept. Itconcerns both the coefficients of the series and the function to which the seriesis asymptotic. To prove that an asymptotic series is asymptotic to the functionf(x) both the coefficients of the series and the function f(x) must be considered.As a consequence, a convergent series need not be asymptotic and an asymptoticseries need not be convergent.

To understand the difference between convergence and asymptotic considerthe asymptotic series

∑∞n=0 anϕn(x), where an are some coefficients and ϕn(x)

an asymptotic sequence as x→ x0 in a domain D. The series is in general onlyformal because it may not converge. The partial sum

SN (x) =

N−1∑n=0

an ϕ(x)

is, nevertheless, well defined for any N . The convergence of the series is con-cerned with the behavior of SN (x) as N → ∞ for fixed x, whereas the asymp-totic property as x→ x0 with the behavior of SN (x) as x→ x0 for fixed N .

A convergent series defines for each x a unique limiting sum S∞(x). However,at difference with an asymptotic expansion, it does not provide information onhow rapidly it converges nor on how well SN (x) approximates S∞(x) for fixedN . An asymptotic series, on the contrary, does not define a unique limitingfunction f(x), but rather a class of asymptotically equivalent functions. Henceit does not provide an arbitrary accurate approximation to f(x) for x 6= x0.Still, the partial sums SN (x) provide good approximations to the value of f(x)for x sufficiently close to x0. The approximation, however, depends on howx is close to x0. This leads to the problem of the optimal truncation of theasymptotic expansion, i.e., the optimal number of terms to be retained in thepartial sum.2

The following example illustrate these differences.

2The optimal truncation can be affected by the presence of transcendentally small sub-dominant functions because they may give finite contributions.

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Example 2.8. Consider the error function, or Gauss error function, defined as

erf(x) =2√π

∫ x

0

e−t2

dt. (13)

Since the Maclaurin series of ez is absolutely convergent for all z ∈ C, expand-ing e−t

2

in power series of −t2 and integrating term by term, we obtain theconvergent power series expansion of erf(x):

erf(x) =2√π

[x− 1

3x3 +

1

2 · 5x5 + . . .

]=

2√π

∞∑n=0

(−1)n

(2n+ 1)n!x2n+1.

The series converges for all x ∈ R because the ratio between the (n+ 1)-th andthe n-th term ∣∣∣∣un+1

un

∣∣∣∣ =2n+ 1

2n+ 3

x2

n+ 1,

is smaller than 1 for all x ∈ R and n large enough. For moderate values of xthe convergence is, however, very slow. For example, 31 terms are needed tohave an approximation to erf(3) with an accuracy of 10−5. A better result canbe obtained using an asymptotic expansion.

To derive an asymptotic expansion of erf(x) we rewrite (13) as,

erf(x) = 1− 2√π

∫ ∞x

e−t2

dt

= 1− 1√π

∫ ∞x2

s−1/2e−s ds.

Integrating by parts twice we get:

erf(x) = 1− 1√π

[e−x

2

x− 1

2

∫ ∞x2

s−3/2e−s ds

]

= 1− 1√π

[e−x

2

x− e−x

2

2x3+

3

4

∫ ∞x2

s−5/2e−s ds

].

The procedure can be easily iterated to any order N . Introducing the functions

Fn(x) =

∫ ∞x2

s−n−1/2e−s ds

= −[s−n−1/2e−s

∣∣∣∞x2

+

(n+

1

2

)∫ ∞x2

s−n−1/2−1e−s ds

]=

e−x2

x2n+1−(n+

1

2

)Fn+1(x),

12

the error function erf(x) can be written as

erf(x) = 1− 1√πF0(x)

= 1− 1√π

[e−x

2

x− 1

2F1(x)

]

= 1− 1√π

[e−x

2

(1

x− 1

2x3

)+

1 · 322

F2(x)

]= . . .

= 1− e−x2

√π

N−1∑n=0

(−1)n(2n− 1)!!

2nx2n+1+RN (x),

where (2n− 1)!! = 1 · 3 · · · · · (2n− 1), and

RN (x) = (−1)N1√π

(2N − 1)!!

2NFN (x).

The series diverges for any value of x because the ratio between the (n+ 1)-thand the n-th term∣∣∣∣un+1

un

∣∣∣∣ =(2n+ 1)!!

