Directory UMM :Data Elmu:jurnal:J-a:Journal of Computational And Applied Mathematics:Vol102.Issue2.1999:
Journal of Computational and Applied Mathematics 102 (1999) 181–194
Asymptotic expansions of integrals:
The term-by-term integration method
Jose L. Lopez ∗
Departamento de Matematica Aplicada, Universidad de Zaragoza, 50009, Zaragoza, Spain
Received 25 May 1998; received in revised form 12 September 1998
Abstract
The classical term-by-term integration technique used for obtaining asymptotic expansions of integrals requires the
integrand to have an uniform asymptotic expansion in the integration variable. A modi cation of this method is presented
in which the uniformity requirement is substituted by a much weaker condition. As we show in some examples, the
relaxation of the uniformity condition provides the term-by-term integration technique a large range of applicability. As a
consequence of this generality, Watson’s lemma and the integration by parts technique applied to Laplace’s and a special
c 1999 Elsevier Science B.V.
family of Fourier’s transforms become corollaries of the term-by-term integration method.
All rights reserved.
AMS classi cation: 41A60; 28A25; 44A05
Keywords: Asymptotic expansions; Integral representations; Term-by-term integration technique; Watson’s lemma;
Integration by parts method
1. Introduction
Asymptotic approximation is an important topic in applied analysis. The solutions to a large kind
of applied problems in
uid mechanics, electromagnetism, statistics and many other elds can, by
means of integral transforms, be represented by integrals. The analytic calculation of these integrals
is usually quite dicult, although they can be approximated in particular limits of interest by means
of asymptotic expansions. A lot of techniques and theories have been proposed during the last
decades for obtaining asymptotic expansions of functions de ned by means of integrals: integration
by parts, Watson’s lemma, stationary phase, steepest descent, Laplace’s method, Mellin transform
techniques, etc. (see, for example, [2, 7, 11] or [12]).
∗
E-mail: [email protected].
c 1999 Elsevier Science B.V. All rights reserved.
0377-0427/99/$ – see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 8 ) 0 0 2 1 6 - 7
182
J.L. Lopez / Journal of Computational and Applied Mathematics 102 (1999) 181–194
Among all of them, the term-by-term integration method (TTIM) is one of the most intuitive
techniques used for deriving asymptotic approximations of integrals. Let us suppose that an asymptotic expansion of the integrand in some asymptotic sequence of the asymptotic variable is known.
Then, if this expansion is uniform in the integration variable, the simple term-by-term integration
of that sequence, if it is well de ned, constitutes an asymptotic expansion of the integral [2, p.
29, Theorem 1.7.5]. The problem is that the uniformity condition is very uncommon in practice.
On the other hand, uniform asymptotic expansions usually present a very complicated form (see,
for example, [10] and references there in for a very complete survey on the subject). Therefore,
even if an uniform asymptotic expansion of the integrand is known, the integration of the terms
of this expansion may be quite dicult and the asymptotic expansion of the integral has not a
computational utility.
Uniformity is a too strong condition to justify the TTIM (integral representations of many special
functions, for example, do not satisfy this requirement). In order to make the TTIM useful for practice, the uniformity condition should be replaced by some weaker (and easier to verify) requirement.
In the new version of the TTIM that we introduce in the next section, this requirement is a certain
bounding condition over the integrand. A large class of integrals satisfy this requirement, which
provides the TTIM a large range of applicability. For example, it is applicable to integral representations of many special functions. In Section 3, new asymptotic expansions of the incomplete beta
function Bx (a; b) and of certain coecients related to the Whittaker function M; (z) are obtained
using this method.
On the other hand, from a theoretical point of view, this generalization of the TTIM led us to
shed some light upon the idea of uni cation of asymptotic methods. This idea was already suggested
in 1963 by Erdelyi and Wyman [4, 13]. In their work, they show in a lucid and expository way
that Darboux’s method, Watson’s lemma, steepest descents and stationary phase (applied to a certain
kind of integrands) can be viewed as particular cases of the method of Laplace. Although from a
more modern point of view, following the work of Wong [12], Watson’s lemma and integration
by parts should be considered as ‘fundamental methods’: steepest descents, Laplace’s, or Perron’s
method, for example, are based on Watson’s lemma, whereas stationary phase or summability methods, for example, are based on the integration by parts technique. In Section 4 we show that the
generalization of the TTIM that we present here is applicable to integrals whose integrand satis es
the conditions required by Watson’s lemma. It is also applicable to Laplace transforms when the
integrand satis es the conditions required by the integration by parts method. Besides, it is applicable to a certain family of Fourier transforms that are classically solved by the integration by parts
technique. These facts suggest the possibility of considering the TTIM as a ‘fundamental’ asymptotic
technique.
2. A new version of the term by term integration method
We will consider complex functions I (z) de ned by means of an integral representation of the
form
I (z) ≡
Z
C
h(w)f(w; z) dw;
(1)
J.L. Lopez / Journal of Computational and Applied Mathematics 102 (1999) 181–194
183
where z ∈ C, C is a complex path and h : C → C and f : C × (C\B(0; r0 )) → C (where B(0; r0 ) denotes
the open disk of center z = 0 and radius r0 ) satisfy the following two hypotheses:
(A) The product h(w)f(w; z) is integrable along the path C for |z| ¿ r0 ¿ 0. The path C may
depend on Arg(z) but not on |z|.
(B) For N = 0; 1; 2; : : : ; N0 (N0 nite or in nite),
f(w; z) =
N
X
an (w)
n=0
z n
+ Ow (z −N +1 );
|z| → ∞;
(2)
where an : C → C, {n } is a sequence of complex numbers satisfying Re(n+1 ) ¿ Re(n ) ¿ 0
∀n ¿ 0 and n 6 N + 1 and the subscript w in the remainder after N terms, Ow (z −N +1 ), means
that it may depend also on w.
The classical TTIM states the following [2, p. 29] (although the method is originally formulated
for arbitrary asymptotic sequences, here we need to consider only sequences of the form {z −n }).
Suppose that the remainder term in expansion (2) of f(w; z) can be bounded by a w-independent
quantity along all the path C, that is,
Ow (z −N +1 ) = O(z −N +1 ) ∀w ∈ C; N = 0; 1; : : : ; N0 ;
(3)
where O(z −N +1 ) is independent of w. Suppose also that
Z
h(w) dw ¡ ∞
and
Z
h(w)an (w) dw ¡ ∞
C
C
∀n 6 N:
(4)
Then, the asymptotic expansion of the integral I (z) with respect to the sequence {z −n } is just given
by introducing expansion (2) into (1) and interchanging the sum and the integral,
Z
h(w)f(w; z) dw =
C
N Z
X
n=0
h(w)an (w) dw
C
1
+ O(z −N +1 );
z n
|z| → ∞:
(5)
The evident advantage of this procedure is its great simplicity. But, if f(w; z) and=or C are not
bounded, it may be dicult to prove that the remainder term Ow (z −N +1 ) is bounded by a
w-independent quantity O(z −N +1 ). This happens, for example, when f(w; z) = f(w=z) and C is unbounded. In this case, Ow (z −N +1 ) = O((w=z)N +1 ) may not be bounded for unbounded values of w.
A simple classical example is the following.
Example 1. The exponential integral [12, p. 14, Eq. (4.1)].
Ei(z) =
Z
z
−∞
ew
dw;
w
|Arg(−z)| ¡ ;
z 6= 0;
(6)
where the integration path is de ned by −∞ ¡ Re(w) 6 Re(z) and Im(w) = Im(z). After the change
of variable w = z − x we obtain
ez
Ei(z) =
z
Z
0
∞
−x
e
x
1−
z
−1
dx:
(7)
184
J.L. Lopez / Journal of Computational and Applied Mathematics 102 (1999) 181–194
Therefore, we can take h(x) ≡ e−x ,
x
x
f(x; z) = f
≡ 1−
z
z
−1
=
N n
X
x
n=0
z
+O
N +1 !
x
z
(8)
and an (x) ≡ xn . This expansion is not uniform in x ∈ [0; ∞) and the classical term-by-term integration
method of [2] cannot be applied.
Integral representations of many other special functions [1] are also examples of failure of the
classical TTIM. The argument is similar: they are of (or can be expressed in) the form
I (z) ≡
Z
h(w)f
C
w
dw;
z
(9)
where C is unbounded and then, uniformity (3) of expansion (Eq. (2)) does not hold. In the
following, we will consider only integrands of form (9) for which hypothesis (B) reads
w
f
z
=
N
X
an
n=0
n
w
z
+O
N +1 !
w
z
|z| → ∞; an ∈ C:
;
(10)
In order to show that the term-by-term integration of expansion (10) in (9) is an asymptotic expansion
of I (z) in the sequence {z −n }, we need to show that the remainder after N terms,
"
w
N (z) ≡ h(w) f
z
C
Z
−
N
X
an
n=0
n #
w
z
dw;
(11)
satis es N (z) = O(z −N +1 ). Uniformity (3) and the rst bound in (4) guarantee this. But condition
(3) may be relaxed by using the following observation.
Remark 1. Hypothesis (10) above means that for N = 0; 1; 2; : : : ; N0 ;
∀ ¿ 1 ∃rN () ¿ 0;
n
N
X
w
w
|z N +1 | f
−
an
6 |aN +1 wN +1 |
z
z
n=0
|z| ¿ rN |w| and
∀w ∈ C;
|z| ¿ r0 :
(12)
Therefore, the remainder in (10) is bounded, at least for |z| ¿ rN |w|, by the quantity |aN +1
(w=z)N +1 |. But, if we can nd a function g : C → R+ , satisfying
f w 6 g(w)
z
∀w ∈C;
|z| ¡ rN |w| and
|z| ¿ r0 ;
(13)
then this remainder is also bounded by a O(z −N +1 )-quantity for |z| ¡ rN |w|. This is so because for
N = 0; 1; 2; : : : ; N0 and ∀|z| ¿ r0 ,
N +1
z
w
f
z
∀w ∈C;
−
and
N
X
n=0
"
#
n !
