5
Asymptotic Formulas and Series
J. BARKLEY ROSSER
PROFESSOR OF MATHEMATICS
CORNELL UNIVERSITY
5.1 Introduction
Asymptotic formulas and series are useful primarily when one is concerned with the behavior of a function as the value of the independent variable becomes large. This sort of thing arises frequently in practical problems. Thus, in considering a physical system, one may wish to investigate its behavior when the number of particles is large, as in the kinetic theory of gases; or when the time is large, as in considerations of how rapidly equilibrium will be approached; or at a considerable distance from the main center of activity, as in the dispersion of waves. In the WKBJ method (see Ref. 8, pages 1092 to 1105), one determines atomic-interaction phenomena—such as phase shift for scattering—by relating the asymptotic formula for long wavelength with that for short wavelength (i.e., for large frequency). One could cite many other instances concerned with situations in which one of the quantities is large. In a considerable number of these, it would be very difficult to investigate the behavior without using asymptotic formulas and series.
In addition to indicating general behavior as the independent variable becomes large, asymptotic series can often be used to get quite accurate numerical estimates with relatively minor calculations. This is usually the case in those regions where computation by ordinary means becomes so laborious as to overtax even a large modern computer.
We shall not attempt to indicate in detail how asymptotic formulas and series are actually used in practical problems. Some such illustrations are to be found in the book by Morse and Feshbach8 and in the one by Jeffreys and Jeffreys.6 We shall give a general summary of the mathematical aspects of the question. Fuller treatments of these aspects have been given by Erdélyi3 and de Bruijn;2 the first of these two authors has compiled extensive bibliographical references,3 which can be consulted if further information is needed.
5.2 Definitions
One can see that
for |z| > 2 by noting that the expression on the right, after the first term, is a geometric series. On replacing the geometric series by its sum, one readily verifies Eq. (5.1). According to the definition that we shall shortly give, the right-hand side of Eq. (5.1) is an asymptotic series for the left-hand side.
One can have asymptotic series that diverge at all points. Note, for example, that n successive integrations by parts will give
One is tempted to let n go to infinity and replace the right-hand side by
Unfortunately, this series diverges for each value of z, so that such a replacement would be invalid.
In spite of this, the series (5.3) is quite useful. We note that the integral term in the right-hand side of Eq. (5.2) is the error that one makes if one uses n terms of the series (5.3) as an approximation for the left-hand side of Eq. (5.2). If z is positive, then clearly the error is positive or negative according as n is even or odd. That is, if we compute the sum of an odd number of terms of the series (5.3), we get a larger value than the left-hand side of Eq. (5.2), but if we compute the sum of an even number of terms, we get a smaller value.
Suppose we desire an estimate for the left-hand side of Eq. (5.2) for z = 10. With z = 10, the sum of the first nine terms of the series (5.3) is 0.91582, while the sum of ten terms is 0.91546. Thus the true value must lie between these two numbers, and we have obtained an estimate of the true value correct to three significant figures.
There is a convergent series for the left-hand side of Eq. (5.2), namely,
where γ is the Euler-Mascheroni constant. To get three-significant-figure accuracy by means of this series would require using more than 40 terms, as against only 10 terms of the series (5.3). As z becomes larger, the superior effectiveness of the series (5.3) becomes even more marked.
The property of the divergent series (5.3) that makes it useful for computation is a somewhat strengthened version of a general property that a series
can have relative to a function f(z). This property is that
for each fixed positive value of N. When the condition (5.5) holds, we say that the series (5.4) is an asymptotic series for f(z) and write
Note that the relationship (5.5) expresses the fact that for |z| large a finite number of terms of the series (5.4) is a good approximation for f(z). Because only a finite number of terms of this series is used, the question of convergence need not arise.
Usually the condition (5.5) will hold not merely for z going to infinity along the positive real axis, but for |z| going to infinity subject to a restriction of the form a ≤ arg z ≤ b, where a and b are a couple of real numbers with a < b. When this happens, we say that the series (5.4) is an asymptotic series for f(z) in the angle a ≤ arg z ≤ b. In this sense, the right-hand side of Eq. (5.1) is an asymptotic series for the left-hand side with no restriction on angle, i.e., for −π ≤ arg z ≤ π.