2n+1x2n+3× 2nx2n+1

(2n− 1)!!=

2n+ 1

2x2, (14)

is larger than 1 whenever 2n+ 1 > 2x2. In spite of that,

|RN (x)| = CN

∣∣∣∣∫ ∞x2

s−N−1/2e−s ds

∣∣∣∣≤ CNx2N+1

∫ ∞x2

e−s ds

≤ CNe−x

2

x2N+1,

where CN = (2N − 1)!!/(2N√π). Thus,

erf(x)−

[1− e−x

2

√π

N−1∑n=0

(−1)n(2n− 1)!!

2n1

x2n+1

]= O

( e−x2

x2N+1

), as x→∞,

and the series gives an asymptotic expansion of erf(x) for large x:

erf(x) ∼ 1− e−x2

√π

∞∑n=0

(−1)n(2n− 1)!!

2n1

x2n+1, as x→∞. (15)

Only the first two terms of the expansion (15) suffice to have an approxi-mation to erf(3) with an accuracy of 10−5. The series, nevertheless, does notprovide an arbitrary accurate approximation to erf(x) and we cannot improve

13

the approximation at will by adding more terms. From (14) it follows that theratio between two successive terms of the series becomes larger than 1 as n & x2.Hence, the asymptotic series (15) has an optimal truncation for N equal to thelargest integer smaller than x2.

In this example we have encountered an asymptotic expansion of the form

f(x) ∼ g(x)

∞∑n=0

an ϕn(x), as x→ x0,

where the function g(x) is bounded as x → x0. In the example g(x) ∝ e−x2

,clearly bounded as x→∞. This reflects the non uniqueness of gauge functions:since g(x) is bounded as x → x0 the functions ϕn(x) and ϕn(x) = g(x)ϕn(x)form two different asymptotic sequence as x → x0. This provides additionalflexibility, but it should be used with care because differences might be numer-ically relevant for x not too close to x0.

Example 2.9. The asymptotic expansion of the function f(x) = 1/(1 − x) asx→ 0 with gauge function ϕn(x) = xn is:

1

1− x∼∞∑n=0

xn, as x→ 0. (16)

The series with gauge functions ϕn(x) = (1 + x)xn is also an asymptotic ex-pansion of f(x) as x → 0 because the function g(x) = 1 + x is bounded asx→ 0:

1

1− x∼∞∑n=0

(1 + x)xn, as x→ 0. (17)

Both series (16) and (17) provide an asymptotic expansion of f(x) = 1/(1− x)as x → 0, but the approximation to f(x) for x close to 0 are different. Thedifference between the two approximations decreases as x approaches 0, butfor small finite value of x it might be numerically relevant. For example, forx = 0.1 the first three terms of (16) give the approximantion 1.11 to the exactvalue 1.11111 . . . , while the first three terms of (17) give 1.221.

2.4. Asymptotic power expansion

Asymptotic expansions of the form

f(x) ∼∞∑n=0

an (x− x0)n, as x→ x0,

are called asymptotic power expansions, or asymptotic power series, and areamong the most common and useful asymptotic expansions.

14

One of the fundamental notions of real analysis is the Taylor expansion of in-finitely differentiable C∞ functions. If f(x) is a smooth, infinitely differentiableC∞ function in a neighborhood of x = x0, then the Taylor theorem ensures that

f(x) =

N∑n=0

f (n)(x0)

n!(x− x0)n + o

(|x− x0|N

), as x→ x0.

Hence, f(x) has the asymptotic power expansion

f(x) ∼∞∑n=0

f (n)(x0)

n!(x− x0)n as x→ x0. (18)

If, and only if, f(x) is analytic at x = x0 the asymptotic power series convergesto f(x) in a neighborhood |x − x0| < r of x0 with r > 0. In this case theseries gives the usual Taylor series representation of the function about x0.Nevertheless, the Taylor expansion of f(x) about the point x = x0 does notprovide an asymptotic expansion of f(x) as x → x1 6= x0, even if convergentfor |x1−x0| < r; partial sums of the Taylor expansion do not usually give goodapproximations to f(x) as x→ x1 6= x0.

If f(x) is C∞ but not analytic at x = x0 the series (18) need not convergein any neighborhood of x0, or if it does converge, its sum need not necessarilybe f(x). See Example 2.5, where the function g(x) is C∞ but not analytic atx = 0 for any c 6= 0.

For asymptotic power expansions the following theorem, due to Borel, pro-vides, in some sense, the inverse of the Taylor theorem. The Borel theoremstates that for any sequence of real numbers an there exists an infinitely differ-entiable C∞ function f(x) such that an = f (n)(x0)/n! and

f(x) ∼∞∑n=0

an (x− x0)n, as x→ x0.