N
X
|an |
w
N +1
an
6 |(rN w) | g(w) +
z
rNn
|z| ¡ rN |w|:
n=0
(14)
J.L. Lopez / Journal of Computational and Applied Mathematics 102 (1999) 181–194
185
Joining inequalities (12) and (14) we obtain, for N = 0; 1; 2; : : : ; N0 , a O(z −N +1 ) bound for the remainder term in (10) uniformly valid ∀|z| ¿ r0 ,
"
#
X
n !
N
+1
N
X
w
|a
|
w
n
an
−
|(rN w)N +1 |
f
6 |z −N +1 | g(w) +
z
z
rNn
n=0
n=0
∀w ∈ C:
(15)
Therefore, the remainder N (z) is bounded by the integral of h(w) times the right-hand side of
the above inequality. If this integral is nite, then N (z) = O(z −N +1 ). We can summarize the above
discussion in the following.
Theorem 1. Consider a complex function I (z) de ned by integral (9) and suppose that the following conditions are satis ed:
(i) The product h(w)f(w=z) is integrable along the path C for |z| ¿ r0 ¿ 0. The path C may
depend on Arg(z) but not on |z|.
(ii) For N = 0; 1; 2; : : : ; N0 (N0 nite or in nite),
w
f
z
=
N
X
n=0
an
n
w
z
+O
N +1 !
w
z
;
|z| → ∞;
(16)
where an ∈ C and {n } is a sequence of complex numbers satisfying Re(n+1 ) ¿ Re(n ) ¿ 0
∀n ¿ 0 and n 6 N + 1.
(iii) There is a function g : C → R+ ; satisfying
f w 6 g(w)
z
∀w ∈ C; |z| ¡ rN |w| and |z| ¿ r0 :
(17)
(iv) For n = 0; 1; 2; : : : ; N + 1;
Z
C
|h(w)wn dw| ¡ ∞
and
Z
C
|h(w)g(w)wn dw| ¡ ∞:
(18)
Then, the asymptotic expansion of I (z) with respect to the sequence {z −n }; for |z| → ∞; is given by
I (z) =
N Z
X
n=0
C
n
h(w)w dw
an
+ N (z);
z n
(19)
where the remainder after N terms, N (z); is bounded by
Z
#
"
N
+1
r N +1
X
N
an
|N (z)| 6
|h(w)| g(w) +
n |wN +1 | dw
z
C
rN
(20)
n=0
and and rN () are de ned in (12).
Remark 2. The veri cation of conditions (iii) and (iv) of Theorem 1 may depend on Arg(z). This
fact is related with the Stokes lines. If expansion (16) is valid only in a certain sector of C; then
the Stokes rays also depend on this sector.
186
J.L. Lopez / Journal of Computational and Applied Mathematics 102 (1999) 181–194
The main di erence between this version and the classical method lies in the substitution uniformity requirement (3) by the much weaker condition (iii). But, if rN () in Eq. (12) is unknown, condition (iii) may not be checked easily. Nevertheless, in practice, an inequality like |f(w=z)| 6 g(w)
usually holds ∀|z| ¿ r0 and hence, the explicit value of rN is not needed for checking condition (iii).
This fact is shown in Example 1.
Example 1 (Continuation). For any r0 ¿ 0 we have that
f x ≡ 1 −
z
x −1
6
z
(
1
|sin(Arg(z))|−1
if |Arg(z)| ¿ =2;
if 0 ¡ |Arg(z)| 6 =2
(21)
and conditions (i)–(iv) of Theorem 1 are satis ed with g(x) ≡ h(Arg(z)), where we have denoted
h(Arg(z)) the second hand of the above inequality. Applying Theorem 1 and after a straightforward
algebra we obtain the well-known result [12, p. 14, Eq. (4.3)]
∞
n!
ez X
;
Ei(z) ∼
z n=0 z n
Arg(z) 6= 0:
(22)
Nevertheless, bound (20) for the remainder term is unpractical unless rN () is known. This problem
can be solved if expansion (16) is convergent in some region |z| ¿ r|w| ¿ 0. Moreover, if (16) is
convergent in this region, then the constant r may be used as rN in (17) to verify condition (iii).
These points are clari ed in the following.
Corollary 1. Assume that:
(ii′ ) The function f(t) admits a convergent series expansion for |t| 6 r −1 ; r ¿ 0;
f (t) =
∞
X
an t n ;
n=0
|t| 6 r −1
(23)
for some r ¿ 0 and conditions (i), (iii) and (iv) of Theorem 1 hold for rN = r. Then, the asymptotic
expansion of I (z) in the sequence {z −n } is given by (19) and an error bound for the remainder
is given by
Z
"
#
∞
r N +1
X
an
|wN +1 || dw|:
|N (z)| 6
|h(w)| g(w) +
r n
C
z
(24)
n=0
Proof. Hypothesis (ii′ ) implies
|z
N +1
n
∞
N
X
X
an
w
w
| f
an
−
6 |(rw)N +1 |
r n
z
z
n=N +1
n=0
∀w ∈ C; |z| ¿ r|w|:
(25)
And condition (iii) with rN = r implies
"
#
n
N
N
X
X
w
a
w
n
|z N +1 | f
an
6 |(rw)N +1 | g(w) +
z
z
r n
n=0
n=0
∀w ∈ C; |z| ¡ r|w|:
(26)
J.L. Lopez / Journal of Computational and Applied Mathematics 102 (1999) 181–194
187
Then, by using inequalities (25) and (26) and the de nition of remainder term (11), bound (24)
follows. This bound is nite by virtue of condition (iv). Therefore, N (z) = O(z −N ) and Eq. (19) is
the asymptotic expansion of I (z) in the sequence {z −n }.
Example 1 (Continuation). Expansion (8) is in fact convergent for |z|=x ¿ (N + 2)=(N + 1),
f (t) ≡ (1 − t)−1 =
∞
X
tn;
|t| 6
n=0
N +1
:
N +2
(27)
Therefore, using Corollary 1 with r ≡ (N + 2)=(N + 1), a bound for the remainder after N terms is
given by
N (z) 6 (h(Arg(z)) + N + 2)eRe(z)+1
(N + 1)!
:
|z|N +2
(28)
Asymptotic approximations of integral representations of many other special functions [1] have
been obtained by using several asymptotic methods. They could also be obtained in a systematic
and easy way using this modi ed TTIM. The procedure would be similar to the employed in
Example 1.
3. Nontrivial examples
The substitution of uniformity condition (3) by weaker condition (17) of Theorem 1 provides the
TTIM with a large range of applicability. Besides, its easy implementation lets us to obtain asymptotic
expansions of more or less sophisticated integrals. In this section we obtain, using Corollary 1, the
asymptotic expansion of the incomplete beta function Bx (a; b) in the sequence {a−n }. In Example 3,
Corollary 1 let us to obtain an asymptotic approximation of the coecients of the expansion of the
Whittaker function M; (z) in power series of from an intricate integral representation.
Example 2. The incomplete beta function Bx (a; b). An integral representation of Bx (a; b) is given
by [11, p. 128],
Bx (a; b) =
Z
x
ya−1 (1 − y)b−1 dy;
0
a ¿ 0; b ¿ 0; 0 6 x 6 1:
(29)
With the change of variable y = xe−t=a we obtain
Bx (a; b) =
xa
a
Z
0
∞
e−t (1 − xe−t=a )b−1 dt:
(30)
Therefore, we have C ≡ [0; ∞), h(t) ≡ e−t and f(t=a) ≡ (1 − xe−t=a )b−1 . The asymptotic expansion of
f(t=a) with respect to the sequence (t=a)n is its Taylor expansion around 0,
t
f
a
=
∞
X
f(n) (0) t n
n=0
n!
a
;
t
¡ −log x:
a
(31)
188
J.L. Lopez / Journal of Computational and Applied Mathematics 102 (1999) 181–194
After some algebra, it may be veri ed that this expansion is given by
(n)
f (0) = (1 − x)
n
X
b−1
k=0
n
(−1) (1 − b)k n; k
x
1−x
k
(32)
;
where the coecients n; k are universal and de ned recursively by
0;0 = 1;
n; k = 0
for k ¡ 0 or k ¿ n;
n; k = n−1; k−1 + k n−1; k
for 0 6 k 6 n:
(33)
Expansion (31) is convergent for |t=a| 6 (−log x)= for any ¿ 1. In order to apply Corollary 1, we
can take r ≡ (−log x)=. The function f(t=a) satis es hypotheses (i) and (ii′ ) with an ≡ f(n) (0)=n!
and n ≡ n. On the other hand, ∀t ∈ [0; ∞), we have that f(t=a) satis es condition (17) ∀ a ¿ 0 with
g(t) ≡ 1 + (1 − x)b−1 ;
(34)
where g(t) and h(t) satisfy (18). Therefore, we can apply Corollary 1 and, after a straightforward
algebra, we obtain,
k
∞
n
X
1
x
xa (1 − x)b−1 X
n
Bx (a; b) ∼
:
(−1)
(1 − b)k n; k
n
a
1
−
x
a
n=0
k=0
#
"
(35)
A bound for the remainder after N terms can be obtained by using formula (24). This expansion
in the sequence {a−n } may be ‘added’ to the list of expansions of the incomplete beta function (see
for example [3, 6, 9, 11, Section 11.3] and references there in).