For complex z not at the origin or on the negative real axis, we can give a meaning to Eq. (5.2) by interpreting the integrals as line integrals along a straight line from z to 1 and thence to infinity along the positive real axis. With this interpretation, Eq. (5.2) continues to hold. Moreover, by making a careful estimate of the size of the integral on the right, one can verify that the series (5.3) is an asymptotic series for the left-hand side of Eq. (5.2) in the angle
for each positive 
. Indeed, if z is in the right-hand half plane, including the imaginary axis except the origin, it can be shown that the error in using a finite number of terms of the series (5.3) is less in absolute value than the first term omitted. Thus, in this domain, the series is very effective for computation with |z| large. The left-hand side of Eq. (5.2) appears in numerous practical applications. In particular, the case in which z is a pure imaginary arises in various optical problems.
Although strictly speaking an asymptotic series must have the form (5.4), one commonly uses more general forms. Thus, instead of using explicitly the series (5.3) one would often write
Equation (5.2) is given in Whittaker and Watson, as are the two following generalized series:
where
and
Here Γ(z) is the gamma function and Jn(z) is the Bessel function of order n. Both expressions (5.7) and (5.8) are valid in the angle (5.6).
Although the sense in which the representation (5.8) holds is a considerable generalization of that for the representation (5.5), one can use the series (5.8) very effectively for computation. Indeed, for large z, it is much more useful than the convergent series
Thus for n = 0 and z as small as 6, we can get five-decimal-place accuracy by using no more than seven terms of the series (5.8) as against at least twelve terms of the series (5.9), despite the remarkably fast convergence of the latter series. As z increases, the superiority of the representation (5.8) over (5.9) improves.
We say that g(z) is an asymptotic formula for f(z) in the angle a ≤ arg z ≤ b if
for a ≤ arg z ≤ b. In such a case, we write
f(z) ~ g(z)
From Eq. (5.2) we get
and from the representation (5.7) we obtain
in the angle (5.6). Since n! = nΓ(n) when n is a positive integer, we deduce the well-known Stirling approximation
One might jump to the conclusion that the first term of an asymptotic series always gives an asymptotic formula. This is not so, as one can see by looking at the series (5.8). Although
is a very useful approximation to Jn(z) for large z, one does not have
since the left-hand side goes to infinity at each of the points
One can have asymptotic formulas and series that hold in a smaller angle than (5.6). In particular, we have
as one can easily show. By substituting these approximations into the series (5.8), for z satisfying (5.11) we get
where
and
The problem of determining the angle in which a given asymptotic series represents a function is quite tricky. One can even have a given function represented by different asymptotic series in different angles. In a given angle, however, an asymptotic series in the strict sense, of the form (5.4), must be unique.
For asymptotic series in the strict sense, of the form (5.4), one can perform the standard operations of adding, subtracting, multiplying, and dividing in the same manner as if the series were convergent. One can also integrate term by term. Differentiation is not always possible. Indeed, there are functions that have asymptotic series of the form (5.4) even though their derivatives do not possess such asymptotic series. Since term-by-term integration is possible, however, one can conclude that, if a function and its derivative both have asymptotic series of the form (5.4), then one can find the series for the derivative by differentiating term by term in the series for the function.
All these items, and similar matters, are discussed in considerable detail by Erdélyi3 and de Bruijn.2
5.3 Integration by Parts
The method of integration by parts that we used in deriving Eq. (5.2) can be generalized. Consider
where g′(x) ≠ 0 for A ≤ x ≤ B. We can rewrite this integral as
whereupon an integration by parts changes it to
Often one of the explicit terms is an asymptotic formula. In any case, the new integral has the same form as the integral (5.13), so that one can repeat the process. Continued repetition commonly produces an asymptotic series.
Thus, starting with
we can take B = ∞, f(x) = 1, and g(x) = −x2 to get
From this, one can conclude that
holds for
if A is positive. Actually, the representation (5.15) holds for complex A and for a wider angle than stated, but the exact specifications are a bit involved, and we shall not give them.