The function f(x) need not to be unique. Hence, unlike convergent Taylorseries, there are no restrictions on the growth rate of the coefficients an of anasymptotic power series, as shown in the following example.

Example 2.10. Consider the integral (Euler, 1754):

f(x) =

∫ ∞0

dte−t

1 + xt, (19)

defined for x ≥ 0. Integrating by parts we get

f(x) = −∫ ∞0

dt1

1 + xt

d

dte−t = 1− x

∫ ∞0

dte−t

(1 + xt)2.

One more integration by parts gives,

f(x) = 1− x+ 2x2∫ ∞0

dte−t

(1 + xt)3,

15

and one more,

f(x) = 1− x+ 2x2 − 2 · 3x3∫ ∞0

dte−t

(1 + xt)4.

Thus, after N integration by parts:

f(x) =

N−1∑n=0

(−1)n n!xn +RN (x),

where

RN (x) = (−1)N N !xN∫ ∞0

dte−t

(1 + xt)N+1.

The coefficients of the power series grow as the factorial of n and the ratiobetween two successive terms |un/un−1| = nx becomes larger than 1 as n > 1/x.The series does not converge for any x > 0, nevertheless, for any N

|RN (x)| ≤ N !xN∫ ∞0

dte−t

(1 + xt)N≤ N !xN

∫ ∞0

dt e−t = N !xN . (20)

Thus RN (x) = o(xN−1) as x→ 0+, and∫ ∞0

dte−t

1 + xt∼∞∑n=0

(−1)nn!xn, as x→ 0+.

The power series is not convergent because the integrand in (19) has a nonin-tegrable singularity at t = −1/x and the integral is not defined for x < 0. Aconvergent power series about x = 0 would converge in a finite interval centredat x = 0, hence also for negative x.

Since the asymptotic power series does not converge, its partial sum doesnot provide an arbitrary accurate approximation to the integral (19) for anyfixed x > 0. However, from (20) it follows that for any fixed N the error ispolynomial in x. What is the optimal truncation of the series, that is the onewhich gives the best approximation? The ratio between two successive terms|un/un−1| = nx grows with n and becomes larger than 1 as n > 1/x. The bestapproximation occurs when N ∼ [1/x], for which RN ∼ (1/x)!x1/x. Using theStirlng’s formula N ! ∼

√2πNNNe−N valid for N � 1, the error of the optimal

truncation reads:

RN (x) ∼√

2π

xe−1/x, as x→ 0+.

Thus, while for arbitrary N the partial sum is polynomially accurate in x, theoptimal truncated sum is exponentially accurate.

2.5. Stokes Phenomenon

The asymptotic expansion as z → z0 of complex function f(z) with anessential singularity in z = z0 is typically valid only in wedge-shaped regions

16

α < arg(z − z0) < β, and the function has different asymptotic expansion indifferent wedges. For definitiveness we shall discuss the case z0 = ∞, howeverthe same scenario appears for essential singularities at any z0 ∈ C.

The change of the form of the asymptotic expansion across the boundaries ofthe wedges is called Stokes phenomenon. The origin of the Stokes phenomenoncan be understood as follows. If the function g(z) is asymptotically equal to thefunction f(z) as z →∞ this means that f(z) = g(z) +h(z) with h(z) subdomi-nant compared with g(z) as z →∞. As z approaches the boundary of the wedgeof validity of the expansion the subdominant h(z) grows in magnitude comparedto the dominant term g(z). At the boundary of the wedge the functions h(z) andg(z) are of the same order of magnitude and the distinction between dominantand subdominant disappears. As z crosses the boundary the functions h(z) andg(z) exchange their role and on the other side g(z) � h(z) as z → ∞. Thisexchange is the Stokes phenomenon, and it is typical of exponential functions.

Example 2.11. Consider the complex function

f(z) = sinh z =ez − e−z

2

In the half plane Re z > 0 both sinh z and ez grows exponentially as z → ∞and

sinh z ∼ 1

2ez, as z →∞, Re z > 0.

In this region the difference h(z) = − 12e−z between sinh z and g(z) = 1

2ez is

transcendentally small as z →∞.In the half plane Re z < 0 the function h(z) = − 1

2e−z is no longer transcen-

dentally small, and

sinh z ∼ −1

2e−z, as z →∞, Re z < 0.

In the region Re z < 0 the function g(z) = 12ez is subdominant.