Example 3. The coecients of the expansion of the Whittaker function M; (z) in power series
of ,
M; (z) = 22 ( + 1)z 1=2
∞
X
Fm() (z)
m=0
(−)m
m!
(36)
are given by [5],
Fm() (z) =
m
ez=2 ( + 1=2) X
m
√
(ln z)m−k Ik() (z);
k
z
k=0
(37)
where the functions Ik() (z) are de ned by the integral representation
(−1)
Ik() (z) = 22
k
2i
f
w
w
≡ 1+
z
2z
Z
e
w=2
w
−(2+1)
C
−(+1=2)
=
dk
dk
+1=2
w
2
∞
X
(−1)n
n=0
n!
+
1
2
f
!
n
w
z
w
2z
dw;
n
;
|w| ¡ |2z|
(38)
(39)
and the contour C starts at −∞ on the real axes, encircles the points 0 and −2z in the counterclockwise direction and returns to the starting point. We are interested in approximating the
J.L. Lopez / Journal of Computational and Applied Mathematics 102 (1999) 181–194
189
coecients Fm() (z) for large values of z. Then, we use Corollary 1 for deriving an asymptotic
expansion of the integrals Ik() (z) for large z. We must take h(w) ≡ ew=2 w−(2+1) (d k =dk )((w=2)+1=2 ),
an ≡ ( + 1=2)n =((−2)n n!), n ≡ n and r ≡ 1. After some algebra we nd that the function g(w) may
be taken
g(w) = eRe(w)=2
where
∞
X
k
wN X
(2w)+1=2 l=0
k−l
k w
ln
gl (w);
l 2
(40)
N
X
|w|n
1 d l
2|Im()|
Re()+l+1
1
gl (w) =
+
e
|2w|
+
(
+
)
n
2
2n n! dl
n!
n=0
n=N +1
dl
1
l ( + 2 )n ;
d
(41)
provided the contour C is such that all its points w satisfy |w| ¿ 1 and |w + 2z| ¿ 1.
Then, using Corollary 1 and after straightforward operations we obtain that the functions Ik() (z)
admit asymptotic expansions
Ik() (z)
∞
k
X
X
k
∼
n=0
j=0
j
d j ( + 12 )n
d(−) j
!
d k−j
dk−j
!
1
(−1)n
:
1
n!z n
( − n + 2 )
(42)
Introducing this expansion into (37) we obtain an asymptotic expansion of Fm() (z). This result may
be checked by substituting now (37) into (36), grouping equal powers of log(z) and reordering
sums. One obtains in this way the familiar asymptotic expansion of the Whittaker function in the
sequence {z −n }.
4. Corollaries of the term-by-term integration method
As we see in Theorem 1 or Corollary 1, the TTIM does not require a too special form for the
integrand, but only the bounding and integrability conditions (iii) and (iv). On the other hand, other
classical techniques (such as those mentioned in the introduction) are adapted to particular forms of
the integrand. Therefore, if those kind of integrands satisfy conditions (i)–(iv) of Theorem 1, the
TTIM becomes a more general technique. In this section we show that, at least, Watson’s lemma
and the integration by parts technique applied to Laplace transforms and a special family of Fourier
transforms are corollaries of this version of the TTIM.
Corollary 2 (Watson’s lemma [7, p. 113]). Suppose that I (z) is de ned by the integral
I (z) =
Z
∞
e−zt q(t) dt;
(43)
0
where
(a) q : C → C has a nite number of discontinuities and in nities in the positive real axis.
(b) As t → 0+ ;
q(t) ∼
∞
X
an t (n+a−b)=b ;
n=0
where an ∈ C; b ¿ 0 and Re(a) ¿ 0. And
(44)
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J.L. Lopez / Journal of Computational and Applied Mathematics 102 (1999) 181–194
(c) Integral (43) has abscissa of convergence ≡ r0 cos(Arg(z)) ¿ 0 (it is integrable for |z| ¿
r0 ¿ 0 and |Arg(z)| ¡ =2).
Then, for |z| → ∞ and |Arg(z)| ¡ =2;
∞
X
I (z) ∼
n=0
n+a
b
an
:
(n+a)=b
z
(45)
And this is actually a strong asymptotic expansion of I (z);
e−Im(a)Arg(z)=b
|N (z)| 6 rN(N +a+1)=b (N +a+1)=b
|z
|
"
#
N
+1
X
K
an
+
(n+a)=b
r0
n=0 rN
((N + Re(a) + 1)=b + 1)
;
(cos(Arg(z))(N +Re(a)+1)=b+1
(46)
where K ¿ 0 is a bound of |q(t)| in a certain interval given below.
Proof. For any T ¿ 0 we write
I (z) =
Z
T
−zt
e
q(t) dt +
0
Z
∞
e
−zt
T
1
q(t) dt =
z
Z
zT
−w
e
0
w
q
z
dw + e
−zT
Z
∞
e−zt q (t + T ) dt: (47)
0
Using condition (c), the second integral in the last line satis es
Z
∞
−zt
e
Z
q (t + T ) dt 6
∞
−T
0
and therefore,
I (z) =
1
z
Z
0
zT
e−w q
|e
−t
q(t + T )| dt 6 e
T
Z
∞
0
|e−t q(t)| dt ¡ ∞
w
dw + O(e−zT ):
z
(48)
(49)
Condition (b) above means that
∞
X
w
w
w
q
∼
an
z
z
z
n=0
(n+a)=b
(50)
and then, hypothesis (iii) of Theorem 1 is satis ed for n ≡ (n + a)=b and f(w=z) ≡ (w=z)q(w=z).
Using condition (a) above, we can choose T such that 0 ¡ T ¡ rst singularity of q(t) apart from
an eventual singularity at t = 0. Therefore,
∀rN−1
¿ 0; ∃K ¿ 0;
q w 6 K
z
for rN−1 6
w
6T
z
(51)
and condition (iii) of Theorem 1 is satis ed with g(w) ≡ K|w|=r0 for any r0 ¿ 0. Condition (iv) of
Theorem 1 is trivially satis ed for the integral in Eq. (49) with h(w) ≡ w−1 e−w and C ≡ [0; zT ). On
the other hand, using condition (c) above, the rst requirement of hypothesis (i) holds, but not the
second one, because C depends on |z|. The independence of |z| is used in the proof of Theorem 1
to obtain that the coecients of expansion (19) and the bound given in Eq. (20) for |N (z)z N +1 | are
independent of |z|. Nevertheless, for the case C ≡ [0; zT ), the dependence of this bound on |z| can
be trivially eliminated,
#
"
N
+1
r (N +a+1)=b Z ∞ exp (iArg(z))
X
K
a
n
N
|e−w |
+
|N (z)| 6
(n+a)=b |w(N +a+1)=b ||dw|;
0
z
r0 w n=0 rN
(52)
J.L. Lopez / Journal of Computational and Applied Mathematics 102 (1999) 181–194
191
and after straightforward operations, we obtain (46). On the other hand, we are going to prove in
what follows that the coecients of expansion (19) are also independent of |z|, but for a quantity
O(e−zT ). This coecients are given by
Z
zT
Z
e−w w(n+a)=b−1 dw =
0
∞ ei Arg(zT )
0
e−w w(n+a)=b−1 dw −
∞ ei Arg(zT )
Z
e−w w(n+a)=b−1 dw:
(53)
zT
The second integral on the right-hand side above can be written
Z
∞ei Arg(zT )
e−w w(n+a)=b−1 dw = Mn (z)e−zT ;
(54)
zT
where
∞ ei Arg(zT )
Z
Mn (z) ≡
e−w (w + zT )(n+a)=b−1 dw:
(55)
0
A bound for the modulus of the integrand in the above integral is given by
(w + zT )(n+a)=b−1 6 e2|Im(a=b)| (|w| + T |z|)In ;
|z| ¿ r0 ;
(56)
where we have abbreviated In ≡ Int(Re((n+a)=n)) and chosen r0 ¿ T −1 . Using (56) and after straightforward operations, we nd that |Mn (z)| is bounded by a O(z In ) quantity. Therefore, the left-hand
side of Eq. (54) is a O(e−zT ) quantity. We can apply Theorem 1 and we obtain (19), where the
integrals inside the brackets equals ((n + a)=b), but for a O(e−zT ) quantity and (45) holds.
Corollary 3 (Integration by parts technique for Laplace transforms [2, p. 78]). Suppose that I (z)
is de ned by the integral
I (z) =
Z
b
e−zt q(t) dt;
(57)
a
where 0 6 a ¡ b 6 ∞; Re(z) ¿ r0 ¿ 0 and q : [a; b] → C has N continuous derivatives in [a; b];
where [a; b] must be set [a; ∞) if b = ∞.
Then, for |z| → ∞;
e−az
I (z) =
z
"
N
X
q(n) (a)
e−az
I (z) =
z
"
N
X
q(n) (a)
n=0
n=0
zn
zn
+ O(z
+ O(z
−(N +1)
−(N +1)
#
e−bz
) −
z
#
)
"
N
X
q(n) (b)
n=0
zn
+ O(z
if b = ∞
−(N +1)
#
)
if b ¡ ∞;
(58)
(59)
and the remainder term N (z) is bounded by
"
!
N
+1
r N +1
X
|q(n) (a)|
N
1 K +
|e−az |
|N (z)| 6
n
z
n!r
N
n=0
!