If A2z is at all large in absolute value, then the series (5.15) is excellent for computing numerical approximations. Indeed, it is far superior to either of the following convergent series:
5.4 The Generalized Watson’s Lemma
If we wish to evaluate
we can proceed formally as follows. Expand f(x) in a Maclaurin series and integrate term by term. The result is
Sometimes this or similar devices are legitimate. Thus, if
we can expand f(x) in a Laurent series in 
and integrate term by term to get
which is a known result. Usually the procedure is not permissible, however, since the resulting series diverges for all z. Even in these cases, the method commonly is legitimate if one interprets the resulting series as an asymptotic series for the original integral. Indeed, if we take A = 0, B = ∞, and g(x) = −x in the integral (5.13), then under very modest assumptions as to the behavior of f(x) and its derivatives, we can infer from repeated integration by parts that
for
To illustrate the use of the foregoing procedure, take
in the right-hand side of Eq. (5.14) to get
Now, by using the series (5.16) with exp (−x2) for f(x), we get
which is the same as (5.15), except that according to the inequalities (5.17) it holds for
−π + ≤ arg (A2z) ≤ π −
The result embodied in the relationships (5.16) and (5.17) is usually called Watson’s lemma. There is a generalized form of it that is useful for getting asymptotic series in a wide variety of cases. This says that
in the angle (5.17). The representation (5.18) holds for each positive K or for K = ∞, subject only to some very mild hypotheses such as that, for some positive z,
exists and that f(w) is analytic in the neighborhood of w = 0. One can also let K be complex, usually at the expense of having the representation (5.18) hold in a smaller angle than (5.17).
To prove the validity of the series (5.18), one can generalize the proof given in Sec. 17.03 of the book by Jeffreys and Jeffreys.6 Note that the right-hand side of (5.18) is what one would get by expanding f(x8) as a Maclaurin series in x8 and integrating term by term from zero to infinity. Indeed, the proof of (5.18) depends on this fact.
For the special cases s = 1 and s = ½, the representation (5.18) has been established by Rosser,10 who used alternative procedures involving integration by parts. This has the advantage of simplifying the estimation of the error that one would commit in using a given finite number of terms of the series.
5.5 Asymptotic Solution of Differential Equations
The method of solution of differential equations by series is well known. To refresh our memories, let us attempt to solve the differential equation
If we differentiate term by term in Eq. (5.20) and substitute into Eq. (5.19), then we get the relationship
by equating to zero the coefficient of zr+8 in the left-hand member. If we are to have A0 ≠ 0, then we must have n2 = s2. When we take s = n, Eq. (5.21) reduces to
Since any solution of Eq. (5.19) remains a solution if multiplied by a constant, the choice of A0 is arbitrary. If we choose
we verify by Eq. (5.22) that the series on the right in Eq. (5.9) is a solution of Eq. (5.19). If we take s = −n, we get a second solution of Eq. (5.19) unless n is an integer, in which case the relationship (5.21) fails at r = 2n.
Because Eq. (5.19) is the differential equation for the Bessel function of order n, these results are consistent with Eq. (5.9).
One might expect that in a similar way one could get an asymptotic series for Jn(z) by differentiating term by term in the series (5.4), substituting into Eq. (5.19), and equating to zero the coefficient of each power of z in the left-hand member. Actually, this is a rather dubious procedure, since, as we remarked earlier, term-by-term differentiation is not always permissible with an asymptotic series. In the present case, there is a more serious difficulty in that Jn(z) does not have an asymptotic series of the form (5.4). Thus it is not surprising that the attempt to find a solution for Eq. (5.19) of the form (5.4) fails.
Upon referring to the representation (5.12), we see that Jn(z) has an asymptotic series of a different form. This suggests that we write
By substituting this into Eq. (5.19), we find that F(z) must satisfy the differential equation
differentiate term by term, and equate the coefficient of z−r on the left to zero. This gives
which immediately verifies the series given in (5.12), except for a constant factor. As remarked earlier, one can determine a solution of Eq. (5.19) only to within a multiplicative constant.