The two leading asymptotic expansions 12ez and − 1

2e−z switch from being

dominant to subdominant on the line Re z = 0 (anti-Stokes lines), where theyare purely oscillatory. They are most unequal in magnitude on the line Im z = 0(Stoke lines), where they are purely real and either exponential growing ordecreasing as z →∞.

The Stokes and anti-Stokes lines are not unique and do not really have a pre-cise definition because the region where the function f(z) has some asymptoticexpansion is rather vague refelcting the non uniqueness of gauge functions. Inthe above example, we could equally well take as Stokes line any line Im z = a.As a general definition, the anti-Stokes lines are roughly where some term in theasymptotic expansion changes from increasing to decreasing. Hence, anti-Stokeslines bound regions where the function has different asymptotic behaviors. TheStokes lines are lines along which some term approaches infinity or zero fastest.

17

3. Elementary Operations on Poicare type Asymptotic Expansions

In singular perturbation methods we usually assume some form of the expan-sions, substitute them into the equations, and perform elementary operationssuch as addition, subtraction, exponentiation, integration and differentiation.Although the expansions are only formal and can be divergent, these operationsare generally carried out without justification. Nevertheless conditions underwhich these operations are justified can be given.

In the following we shall discuss some elementary properties of the Poincaretype expansion (8) of functions f(x; ε). If not specified the gauge functionsϕn(ε) are generic.

3.1. Equating Coefficients.

It is not strictly correct to write

∞∑n=0

an(x)ϕn(ε) ∼∞∑n=0

bn(x)ϕn(ε), as ε→ 0 (21)

because asymptotic series can only be asymptotic to functions. However whatwe mean with (21) is that

∞∑n=0

an(x)ϕn(ε) and

∞∑n=0

bn(x)ϕn(ε),

are asymptotic to functions which are asymptotically equivalent with respect tothe gauge functions ϕn(ε) as ε → 0. That is, they differ by terms o[ϕn(ε)] asε→ 0 for all n. The uniqueness of the Poincare type expansions given the gaugefunctions ϕn(ε) then implies that the coefficients of ϕn(ε) in the two series mustagree. Hence, we can equate the coefficient of the two expansions in (21) andsay that from (21) it follows that an(x) = bn(x).

3.2. Addition and Subtraction.

Addition and subtraction are in general justified, thus

αf(x; ε) + βg(x; ε) ∼∞∑n=0

[αan(x) + βbn(x)

]ϕn(ε), as ε→ 0.

where α and β are constants and an(x) and bn(x) the expansion coefficients off and g, respectively. The proof is simple.

3.3. Multiplication and Division.

Multiplication of asymptotic expansions is justified only if the result is stillan asymptotic expansion. The formal product of

∑anϕn and

∑bnϕn gener-

ates all products ϕnϕm of gauge functions and generally it is not possible torearrange them into an asymptotic sequence. Multiplication is justified only forasymptotic sequences ϕn such that the products ϕnϕm lead to an asymptotic

18

sequence, or posses asymptotic expansions. An important class of asymptoticexpansions with this property are asymptotic power expansions. In this casethe multiplication is straightforward. Suppose f(x; ε) ∼

∑∞n=0 an(x)εn and

g(x; ε) ∼∑∞n=0 bn(x)εn as ε→ 0, then

f(x; ε) g(x; ε) ∼∞∑n=0

cn(x) εn as ε→ 0,

where

cn(x) =

n∑m=0

am(x) bn−m(x). (22)

Proof. Using (22), and rearranging the sums, the reminder fg −∑Nn=0 cnε

n

takes the form,

fg −N∑n=0

n∑m=0

ambn−m εn = fg −

N∑m=0

amεm

N∑n=m

bn−mεn−m

= g[f −

N∑m=0

am εm]

+

N∑m=0

amεm[g −

N−m∑p=0

bp εp].

Since as ε→ 0

f −N∑n=0

an εn = o

(εN),

g −N−m∑n=0

bn εn = o

(εN−m

),

and moreover g(x; ε) − b0(x) = O(ε), both terms in the last equality are o(εN )as ε→ 0. Thus,

f(x; ε) g(x; ε)−N∑n=0

cn(x) εn = o(εN), as ε→ 0,

concluding the proof.

The division of asymptotic expansions can be handled treating it as theinverse of the multiplication, hence division is not justified in general. In theparticular case of asymptotic power expansions assuming b0(x) 6= 0, and using(22), it follows that:

f(x; ε)

g(x; ε)∼∞∑n=0

dn(x) εn as ε→ 0,

19

with d0(x) = a0(x)/b0(x) and

dn(x) =1

b0(x)

[an(x)−

n−1∑m=0

dm(x) bn−m(x)], n ≥ 1.