#
N
+1
X
|q(n) (b)|
(N + 1)!
+ 2 K +
n=0
n!rNn
|e−bz |
(cos(Arg(z)))N +2
;
(60)
192
J.L. Lopez / Journal of Computational and Applied Mathematics 102 (1999) 181–194
where 1 = 2 = 1 and K ¿ 1 is a bound for |q(t)| in [a; b] if b ¡ ∞ and 1 = 2N +2 ; 2 = 0 and K ¿ 1
is a bound for |e−zt q(t)| in [a; ∞) if b = ∞.
Proof. We consider rst the case b ¡ ∞. We extend the function q(t) to a C (N ) -function in [a; ∞)
with q(t) ≡ 0 in a neighborhood of ∞. The explicit extension of q(t) to [b; ∞) is not required.
Nevertheless, a construction of this extension is indicated, for example, in [8, p. 418, Examples 11
and 12]. Then, we can write
I (z) =
∞
Z
e
−zt
a
q(t) dt −
Z
∞
e−zt q(t) dt
(61)
b
and perform the changes of variable t = a + w=z and b + w=z in each one of the respective integrals
on the right-hand side of the above equation,
e−az
I (z) =
z
∞ exp(i Arg(z))
Z
e
−w
0
w
e−bz
+ a dw −
q
z
z
Z
∞ exp(i Arg(z))
e
−w
0
w
+ b dw:
q
z
(62)
But, using that q(t) has N continuous derivatives in the interval [a; b],
w
+t
q
z
=
N
X
q(n) (t) w n
n=0
n!
z
+O
N +1 !
w
z
∀t ∈ [a; b] and ∀w ∈ [0; ∞ ei Arg(z) ):
(63)
Therefore, the integrands in the right-hand side of Eq. (62) satisfy hypothesis (ii) of Theorem 1
up to N terms with f(w=z) ≡ q(w=z + a), an ≡ q(n) (a)=n! and f(w=z) ≡ q(w=z + b), an ≡ q(n) (b)=n!,
respectively, and n ≡ n.
On the other hand, using that q(t) is a continuous function in [a; ∞) and q(t) ≡ 0 around ∞, we
have that ∃K ¿ 1 and K ¡ ∞ satisfying |q(t)| 6 K ∀t ∈ [a; ∞). Therefore, f(w=z) 6 K ∀w ∈ [0;
∞ ei Arg(z) ) and conditions (i), (iii) and (iv) of Theorem 1 are satis ed with C ≡ [0; ∞ ei Arg(z) ),
g(w) ≡ K for any r0 ¿ 0 and h(w) ≡ e−w for each one of the integrals of Eq. (62). Then, using (63)
and applying this theorem we obtain (58) from (62). The bound (60) follows trivially from (20).
The proof is similar for the case b = ∞ setting the second integral in Eq. (61) equal to zero. But in
this case, |q(t)| may not be bounded in [0; ∞). Nevertheless, |e−zt q(t)| must be bounded ∀t ∈ [a; ∞)
by some constant K ¿ 1 ∀|z| ¿ r0 . Therefore, |e−r0 w=z q(w=z + a)| 6 K ∀|z| ¿ r0 , ∀w ∈ [0; ∞ ei Arg(z) )
and then, |q(w=z + a)| 6 KeRe(w)=2 ∀|z| ¿ 2r0 , ∀w ∈ [0; ∞ ei Arg(z) ) and conditions of Theorem 1 are
satis ed. Bound (60) follows from Eq. (20) for g(w) ≡ KeRe(w)=2 .
Corollary 4 (Integration by parts technique for Fourier transforms, [2, p. 78, 12, p. 15]). Suppose
that I (x) is de ned by the integral
I (x) =
Z
b
eixt q(t) dt;
(64)
a
where 0 6 a ¡ ∞; 0 ¡ b ¡ ∞; x ∈ R; |x| ¿ x0 and q : [a; b] → C has N continuous derivatives in
the interval [a; b].
Then, for |x| → ∞;
ieiax
I (x) =
x
"
N
X
q(n) (a)
n=0
#
ieibx
−(N +1)
+
O(x
)
−
(−ix)n
x
"
N
X
q(n) (b)
n=0
(−ix)n
#
+ O(x−(N +1) )
(65)
J.L. Lopez / Journal of Computational and Applied Mathematics 102 (1999) 181–194
193
and the remainder term N (z) is bounded by
rN
|N (z)| 6
|x|
N +1 "
2K +
N
+1
X
n=0
#
|q(n) (a)| + |q(n) (b)|
(N + 1)!
n!rNn
(66)
Proof. Similar to that of the above corollary setting z ≡ − ix.
We have excluded the case b = ∞ in this corollary because in this case, the bound |f(iw=x)| 6 K is
not enough to guarantee the ful llment of condition (iv) of Theorem 1. Nevertheless,
Eqs. (65)–(66) remain valid for b = ∞ if the conditions of the following corollary are satis ed.
Corollary 5. Suppose that I (x) is de ned by Eq. (64) with b ≡ ∞ and
(a) ∃ ∈ (0; =2) if x ¿ 0 or ∈ (−=2; 0) if x ¡ 0 satisfying
Z
∞ ei
0
q(w + a)eixw dw ¡ ∞
∀|x| ¿ x0 :
(67)
(b) The function q(w + a) is analytic in the sector S ≡ {w ∈ C; 0 6 |Arg(w)| 6 }.
Then, for |x| → ∞; Eq. (65) and the bound (66) hold setting q(n) (b) = 0 ∀n ¿ 0.
Proof. We perform changes of variable t = a + s in integral (64) for b ≡ ∞. Then, using condition
(b) and the residue theorem,
I (x) = eiax
Z
∞ exp(i )
eixs q(s + a) ds:
(68)
0
With the change of variable s = iw=x, the above equation reads
iw
ieiax ∞ expi( ∓=2) −w
+ a dw;
e q
(69)
I (x) =
x 0
x
where ∓ stands for x ¿ 0 and x ¡ 0, respectively. Using condition (b) again, the integrand of the
above equation satis es hypothesis (ii) of Theorem 1 with an ≡ in q(n) (a)=n!, n ≡ n and f(w=x) ≡
q(iw=x + a),
Z
iw
q
+a
x
=
N
X
q(n) (a) iw n
n=0
n!
x
+O
N +1 !
w
x
:
(70)
On the other hand, using conditions (a) and (b), ∃K ¿ 1 and K ¡ ∞ satisfying |q(s+a)eixs | 6 K ∀s ∈
[0; ∞ei ) and |x| ¿ x0 . Therefore, |q(s + a)| 6 Ke|x0 s sin | ∀s ∈ [0; ∞ei ). Then, |f(w=x)| 6 Ke|w sin |=2
∀w ∈ [0; ∞ei( ∓=2) ) and |x| ¿ 2x0 and the integral of Eq. (69) satis es also conditions (i), (iii) and
(iv) of Theorem 1 with g(w) ≡ Ke|w sin |=2 , h(w) ≡ e−w , r0 ≡ 2x0 and C ≡ [0; ∞ei( ∓=2) ). Then, using
(70) and applying this theorem, we obtain (65) for q(n) (b) ≡ 0 ∀n ¿ 0.
5. Final comments and summary
A modi cation of the TTIM [2, p. 29, Theorem 1.7.5] for obtaining asymptotic expansions of
integrals in some complex parameter z (for large |z|), has been introduced in Theorem 1 and
194
J.L. Lopez / Journal of Computational and Applied Mathematics 102 (1999) 181–194
Corollary 1. It basically demands the integrand to satisfy two conditions. The rst one is the knowledge of the expansion (convergent or asymptotic) of the integrand in power series of w=z, w being
the integration variable. The second requirement is the ful llment of bound (17) and integrability conditions (18). These properties are much more frequent in practice than uniformity condition
(3) required by the classical TTIM. Therefore, this version happens to be more useful for solving
practical problems, as it has been shown in Example 1.
As it does not demand a too special form for the integrand, one of the advantages of this version
of the TTIM is its wide range of applicability. The other advantage is that it maintains the simplicity
of the classical method: once bounding condition (17) is found and (18) is checked, the procedure
to derive the asymptotic expansion is trivial, just expand the integrand in the sequence {(w=z)n }
and interchange the integral and sum operations. This has been shown in Examples 2 and 3.
We have shown in Corollaries 2–5 that Watson’s lemma and the integration by parts technique
applied to certain families of integrals are corollaries of the TTIM. On the other hand, several
classical asymptotic techniques such as steepest descent, stationary phase, summability, Laplace’s
or Perron’s methods are based on Watson’s lemma or integration by parts [12]. This suggests the
possibility of obtaining these asymptotic methods as corollaries of the TTIM. This point is subject
of present investigations.
Acknowledgements
The author is grateful to Prof. J. Sesma for enlightening discussions. The nancial support of
Comision Interministerial de Ciencia y Tecnologa is acknowledged.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1972.
N. Blestein, R.A. Handelsman, Asymptotic Expansions of Integrals, Dover Pub., New York, 1986.
B.G.S. Doman, An asymptotic expansion for the incomplete beta function, Math. Comp. 65 (1996) 1283 –1288.
A. Erdelyi, M. Wyman, The asymptotic evaluation of certain integrals, Arch. Rational Mech. Anal. 14 (1963)
217–260.
J.L. Lopez, J. Sesma, The Whittaker function M; (z) as a function of , Constructive Approx. (1999) to be published.
E.C. Molina, Expansions for Laplacian integrals in terms of the incomplete gamma functions, International Congress
of Mathematicians, Zurich, Bell System Techn. J. 11 (1932) 563 –575.