In summary, given a differential equation of the form (5.19), one may seek a solution of the form
where C is a multiplicative constant and K(z) is a multiplicative function. One cannot determine C from the differential equation. Because C is arbitrary, one can take A0 = 1. Then, if K(z) is known, one can determine the other A’s from the differential equation. There is a procedure for determining K(z) also from the differential equation in special cases. It is intricate, and we shall not attempt to describe it. It is discussed briefly in Sec. 17.12 of the book by Jeffreys and Jeffreys,6 and more extensively beginning with Sec. 7.3 of the text by Ince.5
As another example, we shall later consider a certain function G(w) that satisfies the differential equation
wG′″(w) + 2G″(w) + 2G(w) = 0
We shall show that it has an asymptotic expansion of the form
with
As it happens, our method for showing this is such that it is extremely laborious to compute Ar for r larger than 1 or 2. It is fairly easy, however, to compute Ar for larger r by using the differential equation. We take β as the independent variable, and set
Then H(β) satisfies the differential equation
Into this we substitute
differentiate term by term, and equate the coefficient of β−2r on the left to zero. This gives
Starting with the known value of unity for A0, we compute
5.6 Other Methods of Deriving Asymptotic Series
In Chap. 3 of the book by de Bruijn2 are listed a number of special means for deriving asymptotic series. We shall not attempt even to list them but shall note one that is fairly widely known. It involves use of the Euler-Maclaurin sum formula. It is applicable only in certain special cases, but when it is applicable, it generally gives very effective series. The Euler-Maclaurin sum formula is described in the book by Whittaker and Watson13 and also in the one by Jeffreys and Jeffreys.6 In the latter reference, it is used in deriving several asymptotic series, including the well-known one
which is valid in the angle (5.6). Here Br is the rth Bernoulli number, with
Many other uses of the Euler-Maclaurin sum formula to derive asymptotic series are given by Rosser.9
5.7 Eulerizing
In Sec. 5.2, we get a three-decimal estimate for the left-hand side of Eq. (5.2) with z = 10 by adding together nine or ten terms of the series (5.3). If one wishes more accuracy, one cannot get it merely by directly adding together more terms. The terms after the tenth get numerically larger and larger so that, while the partial sums are alternately too large and too small, they get farther away from the true value.
Quite a great deal more accuracy can be obtained, however, if one transforms the remainder of the series instead of merely trying to add its terms. The Euler transformation
is commonly useful for this purpose. We note that the coefficients on the right are defined by
so that the computation of these coefficients can be readily accomplished by repeatedly taking differences.
A discussion of Eq. (5.23) together with many illustrations of its use in connection with asymptotic series is given by Rosser.9 Such use of Eq. (5.23) is often called “eulerizing.”
Let us now see how eulerizing can be employed to give more accuracy in the case we considered above. The sum of the first nine terms was exactly 0.9158192. With z = 10, the next terms are
Let us now apply Eq. (5.23) to give an estimate for 108 times the quantity written in parentheses. That is, we try to identify
with the left-hand side of Eq. (5.23). For this, we take x = −1 and
By using the approximations listed above, we need to use only simple differencing operations to compute the following approximations:
Then the right-hand side of Eq. (5.23) takes the form
So our revised estimate for the left-hand side of Eq. (5.2) is
0.91581920 − 0.00018594 = 0.91563326
An accurate value, computed by using a large number of terms of the convergent series, is 0.91563334, so that our estimate is in error only by eight units in the eighth decimal place.
This agreement is all the more surprising when we note that, strictly speaking, Eq. (5.23) is valid only when applied to convergent series, and not always then. When Eq. (5.23) is applied to an asymptotic series, however, it generally yields another asymptotic series, although an instance has been given by Rosser9 in which this does not happen. In cases when the use of Eq. (5.23) yields an asymptotic series, one can often apply Eq. (5.23) a second time, to the transformed series, to get still greater accuracy.
5.8 Continued Fractions
A considerable discussion of means for obtaining more accurate numerical values by getting sharp numerical estimates of the error after summing several terms of an asymptotic series has been given by Rosser.9,10 The simplest of these, often highly effective, is eulerizing, which we have just discussed. The most powerful and widely useful method involves the use of continued fractions. Unfortunately, the theory is quite involved, so that it is not a method that can be readily applied without considerable preliminary study. We shall not attempt an explanation; indeed, we shall not even say what a continued fraction is. For some discussion and illustrations, the reader may consult Ref. 9. A full and extensive treatment is given by Wall.12
We shall, however, illustrate one of the key ideas in the theory of continued fractions. Note that, as in Eq. (5.1), the quotient of two polynomials gives a series of the form (5.4) that converges for all sufficiently large z. If we choose two polynomials such that the first n terms of their quotient exactly equal the first n terms of the series (5.4), then the quotient should be a good approximation. Indeed, it usually is.