If the expansion coefficients an(x) and/or bn(x) are functions of x the divisionof uniformly valid asymptotic expansions might lead to a non-uniformly validexpansion. When this occurs the division is not justified anymore. The Example2.7, with f(x; ε) ∼ 1 and g(x; ε) ∼ x+ ε as ε→ 0+, illustrates this point.

3.4. Exponentiation.

Exponentiation of asymptotic expansions is generally not justified. As formultiplication, exponentiation produces all products of gauge functions. Thusthe exponentiation can be justified only for asymptotic expansions such that theproducts ϕnϕm form an asymptotic sequence, or posses asymptotic expansions.Moreover, if the coefficients of the asymptotic expansion are functions of x theexponentiation might results in a non-uniform expansion. For instance, Example2.7 shows that the asymptotic expansion of (x+ε)−1 as ε→ 0+ is not uniformlyvalid for x ≥ 0.

Example 3.1. Consider the function f(x; ε) with the uniform asymptotic ex-pansion

f(x; ε) ∼ x+ ε as ε→ 0+,

in the domain D : x ≥ 0. Then√f(x; ε) ∼

√x+ ε ∼

√x

[1 +

ε

2x− ε2

8x2+ . . .

]as ε→ 0+.

This expansion is valid uniformly in the domain D0 : 0 < x0 ≤ x, where x0is a positive constant, but not in the domain D. The uniform validity of theasymptotic expansion breaks down when x/ε = O(1), and indeed when x = 0the expansion has a completely different form:√

f(0; ε) ∼√ε as ε→ 0+.

3.4.1. Integration.

Asymptotic expansions can be generally integrated term-by-term with re-spect to the expansion parameter ε and/or the variable x.

If f(x; ε) and the gauge functions ϕn(ε) are integrable functions of ε in theinterval I : ε ∈ [0; ε0] where ε0 is constant, then term-by-term integration isjustified and ∫ ε

0

dε′ f(x; ε′) ∼∞∑n=0

an(x)

∫ ε

0

dε′ ϕn(ε′) as ε→ 0.

20

Proof. The proof is straightforward. Since f −∑Nn=0 anϕn = o(ϕN ) as ε → 0,

then:∫ ε

0

dε′ f(x; ε′)−N∑n=0

an(x)

∫ ε

0

dε′ ϕn(ε′) =

∫ ε

0

dε′[f(x; ε′)−

N∑n=0

an(x)ϕn(ε′)]

=

∫ ε

0

dε′ o[ϕN (ε′)

]= o

[∫ ε

0

dε′ϕN (ε′)

], as ε→ 0+.

In the case of asymptotic power expansions

f(x; ε) ∼∞∑n=0

an(x) εn as ε→ 0,

termwise integration is always justified and leads to∫ ε

0

dε′ f(x; ε′) ∼∞∑n=0

an(x)

n+ 1εn+1, as ε→ 0.

On the contrary, the first two terms of the asymptotic expansion

f(x; ε) ∼∞∑n=0

an(x) ε−n, as ε→ +∞,

are not integrable at ε = +∞, and termwise integration of the series is notjustified. Nevertheless,∫ ∞

ε

dt[f(x; t)− a0(x)− a1(x) t−1

]∼∞∑n=2

an(x)

n− 1ε1−n, as ε→ +∞.

The proof is straightforward and is left as exercise.The extension to asymptotic expansions with gauge functions ϕn(ε) = εαn

as ε→ 0+, or ϕn(ε) = ε−αn as ε→ +∞, with αn+1 > αn is trivial.Similarly, if f(x; ε) and an(x) are integrable functions of x in some interval

I : x ∈ [x0, x1], then for any x ∈ I∫ x

x0

dx′ f(x′; ε) ∼∞∑n=0

ϕn(ε)

∫ x

x0

dx′ an(x′), as ε→ 0.

The proof is straightforward.

21

3.5. Differentiation.

Termwise differentiation of uniform asymptotic expansions, either with re-spect to the perturbation parameter ε or with respect to the variable x, mightlead to non-uniformly valid asymptotic expansions. When this occurs the dif-ferentiation of the asymptotic expansion term-by-term is not justified.

Example 3.2. Consider the function f(x; ε) with the uniform asymptotic ex-pansion

f(x; ε) ∼ 1 + ε√x as ε→ 0+,

in the domain D : x ≥ 0. Then

∂

∂xf(x; ε) ∼ ε

2√x

as ε→ 0+.