F.W.J. Olver, Asymptotics and special functions, Academic Press, New York, 1974.
W. Rudin, Real and Complex Analysis, 2nd Ed., MacGraw-Hill, New York, 1974.
N.M. Temme, Incomplete Laplace integrals: uniform asymptotic expansions with applications to the incomplete beta
function, SIAM J. Math. Anal. 18 (1987) 1638 –1663.
N.M. Temme, Uniform asymptotic expansions of integrals: a selection of problems, J. Comput. Appl. Math. 65
(1995) 395– 417.
N.M. Temme, Special functions, An Introduction to the Classical Functions of Mathematical Physics, Wiley,
New York, 1996.
R. Wong, Asymptotic Approximations of Integrals, Academic Press, New York, 1989.
M. Wyman, The method of Laplace, Trans. Roy. Soc. Canada 2 (1963) 227–256.
Asymptotic expansions of integrals:
The term-by-term integration method
Jose L. Lopez ∗
Departamento de Matematica Aplicada, Universidad de Zaragoza, 50009, Zaragoza, Spain
Received 25 May 1998; received in revised form 12 September 1998
Abstract
The classical term-by-term integration technique used for obtaining asymptotic expansions of integrals requires the
integrand to have an uniform asymptotic expansion in the integration variable. A modi cation of this method is presented
in which the uniformity requirement is substituted by a much weaker condition. As we show in some examples, the
relaxation of the uniformity condition provides the term-by-term integration technique a large range of applicability. As a
consequence of this generality, Watson’s lemma and the integration by parts technique applied to Laplace’s and a special
c 1999 Elsevier Science B.V.
family of Fourier’s transforms become corollaries of the term-by-term integration method.
All rights reserved.
AMS classi cation: 41A60; 28A25; 44A05
Keywords: Asymptotic expansions; Integral representations; Term-by-term integration technique; Watson’s lemma;
Integration by parts method
1. Introduction
Asymptotic approximation is an important topic in applied analysis. The solutions to a large kind
of applied problems in
uid mechanics, electromagnetism, statistics and many other elds can, by
means of integral transforms, be represented by integrals. The analytic calculation of these integrals
is usually quite dicult, although they can be approximated in particular limits of interest by means
of asymptotic expansions. A lot of techniques and theories have been proposed during the last
decades for obtaining asymptotic expansions of functions de ned by means of integrals: integration
by parts, Watson’s lemma, stationary phase, steepest descent, Laplace’s method, Mellin transform
techniques, etc. (see, for example, [2, 7, 11] or [12]).
∗
E-mail: [email protected].
c 1999 Elsevier Science B.V. All rights reserved.
0377-0427/99/$ – see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 8 ) 0 0 2 1 6 - 7
182
J.L. Lopez / Journal of Computational and Applied Mathematics 102 (1999) 181–194
Among all of them, the term-by-term integration method (TTIM) is one of the most intuitive
techniques used for deriving asymptotic approximations of integrals. Let us suppose that an asymptotic expansion of the integrand in some asymptotic sequence of the asymptotic variable is known.
Then, if this expansion is uniform in the integration variable, the simple term-by-term integration
of that sequence, if it is well de ned, constitutes an asymptotic expansion of the integral [2, p.
29, Theorem 1.7.5]. The problem is that the uniformity condition is very uncommon in practice.
On the other hand, uniform asymptotic expansions usually present a very complicated form (see,
for example, [10] and references there in for a very complete survey on the subject). Therefore,
even if an uniform asymptotic expansion of the integrand is known, the integration of the terms
of this expansion may be quite dicult and the asymptotic expansion of the integral has not a
computational utility.
Uniformity is a too strong condition to justify the TTIM (integral representations of many special
functions, for example, do not satisfy this requirement). In order to make the TTIM useful for practice, the uniformity condition should be replaced by some weaker (and easier to verify) requirement.
In the new version of the TTIM that we introduce in the next section, this requirement is a certain
bounding condition over the integrand. A large class of integrals satisfy this requirement, which
provides the TTIM a large range of applicability. For example, it is applicable to integral representations of many special functions. In Section 3, new asymptotic expansions of the incomplete beta
function Bx (a; b) and of certain coecients related to the Whittaker function M; (z) are obtained
using this method.
On the other hand, from a theoretical point of view, this generalization of the TTIM led us to
shed some light upon the idea of uni cation of asymptotic methods. This idea was already suggested
in 1963 by Erdelyi and Wyman [4, 13]. In their work, they show in a lucid and expository way
that Darboux’s method, Watson’s lemma, steepest descents and stationary phase (applied to a certain
kind of integrands) can be viewed as particular cases of the method of Laplace. Although from a
more modern point of view, following the work of Wong [12], Watson’s lemma and integration
by parts should be considered as ‘fundamental methods’: steepest descents, Laplace’s, or Perron’s
method, for example, are based on Watson’s lemma, whereas stationary phase or summability methods, for example, are based on the integration by parts technique. In Section 4 we show that the
generalization of the TTIM that we present here is applicable to integrals whose integrand satis es
the conditions required by Watson’s lemma. It is also applicable to Laplace transforms when the
integrand satis es the conditions required by the integration by parts method. Besides, it is applicable to a certain family of Fourier transforms that are classically solved by the integration by parts
technique. These facts suggest the possibility of considering the TTIM as a ‘fundamental’ asymptotic
technique.
2. A new version of the term by term integration method
We will consider complex functions I (z) de ned by means of an integral representation of the
form
I (z) ≡
Z
C
h(w)f(w; z) dw;
(1)
J.L. Lopez / Journal of Computational and Applied Mathematics 102 (1999) 181–194
183
where z ∈ C, C is a complex path and h : C → C and f : C × (C\B(0; r0 )) → C (where B(0; r0 ) denotes
the open disk of center z = 0 and radius r0 ) satisfy the following two hypotheses:
(A) The product h(w)f(w; z) is integrable along the path C for |z| ¿ r0 ¿ 0. The path C may
depend on Arg(z) but not on |z|.
(B) For N = 0; 1; 2; : : : ; N0 (N0 nite or in nite),
f(w; z) =
N
X
an (w)
n=0
z n
+ Ow (z −N +1 );
|z| → ∞;
(2)
where an : C → C, {n } is a sequence of complex numbers satisfying Re(n+1 ) ¿ Re(n ) ¿ 0
∀n ¿ 0 and n 6 N + 1 and the subscript w in the remainder after N terms, Ow (z −N +1 ), means
that it may depend also on w.
The classical TTIM states the following [2, p. 29] (although the method is originally formulated
for arbitrary asymptotic sequences, here we need to consider only sequences of the form {z −n }).
Suppose that the remainder term in expansion (2) of f(w; z) can be bounded by a w-independent
quantity along all the path C, that is,
Ow (z −N +1 ) = O(z −N +1 ) ∀w ∈ C; N = 0; 1; : : : ; N0 ;
(3)
where O(z −N +1 ) is independent of w. Suppose also that
Z
h(w) dw ¡ ∞
and
Z
h(w)an (w) dw ¡ ∞
C
C
∀n 6 N:
(4)
Then, the asymptotic expansion of the integral I (z) with respect to the sequence {z −n } is just given
by introducing expansion (2) into (1) and interchanging the sum and the integral,
Z
h(w)f(w; z) dw =
C
N Z
X
n=0
h(w)an (w) dw
C
1
+ O(z −N +1 );
z n
|z| → ∞:
(5)
The evident advantage of this procedure is its great simplicity. But, if f(w; z) and=or C are not
bounded, it may be dicult to prove that the remainder term Ow (z −N +1 ) is bounded by a
w-independent quantity O(z −N +1 ). This happens, for example, when f(w; z) = f(w=z) and C is unbounded. In this case, Ow (z −N +1 ) = O((w=z)N +1 ) may not be bounded for unbounded values of w.
A simple classical example is the following.
Example 1. The exponential integral [12, p. 14, Eq. (4.1)].
Ei(z) =
Z
z
−∞
ew
dw;
w
|Arg(−z)| ¡ ;
z 6= 0;
(6)
where the integration path is de ned by −∞ ¡ Re(w) 6 Re(z) and Im(w) = Im(z). After the change
of variable w = z − x we obtain
ez
Ei(z) =
z
Z
0
∞
−x
e
x
1−
z
−1
dx:
(7)
184
J.L. Lopez / Journal of Computational and Applied Mathematics 102 (1999) 181–194
Therefore, we can take h(x) ≡ e−x ,
x
x
f(x; z) = f
≡ 1−
z
z
−1
=
N n
X
x
n=0
z
+O
N +1 !
x
z
(8)
and an (x) ≡ xn . This expansion is not uniform in x ∈ [0; ∞) and the classical term-by-term integration
method of [2] cannot be applied.
Integral representations of many other special functions [1] are also examples of failure of the
classical TTIM. The argument is similar: they are of (or can be expressed in) the form
I (z) ≡
Z
h(w)f
C
w
dw;
z
(9)
where C is unbounded and then, uniformity (3) of expansion (Eq. (2)) does not hold. In the
following, we will consider only integrands of form (9) for which hypothesis (B) reads
w
f
z
=
N
X
an
n=0
n
w
z
+O
N +1 !
w
z
|z| → ∞; an ∈ C:
;
(10)
In order to show that the term-by-term integration of expansion (10) in (9) is an asymptotic expansion
of I (z) in the sequence {z −n }, we need to show that the remainder after N terms,
"
w
N (z) ≡ h(w) f
z
C
Z
−
N
X
an
n=0
n #
w
z
dw;
(11)
satis es N (z) = O(z −N +1 ). Uniformity (3) and the rst bound in (4) guarantee this. But condition
(3) may be relaxed by using the following observation.