As an example, note that the series in (5.1) agrees exactly with the series (5.3) for the first three terms. Taking z = 10 in the left-hand side of Eq. (5.1) gives
This is a fair approximation to the true value, which is 0.9156 to four decimal places.
To get a quotient of two quadratic polynomials, we write formally
multiply out on the right, and equate coefficients of the first five powers of z on the two sides. Solving the resulting equations gives a = 5, b = 2, c = 6, and d = 6. We have, in fact,
for 
. The first five terms on the right agree exactly with the first five terms of the series (5.3), and the sixth terms are similar. If we take z = 10 in the left-hand side above, we get
as compared with the more accurate five-decimal-place approximation 0.91563. This is considerably closer than one could come directly with any partial sum of the series (5.3).
Quotients of polynomials such as we have illustrated arise automatically as convergents of continued-fraction expansions. One of the tremendous advantages of continued-fraction expansions is that in a large number of cases a divergent asymptotic series will transform into a convergent continued fraction. In such a case, one can get as accurate an approximation as desired by going to a quotient of polynomials of sufficiently high degree.
5.9 Laplace’s Method
One often wishes to determine an asymptotic formula or asymptotic series for
or
Of these integrals, (5.24) is actually a special case of (5.25), as can be seen by taking
f(t) = g(t) + g(−t)
in (5.25). If f(t) is analytic in the neighborhood of the origin, one can get an asymptotic series in 
for the integral (5.25) by expanding f(t) as a Maclaurin series in t and integrating term by term. One can get the same expansion by using the generalized Watson’s lemma; specifically, substitute x = t2 in the integral (5.25) and then take s = ½ and r = −½ in the expansion (5.18).
Various classical procedures such as Laplace’s method, the method of steepest descent, and the saddle-point method amount to nothing more than applying various ingenious devices to reduce the function in question to one of the two forms (5.24) or (5.25). A common instance of this arises when one is considering the integral (5.13) and g′(x) = 0 for a unique x with A ≤ x ≤ B, so that one cannot use integration by parts as discussed in Sec. 5.3.
We shall illustrate by showing how to derive the expansion (5.7). We start with the Euler formula
By putting t = zx, we get
We now let
It turns out that x is an analytic function of t near the origin, and we have
This has the form (5.24). If we expand dx/dt as a Maclaurin series in t and integrate term by term, we get the expansion (5.7), valid in the angle (5.17).
A detailed discussion of this particular expansion is given by Rosser.10 Its validity in the larger angle (5.6) is established by Whittaker and Watson.13
If g′(x) = 0 for several points in the interval (A,B), we break the integral (5.13) into several integrals, with g′(x) = 0 at most once in each interval. Then we treat the various integrals separately.
The procedure illustrated above can be employed to devise a proof for results such as the following, use of which is commonly called Laplace’s method.
THEOREM. Let A, B, and C be real numbers satisfying A < C < B Let f(x) be a function, not necessarily real, that is continuous for A ≤ x ≤ B. and analytic at x = C. Let g(x) be a real function, analytic at x = C, that has a continuous second derivative for A < x < B. Let g′(C) = 0, g″(C) < 0, f(C) ≠ 0; let also g′(x) ≠ 0 for A < x < C and for C < x < B. Then an asymptotic formula for
is
valid for
while in the same angle each of
has the asymptotic formula
For example, if we take A = 0, B = ∞, C = 1,
f(x) = 1g(x) = 1 − x + log x
in the theorem, we get
Certain intuitive considerations are quite helpful in understanding the above theorem. The hypotheses of the theorem ensure that g(x) takes its maximum at x = C. Thus, since 
, we see that
|ezg(x)|
also takes its maximum at x = C. Furthermore, since g(x) is in the exponent, this maximum will be very sharp at x = C. Consequently, most of the value of the integral comes from the values of x near C. So we replace f(x) and g(x) by their values near x = C, specifically
f(C)for f(x)andg(C) + ½g″(C)(x − C)2for g(x)
Then our integral assumes the form
For large z, the value of this integral is changed only slightly if we replace A and B by −∞ and +∞, respectively. This gives the asymptotic formula stated in the theorem.