This expansion is valid uniformly in the domain D0 : 0 < x0 ≤ x, where x0 is apositive constant, but not in the domain D.

In addition to this, subdominant terms might become relevant under differ-entiation as illustrated by the following example.

Example 3.3. The functions f(x) and

g(x) = f(x) + e−x−2

sin ex−2

,

have the same asymptotic power expansion as x → 0 because e−x−2

is tran-scendentally small with respect to any power xn as x→ 0 and the last term issubdominant. However their derivatives f ′(x) and

g′(x) = f ′(x) + 2e−x

−2

x3sin ex

−2

− 2

x3cos ex

−2

,

do not have the same power expansion as x→ 0 because x−3 is not subdominantto xn as x→ 0.

To differentiate term-by-term the asymptotic expansion

f(x; ε) ∼∞∑n=0

an(x)ϕn(ε), as ε→ 0, (23)

some additional conditions are needed. For instance, assume it is known thatthe partial derivative ∂f(x; ε)/∂ε exists and possess an asymptotic expansionas ε → 0 with some gauge functions ψn(ε) and expansion coefficients an(x).If ∂f(x; ε)/∂ε and ψn(ε) are integrable functions of ε in the interval I : 0 ≤ε ≤ ε0, then the asymptotic expansion can be integrated term-by-term withrespect to ε leading to the asymptotic expansion (23) of f(x; ε) with ϕn(ε) =∫ εdε′ψn(ε′). Under these conditions termwise differentiation with respect to ε

of the asymptotic expansion (23) is justified and leads to:

∂f(x; ε)

∂ε∼∞∑n=0

an(x)ϕ′n(ε), as ε→ 0.

22

Similarly, if it is known that ∂f(x; ε)/∂x exists and possess an asymptoticexpansion as ε→ 0 with gauge functions ϕn(ε) and expansion coefficients bn(x).And, moreover, ∂f(x; ε)/∂x and bn(x) are integrable functions of x in the inter-val I : x0 ≤ x ≤ x1, then the asymptotic expansion can be integrated term-by-term with respect to x obtaining the asymptotic expansion (23) of f(x; ε) withan(x) =

∫ xdx′bn(x′) for x ∈ I . Thus, if these conditions are satisfied termwise

differentiation of the asymptotic expansion (23) with respect to x is justifiedand leads to:

∂f(x; ε)

∂x∼∞∑n=0

a′n(x)ϕn(ε), as ε→ 0.

We shall came back on this point when discussing the asymptotic expansionof the solutions of differential equations.

4. Asymptotic Expansion of Integrals

Asymptotic expansion of integrals is a collection of techniques originallydeveloped because many special functions used in Physics and Applied Mathe-matics have integral representations. Nowadays asymptotic expansion of inte-grals find applications in different fields, such as Statistical Mechanics and FieldTheory. We shall not discuss asymptotic expansion of integrals here, rather wediscuss two examples to illustrate the use of asymptotic expansions.

4.1. Generalized Stieltjeis Series and Integrals.

Consider the function f(x) defined by the Stieltjeis integral :

f(x) =

∫ ∞0

dtρ(t)

1 + xt, x ≥ 0 (24)

where ρ(t) is a generic non-negative function:

ρ(t) ≥ 0, t ≥ 0.

We have already encountered this type of integral in Example 2.10 with ρ(t) =e−t. There we obtained the asymptotic power expansion of f(x) as x→ 0+ bysuccessive integration by parts. For a generic function ρ(t) this procedure cannotbe applied. Nevertheless, an asymptotic power expansion of f(x) as x→ 0+ canbe obtained by substituting into the integrand the asymptotic powers expansion

1

1 + xt∼∞∑n=0

(−xt)n, as x→ 0+,

and integrating term-by-term. Thus, if the moments

an =

∫ ∞0

dt ρ(t) tn n ≥ 0 (25)

23

exists and are finite for any n, the function f(x) as x→ 0+ has the asymptoticexpansion:

f(x) ∼∞∑n=0

(−1)nanxn, as x→ 0+, (26)

called Stieltjeis Series. The series does not converge for any function ρ(t). Aconvergent power series in x converges inside a disk of finite radius centered atx = 0. The Stieltjeis integral (24) is not defined for negative x because thesingularity at t = −1/x is not integrable; thus, the Stieltjeis Series (26) cannotbe convergent.