Remark 1. Hypothesis (10) above means that for N = 0; 1; 2; : : : ; N0 ;
∀ ¿ 1 ∃rN () ¿ 0;
n
N
X
w
w
|z N +1 | f
−
an
6 |aN +1 wN +1 |
z
z
n=0
|z| ¿ rN |w| and
∀w ∈ C;
|z| ¿ r0 :
(12)
Therefore, the remainder in (10) is bounded, at least for |z| ¿ rN |w|, by the quantity |aN +1
(w=z)N +1 |. But, if we can nd a function g : C → R+ , satisfying
f w 6 g(w)
z
∀w ∈C;
|z| ¡ rN |w| and
|z| ¿ r0 ;
(13)
then this remainder is also bounded by a O(z −N +1 )-quantity for |z| ¡ rN |w|. This is so because for
N = 0; 1; 2; : : : ; N0 and ∀|z| ¿ r0 ,
N +1
z
w
f
z
∀w ∈C;
−
and
N
X
n=0
"
#
n !
N
X
|an |
w
N +1
an
6 |(rN w) | g(w) +
z
rNn
|z| ¡ rN |w|:
n=0
(14)
J.L. Lopez / Journal of Computational and Applied Mathematics 102 (1999) 181–194
185
Joining inequalities (12) and (14) we obtain, for N = 0; 1; 2; : : : ; N0 , a O(z −N +1 ) bound for the remainder term in (10) uniformly valid ∀|z| ¿ r0 ,
"
#
X
n !
N
+1
N
X
w
|a
|
w
n
an
−
|(rN w)N +1 |
f
6 |z −N +1 | g(w) +
z
z
rNn
n=0
n=0
∀w ∈ C:
(15)
Therefore, the remainder N (z) is bounded by the integral of h(w) times the right-hand side of
the above inequality. If this integral is nite, then N (z) = O(z −N +1 ). We can summarize the above
discussion in the following.
Theorem 1. Consider a complex function I (z) de ned by integral (9) and suppose that the following conditions are satis ed:
(i) The product h(w)f(w=z) is integrable along the path C for |z| ¿ r0 ¿ 0. The path C may
depend on Arg(z) but not on |z|.
(ii) For N = 0; 1; 2; : : : ; N0 (N0 nite or in nite),
w
f
z
=
N
X
n=0
an
n
w
z
+O
N +1 !
w
z
;
|z| → ∞;
(16)
where an ∈ C and {n } is a sequence of complex numbers satisfying Re(n+1 ) ¿ Re(n ) ¿ 0
∀n ¿ 0 and n 6 N + 1.
(iii) There is a function g : C → R+ ; satisfying
f w 6 g(w)
z
∀w ∈ C; |z| ¡ rN |w| and |z| ¿ r0 :
(17)
(iv) For n = 0; 1; 2; : : : ; N + 1;
Z
C
|h(w)wn dw| ¡ ∞
and
Z
C
|h(w)g(w)wn dw| ¡ ∞:
(18)
Then, the asymptotic expansion of I (z) with respect to the sequence {z −n }; for |z| → ∞; is given by
I (z) =
N Z
X
n=0
C
n
h(w)w dw
an
+ N (z);
z n
(19)
where the remainder after N terms, N (z); is bounded by
Z
#
"
N
+1
r N +1
X
N
an
|N (z)| 6
|h(w)| g(w) +
n |wN +1 | dw
z
C
rN
(20)
n=0
and and rN () are de ned in (12).
Remark 2. The veri cation of conditions (iii) and (iv) of Theorem 1 may depend on Arg(z). This
fact is related with the Stokes lines. If expansion (16) is valid only in a certain sector of C; then
the Stokes rays also depend on this sector.
186
J.L. Lopez / Journal of Computational and Applied Mathematics 102 (1999) 181–194
The main di erence between this version and the classical method lies in the substitution uniformity requirement (3) by the much weaker condition (iii). But, if rN () in Eq. (12) is unknown, condition (iii) may not be checked easily. Nevertheless, in practice, an inequality like |f(w=z)| 6 g(w)
usually holds ∀|z| ¿ r0 and hence, the explicit value of rN is not needed for checking condition (iii).
This fact is shown in Example 1.
Example 1 (Continuation). For any r0 ¿ 0 we have that
f x ≡ 1 −
z
x −1
6
z
(
1
|sin(Arg(z))|−1
if |Arg(z)| ¿ =2;
if 0 ¡ |Arg(z)| 6 =2
(21)
and conditions (i)–(iv) of Theorem 1 are satis ed with g(x) ≡ h(Arg(z)), where we have denoted
h(Arg(z)) the second hand of the above inequality. Applying Theorem 1 and after a straightforward
algebra we obtain the well-known result [12, p. 14, Eq. (4.3)]
∞
n!
ez X
;
Ei(z) ∼
z n=0 z n
Arg(z) 6= 0:
(22)
Nevertheless, bound (20) for the remainder term is unpractical unless rN () is known. This problem
can be solved if expansion (16) is convergent in some region |z| ¿ r|w| ¿ 0. Moreover, if (16) is
convergent in this region, then the constant r may be used as rN in (17) to verify condition (iii).
These points are clari ed in the following.
Corollary 1. Assume that:
(ii′ ) The function f(t) admits a convergent series expansion for |t| 6 r −1 ; r ¿ 0;
f (t) =
∞
X
an t n ;
n=0
|t| 6 r −1
(23)
for some r ¿ 0 and conditions (i), (iii) and (iv) of Theorem 1 hold for rN = r. Then, the asymptotic
expansion of I (z) in the sequence {z −n } is given by (19) and an error bound for the remainder
is given by
Z
"
#
∞
r N +1
X
an
|wN +1 || dw|:
|N (z)| 6
|h(w)| g(w) +
r n
C
z
(24)
n=0
Proof. Hypothesis (ii′ ) implies
|z
N +1
n
∞
N
X
X
an
w
w
| f
an
−
6 |(rw)N +1 |
r n
z
z
n=N +1
n=0
∀w ∈ C; |z| ¿ r|w|:
(25)
And condition (iii) with rN = r implies
"
#
n
N
N
X
X
w
a
w
n
|z N +1 | f
an
6 |(rw)N +1 | g(w) +
z
z
r n
n=0
n=0
∀w ∈ C; |z| ¡ r|w|:
(26)
J.L. Lopez / Journal of Computational and Applied Mathematics 102 (1999) 181–194
187
Then, by using inequalities (25) and (26) and the de nition of remainder term (11), bound (24)
follows. This bound is nite by virtue of condition (iv). Therefore, N (z) = O(z −N ) and Eq. (19) is
the asymptotic expansion of I (z) in the sequence {z −n }.
Example 1 (Continuation). Expansion (8) is in fact convergent for |z|=x ¿ (N + 2)=(N + 1),
f (t) ≡ (1 − t)−1 =
∞
X
tn;
|t| 6
n=0
N +1
:
N +2
(27)
Therefore, using Corollary 1 with r ≡ (N + 2)=(N + 1), a bound for the remainder after N terms is
given by
N (z) 6 (h(Arg(z)) + N + 2)eRe(z)+1
(N + 1)!
:
|z|N +2
(28)
Asymptotic approximations of integral representations of many other special functions [1] have
been obtained by using several asymptotic methods. They could also be obtained in a systematic
and easy way using this modi ed TTIM. The procedure would be similar to the employed in
Example 1.
3. Nontrivial examples
The substitution of uniformity condition (3) by weaker condition (17) of Theorem 1 provides the
TTIM with a large range of applicability. Besides, its easy implementation lets us to obtain asymptotic
expansions of more or less sophisticated integrals. In this section we obtain, using Corollary 1, the
asymptotic expansion of the incomplete beta function Bx (a; b) in the sequence {a−n }. In Example 3,
Corollary 1 let us to obtain an asymptotic approximation of the coecients of the expansion of the
Whittaker function M; (z) in power series of from an intricate integral representation.
Example 2. The incomplete beta function Bx (a; b). An integral representation of Bx (a; b) is given
by [11, p. 128],
Bx (a; b) =
Z
x
ya−1 (1 − y)b−1 dy;
0
a ¿ 0; b ¿ 0; 0 6 x 6 1:
(29)
With the change of variable y = xe−t=a we obtain
Bx (a; b) =
xa
a
Z
0
∞
e−t (1 − xe−t=a )b−1 dt:
(30)
Therefore, we have C ≡ [0; ∞), h(t) ≡ e−t and f(t=a) ≡ (1 − xe−t=a )b−1 . The asymptotic expansion of
f(t=a) with respect to the sequence (t=a)n is its Taylor expansion around 0,
t
f
a
=
∞
X
f(n) (0) t n
n=0
n!
a
;
t
¡ −log x:
a
(31)
188
J.L. Lopez / Journal of Computational and Applied Mathematics 102 (1999) 181–194
After some algebra, it may be veri ed that this expansion is given by
(n)
f (0) = (1 − x)
n
X
b−1
k=0
n
(−1) (1 − b)k n; k
x
1−x
k
(32)
;
where the coecients n; k are universal and de ned recursively by
0;0 = 1;
n; k = 0
for k ¡ 0 or k ¿ n;
n; k = n−1; k−1 + k n−1; k
for 0 6 k 6 n:
(33)
Expansion (31) is convergent for |t=a| 6 (−log x)= for any ¿ 1. In order to apply Corollary 1, we
can take r ≡ (−log x)=. The function f(t=a) satis es hypotheses (i) and (ii′ ) with an ≡ f(n) (0)=n!
and n ≡ n. On the other hand, ∀t ∈ [0; ∞), we have that f(t=a) satis es condition (17) ∀ a ¿ 0 with
g(t) ≡ 1 + (1 − x)b−1 ;
(34)
where g(t) and h(t) satisfy (18). Therefore, we can apply Corollary 1 and, after a straightforward
algebra, we obtain,
k
∞
n
X
1
x
xa (1 − x)b−1 X
n
Bx (a; b) ∼
:
(−1)
(1 − b)k n; k
n
a
1
−
x
a
n=0
k=0
#
"
(35)
A bound for the remainder after N terms can be obtained by using formula (24). This expansion
in the sequence {a−n } may be ‘added’ to the list of expansions of the incomplete beta function (see
for example [3, 6, 9, 11, Section 11.3] and references there in).