By means of a proof based on these ideas, the theorem can be strengthened in various ways. For instance, the hypotheses that f(x) and g(x) are analytic at x = C are quite unnecessary. One can strengthen the theorem still further by weakening the hypotheses of continuity and existence of derivatives. Various results of this sort are given by Rosser.11
Of much more practical use is the fact that the foregoing theorem continues to hold even if f and g are allowed to be functions of both x and z instead of merely x, provided the dependence on z is rather mild. For details, the reader should consult Ref. 11, but we shall cite one useful result. If there are constants k, a, and b such that
uniformly in the region
and |z| sufficiently large, then the above theorem continues to hold with k in place of f(C), a in place of g(C), and b in place of g″(C).
We shall illustrate with a simple example. Consider
Take y = zx and get
Now take A = 0, B = ∞, C = 1,
and get k = 1, a = −1, and b = −1. So an asymptotic formula for I(z) is
The efficiency of this formula is illustrated in Table 5.1.
Table 5.1 Efficiency of the Asymptotic Formula J(z) for the Integral I(z)
5.10 The Method of Stationary Phase
If A and B are both finite, then the theorem of the previous section holds even for z on the imaginary axis, and indeed without the hypotheses that f(x) and g(x) are analytic at C; see the discussion by Erdélyi3 under the heading “The method of stationary phase.” This is the name given to the employment of the theorem when z is a pure imaginary. Specifically, let the hypotheses of the theorem hold, except perhaps those concerning analyticity. Also let A and B both be finite. Then, for t real and positive,
and
In the method of stationary phase, one can let g″(C) > 0, since it suffices to change the signs of j, g(x), g(C), and g″(C) simultaneously in the above results to recover the case g″(C) < 0.
As in the Laplace method, each of the integrals, from A to C and from C to B, has as an asymptotic formula one-half the formula given for the integral from A to B.
The intuitive argument for the method of stationary phase goes as follows: At x ≠ C, g(x) is changing. Then for large t, tg(x) is changing very rapidly. Thus ejtg(x) is oscillating very rapidly. So when we evaluate
the alternate plus and minus values produced by the oscillation tend to cancel out. At x = C, g(x) is stationary, and ejtg(x) does not oscillate. Thus there is not the cancellation that there is at other values of x. So the value of the integral depends primarily on what f(x) and g(x) do near x = C. Specifically, we make the same replacements for f(x) and g(x) as in our intuitive discussion of the previous section, and then replace A and B by −∞ and +∞. This gives the results stated.
If f(x) is positive and g(x) and t are real, then f(x)ejtg(x) represents an oscillation of amplitude f(x) and phase tg(x). Hence the name “stationary phase,” since x = C is the point at which g(x) is stationary.
We illustrate the method of stationary phase by finding the asymptotic behavior of Jn(t) for t positive and n a nonnegative integer. It is shown by Whittaker and Watson13 that in this case
We take A = 0, B = π, C = π/2, f(x) = e−jnx, and g(x) = sin x. This gives
By rewriting the latter as
we see that Jn(t) behaves like
for large positive t. In this connection, we refer to the expansion (5.8).
5.11 The Method of Steepest Descent
The method of steepest descent and the saddle-point method are essentially the same, and they amount to reducing the quantity to be studied to one of the forms (5.24) or (5.25). Unfortunately, there is no general procedure for discovering how to make the required reduction. Illustrations, with little motivation, are given by Jeffreys and Jeffreys6 and by Erdélyi.3 In Chap. 5 of the book by de Bruijn,2 there is a serious discussion of heuristic considerations that would be helpful in undertaking to carry out reductions to one of the foregoing forms. Some slightly different considerations are put forth by Chester and Friedman.1 Anyone who is faced with the need for expanding a given function in an asymptotic series, and suspects that the method of steepest descent might be of value in doing it, would do well to consult one of the sources cited, or indeed several of them. He will still find that the method requires considerable resourcefulness and ingenuity on his own part.
We shall not attempt any motivation, but shall remark only that if one has an integral of the form (5.13) to deal with, one attempts to transform the path of integration in the complex plane so that it passes through a “saddle point.” One feature of saddle points that helps in finding them is that g′(z) = 0 at a saddle point. We shall give a particularly simple example.