Proof. The proof that series (26) is an asymptotic power expansion as x→ 0+

is simple. By using (25) and the identity

N∑n=0

zn =1− zN+1

1− z,

valid for any N , we have

f(x)−N∑n=0

(−1)nanxn =

∫ ∞0

dt ρ(t)[ 1

1 + xt−

N∑n=0

(−xt)n]

=

∫ ∞0

dt ρ(t)[ 1

1 + xt− 1− (−xt)N+1

1 + xt

]=

∫ ∞0

dtρ(t)

1 + xt(−xt)N+1.

Then ∣∣∣f(x)−N∑n=0

(−1)nanxn∣∣∣ ≤ xN+1

∫ ∞0

dt ρ(t) tN+1 = aN+1xN+1

and

f(x)−N∑n=0

(−1)nanxn = o

(xN), as x→ 0+,

which concludes the proof.

4.2. Gamma Function.

The gamma function Γ(ν) is defined for all complex numbers ν with realpositive part as

Γ(ν) =

∫ ∞0

dt tν−1e−t, Re ν > 0. (27)

The gamma function is then extended via an analytic continuation to the wholecomplex plane except to non-positive integers, where it has simple poles. Forsimplicity we consider the case of real positive large ν.

24

By using the functional relation νΓ(ν) = Γ(1 + ν) and introducing the newvariable t = ν(y + 1), Γ(ν) can be written as

Γ(ν) =1

νΓ(1 + ν) = νν e−ν

∫ ∞−1

dy eνh(y),

whereh(y) = ln(1 + y)− y.

The function h(y) has a global maximum at y = 0, see Figure 1. When ν � 1

��

��

��

��

��

��

�� �� �� �� �� �� �� ��

����

�

Figure 1: Function h(y) = ln(1 + y)− y.

only the neighborhood of y = 0 contributes significantly to the integral becausecontributions from y not in the neighborhood 0 are exponentially depressed.Then Γ(ν) can be estimated for ν � 1 by restricting the integration to the neigh-borhood of y = 0, where h(y) attains its maximum value (Laplace’s method).We thus divide the interval of integration into three regions:∫ ∞

−1dy eνh(y) =

∫ −ε−1

dy eνh(y) +

∫ ε

−εdy eνh(y) +

∫ ∞ε

dy eνh(y),

were ε� 1 is arbitrary, but sufficiently small so that in the interval −ε ≤ y ≤ εthe function h(y) can be replaced by the Maclaurin expansion:

h(y) = −1

2y2 +

1

3y3 − 1

4y4 +O(y5). (28)

The parameter ε may depend on ν, but for the moment it is sufficient to knowthat ε � 1 so that (28) can be used. The functional dependence of ε on ν is

25

fixed later to make the whole calculation consistent. As ν → +∞ the first andthird integrals are of the order∫ −ε

−1dy eνh(y) = O

(eνh(−ε)

)= O

(e−νε

2/2),

∫ ∞ε

dy eνh(y) = O(

eνh(ε))

= O(e−νε

2/2).

Provided ε√ν � 1, i.e., ν−1/2 � ε � 1, their contribution is exponentially

small as ν � 1.Replacing h(y) into the second integral by the expansion (28), and introduc-

ing the variable x = y√ν, then:∫ ∞

−1dy eνh(y) ∼ ν−1/2

∫ ε√ν

−ε√ν

dx e− 1

2x2+ 1

3√νx3− 1

4ν x4+O(x5/ν3/2)

+O(e−νε

2/2).

(29)Now we use the arbitrariness of ε and take ε = ν−a with 1/3 < a < 1/2. Thischoice not only ensures that ε � 1 and ε

√ν � 1 as ν → ∞, but also that for

|x| ≤ ε√ν:

xm

νm/2−1� xn

νn/2−1� 1 m > n, n,m = 3, 4, 5, . . . .

The exponential under the integral in (29) is then of the form exp[−x2/2+gν(x)]with gν(x) � 1 as ν → +∞ in the whole domain of integration [−ε

√ν, ε√ν].

Expanding the exponential exp[gν(x)] in powers of gν(x) we obtain:∫ ∞−1

dy eνh(y) ∼ ν−1/2∫ ε√ν

−ε√ν

dx e−12x

2[1 + gν(x) +

1

2gν(x)2 + . . .

]+O

(e−νε

2/2).

The integration limits±ε√ν can be safely extended to±∞ with an error of order

O(e−νε

2/2), exponentially small as ν → +∞. Thus, introducing the Gaussian

average:

〈f〉 =

∫ +∞

−∞

dx√2π

e−12x

2

f(x),

the asymptotic expansion of the function Γ(ν) as ν → +∞ can be written as

Γ(ν) ∼ νν−1/2 e−ν√

2π[1 + 〈gν〉+

1

2〈g2ν〉+

1

3!〈g3ν〉+ . . .