Example 3. The coecients of the expansion of the Whittaker function M; (z) in power series
of ,
M; (z) = 22 ( + 1)z 1=2
∞
X
Fm() (z)
m=0
(−)m
m!
(36)
are given by [5],
Fm() (z) =
m
ez=2 ( + 1=2) X
m
√
(ln z)m−k Ik() (z);
k
z
k=0
(37)
where the functions Ik() (z) are de ned by the integral representation
(−1)
Ik() (z) = 22
k
2i
f
w
w
≡ 1+
z
2z
Z
e
w=2
w
−(2+1)
C
−(+1=2)
=
dk
dk
+1=2
w
2
∞
X
(−1)n
n=0
n!
+
1
2
f
!
n
w
z
w
2z
dw;
n
;
|w| ¡ |2z|
(38)
(39)
and the contour C starts at −∞ on the real axes, encircles the points 0 and −2z in the counterclockwise direction and returns to the starting point. We are interested in approximating the
J.L. Lopez / Journal of Computational and Applied Mathematics 102 (1999) 181–194
189
coecients Fm() (z) for large values of z. Then, we use Corollary 1 for deriving an asymptotic
expansion of the integrals Ik() (z) for large z. We must take h(w) ≡ ew=2 w−(2+1) (d k =dk )((w=2)+1=2 ),
an ≡ ( + 1=2)n =((−2)n n!), n ≡ n and r ≡ 1. After some algebra we nd that the function g(w) may
be taken
g(w) = eRe(w)=2
where
∞
X
k
wN X
(2w)+1=2 l=0
k−l
k w
ln
gl (w);
l 2
(40)
N
X
|w|n
1 d l
2|Im()|
Re()+l+1
1
gl (w) =
+
e
|2w|
+
(
+
)
n
2
2n n! dl
n!
n=0
n=N +1
dl
1
l ( + 2 )n ;
d
(41)
provided the contour C is such that all its points w satisfy |w| ¿ 1 and |w + 2z| ¿ 1.
Then, using Corollary 1 and after straightforward operations we obtain that the functions Ik() (z)
admit asymptotic expansions
Ik() (z)
∞
k
X
X
k
∼
n=0
j=0
j
d j ( + 12 )n
d(−) j
!
d k−j
dk−j
!
1
(−1)n
:
1
n!z n
( − n + 2 )
(42)
Introducing this expansion into (37) we obtain an asymptotic expansion of Fm() (z). This result may
be checked by substituting now (37) into (36), grouping equal powers of log(z) and reordering
sums. One obtains in this way the familiar asymptotic expansion of the Whittaker function in the
sequence {z −n }.
4. Corollaries of the term-by-term integration method
As we see in Theorem 1 or Corollary 1, the TTIM does not require a too special form for the
integrand, but only the bounding and integrability conditions (iii) and (iv). On the other hand, other
classical techniques (such as those mentioned in the introduction) are adapted to particular forms of
the integrand. Therefore, if those kind of integrands satisfy conditions (i)–(iv) of Theorem 1, the
TTIM becomes a more general technique. In this section we show that, at least, Watson’s lemma
and the integration by parts technique applied to Laplace transforms and a special family of Fourier
transforms are corollaries of this version of the TTIM.
Corollary 2 (Watson’s lemma [7, p. 113]). Suppose that I (z) is de ned by the integral
I (z) =
Z
∞
e−zt q(t) dt;
(43)
0
where
(a) q : C → C has a nite number of discontinuities and in nities in the positive real axis.
(b) As t → 0+ ;
q(t) ∼
∞
X
an t (n+a−b)=b ;
n=0
where an ∈ C; b ¿ 0 and Re(a) ¿ 0. And
(44)
190
J.L. Lopez / Journal of Computational and Applied Mathematics 102 (1999) 181–194
(c) Integral (43) has abscissa of convergence ≡ r0 cos(Arg(z)) ¿ 0 (it is integrable for |z| ¿
r0 ¿ 0 and |Arg(z)| ¡ =2).
Then, for |z| → ∞ and |Arg(z)| ¡ =2;
∞
X
I (z) ∼
n=0
n+a
b
an
:
(n+a)=b
z
(45)
And this is actually a strong asymptotic expansion of I (z);
e−Im(a)Arg(z)=b
|N (z)| 6 rN(N +a+1)=b (N +a+1)=b
|z
|
"
#
N
+1
X
K
an
+
(n+a)=b
r0
n=0 rN
((N + Re(a) + 1)=b + 1)
;
(cos(Arg(z))(N +Re(a)+1)=b+1
(46)
where K ¿ 0 is a bound of |q(t)| in a certain interval given below.
Proof. For any T ¿ 0 we write
I (z) =
Z
T
−zt
e
q(t) dt +
0
Z
∞
e
−zt
T
1
q(t) dt =
z
Z
zT
−w
e
0
w
q
z
dw + e
−zT
Z
∞
e−zt q (t + T ) dt: (47)
0
Using condition (c), the second integral in the last line satis es
Z
∞
−zt
e
Z
q (t + T ) dt 6
∞
−T
0
and therefore,
I (z) =
1
z
Z
0
zT
e−w q
|e
−t
q(t + T )| dt 6 e
T
Z
∞
0
|e−t q(t)| dt ¡ ∞
w
dw + O(e−zT ):
z
(48)
(49)
Condition (b) above means that
∞
X
w
w
w
q
∼
an
z
z
z
n=0
(n+a)=b
(50)
and then, hypothesis (iii) of Theorem 1 is satis ed for n ≡ (n + a)=b and f(w=z) ≡ (w=z)q(w=z).
Using condition (a) above, we can choose T such that 0 ¡ T ¡ rst singularity of q(t) apart from
an eventual singularity at t = 0. Therefore,
∀rN−1
¿ 0; ∃K ¿ 0;
q w 6 K
z
for rN−1 6
w
6T
z
(51)
and condition (iii) of Theorem 1 is satis ed with g(w) ≡ K|w|=r0 for any r0 ¿ 0. Condition (iv) of
Theorem 1 is trivially satis ed for the integral in Eq. (49) with h(w) ≡ w−1 e−w and C ≡ [0; zT ). On
the other hand, using condition (c) above, the rst requirement of hypothesis (i) holds, but not the
second one, because C depends on |z|. The independence of |z| is used in the proof of Theorem 1
to obtain that the coecients of expansion (19) and the bound given in Eq. (20) for |N (z)z N +1 | are
independent of |z|. Nevertheless, for the case C ≡ [0; zT ), the dependence of this bound on |z| can
be trivially eliminated,
#
"
N
+1
r (N +a+1)=b Z ∞ exp (iArg(z))
X
K
a
n
N
|e−w |
+
|N (z)| 6
(n+a)=b |w(N +a+1)=b ||dw|;
0
z
r0 w n=0 rN
(52)
J.L. Lopez / Journal of Computational and Applied Mathematics 102 (1999) 181–194
191
and after straightforward operations, we obtain (46). On the other hand, we are going to prove in
what follows that the coecients of expansion (19) are also independent of |z|, but for a quantity
O(e−zT ). This coecients are given by
Z
zT
Z
e−w w(n+a)=b−1 dw =
0
∞ ei Arg(zT )
0
e−w w(n+a)=b−1 dw −
∞ ei Arg(zT )
Z
e−w w(n+a)=b−1 dw:
(53)
zT
The second integral on the right-hand side above can be written
Z
∞ei Arg(zT )
e−w w(n+a)=b−1 dw = Mn (z)e−zT ;
(54)
zT
where
∞ ei Arg(zT )
Z
Mn (z) ≡
e−w (w + zT )(n+a)=b−1 dw:
(55)
0
A bound for the modulus of the integrand in the above integral is given by
(w + zT )(n+a)=b−1 6 e2|Im(a=b)| (|w| + T |z|)In ;
|z| ¿ r0 ;
(56)
where we have abbreviated In ≡ Int(Re((n+a)=n)) and chosen r0 ¿ T −1 . Using (56) and after straightforward operations, we nd that |Mn (z)| is bounded by a O(z In ) quantity. Therefore, the left-hand
side of Eq. (54) is a O(e−zT ) quantity. We can apply Theorem 1 and we obtain (19), where the
integrals inside the brackets equals ((n + a)=b), but for a O(e−zT ) quantity and (45) holds.
Corollary 3 (Integration by parts technique for Laplace transforms [2, p. 78]). Suppose that I (z)
is de ned by the integral
I (z) =
Z
b
e−zt q(t) dt;
(57)
a
where 0 6 a ¡ b 6 ∞; Re(z) ¿ r0 ¿ 0 and q : [a; b] → C has N continuous derivatives in [a; b];
where [a; b] must be set [a; ∞) if b = ∞.
Then, for |z| → ∞;
e−az
I (z) =
z
"
N
X
q(n) (a)
e−az
I (z) =
z
"
N
X
q(n) (a)
n=0
n=0
zn
zn
+ O(z
+ O(z
−(N +1)
−(N +1)
#
e−bz
) −
z
#
)
"
N
X
q(n) (b)
n=0
zn
+ O(z
if b = ∞
−(N +1)
#
)
if b ¡ ∞;
(58)
(59)
and the remainder term N (z) is bounded by
"
!
N
+1
r N +1
X
|q(n) (a)|
N
1 K +
|e−az |
|N (z)| 6
n
z
n!r
N
n=0
!
#
N
+1
X
|q(n) (b)|
(N + 1)!
+ 2 K +
n=0
n!rNn
|e−bz |
(cos(Arg(z)))N +2
;
(60)
192
J.L. Lopez / Journal of Computational and Applied Mathematics 102 (1999) 181–194
where 1 = 2 = 1 and K ¿ 1 is a bound for |q(t)| in [a; b] if b ¡ ∞ and 1 = 2N +2 ; 2 = 0 and K ¿ 1
is a bound for |e−zt q(t)| in [a; ∞) if b = ∞.
Proof. We consider rst the case b ¡ ∞. We extend the function q(t) to a C (N ) -function in [a; ∞)
with q(t) ≡ 0 in a neighborhood of ∞. The explicit extension of q(t) to [b; ∞) is not required.
Nevertheless, a construction of this extension is indicated, for example, in [8, p. 418, Examples 11
and 12]. Then, we can write
I (z) =
∞
Z
e
−zt
a
q(t) dt −
Z
∞
e−zt q(t) dt
(61)
b
and perform the changes of variable t = a + w=z and b + w=z in each one of the respective integrals
on the right-hand side of the above equation,
e−az
I (z) =
z
∞ exp(i Arg(z))
Z
e
−w
0
w
e−bz
+ a dw −
q
z
z
Z
∞ exp(i Arg(z))
e
−w
0
w
+ b dw:
q
z
(62)
But, using that q(t) has N continuous derivatives in the interval [a; b],
w
+t
q
z
=
N
X
q(n) (t) w n
n=0
n!
z
+O
N +1 !
w
z
∀t ∈ [a; b] and ∀w ∈ [0; ∞ ei Arg(z) ):
(63)
Therefore, the integrands in the right-hand side of Eq. (62) satisfy hypothesis (ii) of Theorem 1
up to N terms with f(w=z) ≡ q(w=z + a), an ≡ q(n) (a)=n! and f(w=z) ≡ q(w=z + b), an ≡ q(n) (b)=n!,
respectively, and n ≡ n.
On the other hand, using that q(t) is a continuous function in [a; ∞) and q(t) ≡ 0 around ∞, we
have that ∃K ¿ 1 and K ¡ ∞ satisfying |q(t)| 6 K ∀t ∈ [a; ∞). Therefore, f(w=z) 6 K ∀w ∈ [0;
∞ ei Arg(z) ) and conditions (i), (iii) and (iv) of Theorem 1 are satis ed with C ≡ [0; ∞ ei Arg(z) ),
g(w) ≡ K for any r0 ¿ 0 and h(w) ≡ e−w for each one of the integrals of Eq. (62). Then, using (63)
and applying this theorem we obtain (58) from (62). The bound (60) follows trivially from (20).
The proof is similar for the case b = ∞ setting the second integral in Eq. (61) equal to zero. But in
this case, |q(t)| may not be bounded in [0; ∞). Nevertheless, |e−zt q(t)| must be bounded ∀t ∈ [a; ∞)
by some constant K ¿ 1 ∀|z| ¿ r0 . Therefore, |e−r0 w=z q(w=z + a)| 6 K ∀|z| ¿ r0 , ∀w ∈ [0; ∞ ei Arg(z) )
and then, |q(w=z + a)| 6 KeRe(w)=2 ∀|z| ¿ 2r0 , ∀w ∈ [0; ∞ ei Arg(z) ) and conditions of Theorem 1 are
satis ed. Bound (60) follows from Eq. (20) for g(w) ≡ KeRe(w)=2 .
Corollary 4 (Integration by parts technique for Fourier transforms, [2, p. 78, 12, p. 15]). Suppose
that I (x) is de ned by the integral
I (x) =
Z
b
eixt q(t) dt;
(64)
a
where 0 6 a ¡ ∞; 0 ¡ b ¡ ∞; x ∈ R; |x| ¿ x0 and q : [a; b] → C has N continuous derivatives in
the interval [a; b].
Then, for |x| → ∞;
ieiax
I (x) =
x
"
N
X
q(n) (a)
n=0
#
ieibx
−(N +1)
+
O(x
)
−
(−ix)n
x
"
N
X
q(n) (b)
n=0
(−ix)n
#
+ O(x−(N +1) )
(65)
J.L. Lopez / Journal of Computational and Applied Mathematics 102 (1999) 181–194
193
and the remainder term N (z) is bounded by
rN
|N (z)| 6
|x|
N +1 "
2K +
N
+1
X
n=0
#
|q(n) (a)| + |q(n) (b)|
(N + 1)!
n!rNn
(66)
Proof. Similar to that of the above corollary setting z ≡ − ix.
We have excluded the case b = ∞ in this corollary because in this case, the bound |f(iw=x)| 6 K is
not enough to guarantee the ful llment of condition (iv) of Theorem 1. Nevertheless,
Eqs. (65)–(66) remain valid for b = ∞ if the conditions of the following corollary are satis ed.
Corollary 5. Suppose that I (x) is de ned by Eq. (64) with b ≡ ∞ and
(a) ∃ ∈ (0; =2) if x ¿ 0 or ∈ (−=2; 0) if x ¡ 0 satisfying
Z
∞ ei
0
q(w + a)eixw dw ¡ ∞
∀|x| ¿ x0 :
(67)
(b) The function q(w + a) is analytic in the sector S ≡ {w ∈ C; 0 6 |Arg(w)| 6 }.
Then, for |x| → ∞; Eq. (65) and the bound (66) hold setting q(n) (b) = 0 ∀n ¿ 0.
Proof. We perform changes of variable t = a + s in integral (64) for b ≡ ∞. Then, using condition
(b) and the residue theorem,
I (x) = eiax
Z
∞ exp(i )
eixs q(s + a) ds:
(68)
0
With the change of variable s = iw=x, the above equation reads
iw
ieiax ∞ expi( ∓=2) −w
+ a dw;
e q
(69)
I (x) =
x 0
x
where ∓ stands for x ¿ 0 and x ¡ 0, respectively. Using condition (b) again, the integrand of the
above equation satis es hypothesis (ii) of Theorem 1 with an ≡ in q(n) (a)=n!, n ≡ n and f(w=x) ≡
q(iw=x + a),
Z
iw
q
+a
x
=
N
X
q(n) (a) iw n
n=0
n!
x
+O
N +1 !
w
x
:
(70)
On the other hand, using conditions (a) and (b), ∃K ¿ 1 and K ¡ ∞ satisfying |q(s+a)eixs | 6 K ∀s ∈
[0; ∞ei ) and |x| ¿ x0 . Therefore, |q(s + a)| 6 Ke|x0 s sin | ∀s ∈ [0; ∞ei ). Then, |f(w=x)| 6 Ke|w sin |=2
∀w ∈ [0; ∞ei( ∓=2) ) and |x| ¿ 2x0 and the integral of Eq. (69) satis es also conditions (i), (iii) and
(iv) of Theorem 1 with g(w) ≡ Ke|w sin |=2 , h(w) ≡ e−w , r0 ≡ 2x0 and C ≡ [0; ∞ei( ∓=2) ). Then, using
(70) and applying this theorem, we obtain (65) for q(n) (b) ≡ 0 ∀n ¿ 0.
5. Final comments and summary
A modi cation of the TTIM [2, p. 29, Theorem 1.7.5] for obtaining asymptotic expansions of
integrals in some complex parameter z (for large |z|), has been introduced in Theorem 1 and
194
J.L. Lopez / Journal of Computational and Applied Mathematics 102 (1999) 181–194
Corollary 1. It basically demands the integrand to satisfy two conditions. The rst one is the knowledge of the expansion (convergent or asymptotic) of the integrand in power series of w=z, w being
the integration variable. The second requirement is the ful llment of bound (17) and integrability conditions (18). These properties are much more frequent in practice than uniformity condition
(3) required by the classical TTIM. Therefore, this version happens to be more useful for solving
practical problems, as it has been shown in Example 1.
As it does not demand a too special form for the integrand, one of the advantages of this version
of the TTIM is its wide range of applicability. The other advantage is that it maintains the simplicity
of the classical method: once bounding condition (17) is found and (18) is checked, the procedure
to derive the asymptotic expansion is trivial, just expand the integrand in the sequence {(w=z)n }
and interchange the integral and sum operations. This has been shown in Examples 2 and 3.
We have shown in Corollaries 2–5 that Watson’s lemma and the integration by parts technique
applied to certain families of integrals are corollaries of the TTIM. On the other hand, several
classical asymptotic techniques such as steepest descent, stationary phase, summability, Laplace’s
or Perron’s methods are based on Watson’s lemma or integration by parts [12]. This suggests the
possibility of obtaining these asymptotic methods as corollaries of the TTIM. This point is subject
of present investigations.
Acknowledgements
The author is grateful to Prof. J. Sesma for enlightening discussions. The nancial support of
Comision Interministerial de Ciencia y Tecnologa is acknowledged.
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