],

where

gν(x) =1

3√νx3 − 1

4νx4 +O(x5/ν3/2).

The averages can be evaluated using the well known result for the moments ofthe Gaussian distribution of zero average and variance 1: 〈x2n〉 = (2n−1)!! and〈x2n+1〉 = 0. To the leading order we have

〈gν〉 = − 1

4ν〈x4〉+O(1/ν2) = − 3

4ν+O(1/ν2),

26

〈g2ν〉 =1

9ν〈x6〉+O(1/ν2) =

5 · 39ν

+O(1/ν2).

and 〈g3ν〉 = O(1/ν2), so that

Γ(ν) ∼ νν−1/2 e−ν√

2π[1 +

1

12ν+O(1/ν2)

], as ν → +∞, (30)

known as Stirling’s formula, after James Stirling, even if it was first stated byAbraham de Moivre.

The proof that (30) is an asymptotic expansion of Γ(ν) as ν → +∞ isstraightforward. It is enough to compute the next terms in the expansion andestimate the error when the series is truncated to the first N terms. The con-tributions of order O

(e−νε

2/2)

are transcendentally small with respect to anypower ν−n as ν → +∞ and can be ignored.

The asymptotic expansion (30) of Γ(ν) has been derived using the integralrepresentation (27) and the Laplace’s method; however, the definition (27) isnot the only possible definition of Γ(ν). For example, an asymptotic expansionof Γ(ν) as ν → +∞ can be obtained also from the integral representation

ln Γ(ν) =(ν − 1

2

)ln ν − ν + ln

√2π + 2

∫ ∞0

dtarctg( tν )

e2πt − 1.

Performing the change of variable t = νy, one sees that as ν → +∞ only valuesof y up to y = O(1/ν) contributes significantly to the integral; the others givejust a correction O(e−2πν) to the value of the integral. Then, since 1/ν � 1, wecan replace arctg(x) by the asymptotic expansion (Taylor series)

arctg(x) ∼∞∑n=1

(−1)n−1x2n−1

2n− 1, x→ 0,

and integrate termwise the resulting asymptotic expansion:∫ ∞0

dtarctg( tν )

e2πt − 1∼∞∑n=1

(−1)n−1

(2n− 1) ν2n−1

∫ ∞0

dtt2n−1

e2πt − 1. (31)

We have neglected the O(e−2πν) terms because transcendentally small withrespect to any power ν−n as ν → +∞. Termwise integration is justified becausethe coefficient of the expansion are integrable functions of t. The integral in (31)is equal to ∫ ∞

0

dtt2n−1

e2πt − 1=

(−1)n−1

4nB2n, (32)

where Bn are the Bernoulli numbers given by the coefficients of the expansion

z

ez − 1=

∞∑n=0

Bnzn

n!, 0 < |z| < 2π.

27

Substituting (32) into (31), we have

ln Γ(ν) ∼(ν− 1

2

)ln ν−ν+ln

√2π+

∞∑n=1

B2n

2n(2n− 1) ν2n−1, as ν → +∞. (33)

The first terms of the series using B2 = 1/6, B4 = −1/30, B6 = 1/42, are

ln Γ(ν) ∼(ν−1

2

)ln ν−ν+ln

√2π+

1

12ν− 1

360ν3+

1

1260ν5+O(ν−7), as ν → +∞,

The series (33) is not convergent, but when truncated gives and approximationof ln Γ(ν) as ν → +∞ with error of the order of the first omitted term.

The asymptotic expansion (33) differs from the expansion (30) derived usingthe integral representation (27); nevertheless, they are asymptotically equivalentas ν → +∞. From (33) we have

Γ(ν) ∼ νν−1/2e−ν√

2π exp

[1

12ν+O(ν−3)

], (34)

which differs from (30) by terms O(1/ν2) as ν →∞ because

exp

[1

12ν+O(ν−3)

]− 1− 1/12ν = O(1/ν2) as ν → +∞.

For finite values of ν, however, they give different approximations. For examplethe approximation (30) gives Γ(5) = 23.99 . . . , Γ(6) = 119.99 . . . and Γ(7) =719.95 . . . , while approximation (34) gives Γ(5) = 24.00 . . . , Γ(6) = 120.00 . . .and Γ(7) = 720.00 . . . .

28