The Airy integral is useful in a number of practical applications. For real z, it is defined by
This is clearly equivalent to
where L1 is the imaginary axis. If z is positive, we can justify changing the path of integration to L2, a vertical line through 
. On this line, we have 
, and so get
Expanding the cosine term in a Maclaurin series and integrating term by term gives
Note that the point through which we passed the new path, namely 
, is one of the two points at which the derivative of
is zero. Why we chose this zero rather than the other, why we chose the particular path through 
that we did rather than some other path, and similar questions are illuminated, but not entirely answered, in de Bruijn’s discussion.2
We mention in passing that one can extend the definition of the Airy integral into the complex plane. When one does, the asymptotic series given above is valid in the angle (5.6).
We saw above that one can get an asymptotic formula for the integral (5.13) even when f and g are functions of both x and z, provided the dependence on z is slight. One can get asymptotic series also in this case. There is no well-developed theory for this general case, and one often proceeds by improvising through analogy with simpler situations. A fairly intricate example is given at the end of Ref. 10.
In the method of steepest descent, in which one is trying to reduce a given expression to one of the form (5.24) or (5.25) by deforming the path of integration to pass through a saddle point, one can often greatly simplify the choice of a path by permitting g in the integral (5.24) or f in the integral (5.25) to depend on z as well as t. This is discussed in Sec. 4.4 of the book by de Bruijn2 and in Ref. 11. In the latter, a proof is sketched for the following principle.
where z and w are complex variables, and the integral is taken along some path P of finite length. Let the path begin at a point A such that
g′(A) = 0
and never return to A. Let h(w) and g(w) be analytic in the neighborhood of A. Let h(w) be bounded on P. Let there be real constants a, b, and c, with a ≤ b and c positive, such that
for w on P and a ≤ arg z ≤ b. Let P have the inclination θ at the point A, in the sense that the half ray defined by w = A + xeiθ, with 0 ≤ x, shall be tangent to P at A. Introduce t as a new variable of integration by the relationship
Take α to be the constant such that the Maclaurin series for g(w) − g(A) in terms of t starts with the term −αt2. Under the substitution (5.28), the integral (5.26) takes the form
with
We now have a form analogous to the integral (5.25) and can proceed as suggested for dealing with (5.25). Specifically, form the Maclaurin expansion
Each Kn is a polynomial in z of degree not more than n/3. If we now formally integrate term by term in
and collect terms in like powers of 
, we get a series
with the Bn constant, such that
is an asymptotic series for the integral (5.26) in the angle a ≤ arg z ≤ b.
It may appear that our conditions on the path P have been rather stringent. Actually, we can encompass a wide variety of paths by minor adaptations. For example, if the path ends at A instead of beginning at A, we can reverse the order of integration; then the principle above can be applied and will give an asymptotic series for the negative of our function. If the path passes several times through A, we cut it into several parts; some will begin at A and others will end at A, but we can handle both cases. In case the path passes once through A with no change in direction at A and does not begin or end at A, we could cut the path into two pieces, as proposed above, but it can be shown that the same result would be achieved if we left the path in one piece and merely replaced the integral (5.30) by
If the path passes through several points at which g′(w) = 0, it may be necessary to cut the path into several pieces and treat these pieces separately. Finally, if one has a case in which P is infinite in length, one will have to make some estimate to justify throwing away all but a finite portion; this is usually very easy.
We shall now give an illustration. We should warn the reader that the above principle is advantageous primarily in that it often permits a simple choice of path. The penalty that one pays for this simplicity in the choice of the path is that commonly the calculations become extremely difficult if one wishes more than one or two terms of the series. One can often, however, find a differential equation that is satisfied by the function. Then one can usually get the higher coefficients with comparative ease by applying the method of Sec. 5.5. Our illustration will display these features, and it will also show how one can get easier calculations by using more sophistication in the choice of a path.
Suppose we write
and require the asymptotic behavior of F(x) for large positive x. We easily obtain
We put
t = (2x2)⅓w
and get
This has the form (5.26) with
We can easily take the path of integration to be the unit circle, whereupon it will pass through all three zeros of g′(w), to wit, the three cube roots of −1. Of these, −1 itself causes no trouble, but the other two interfere with each other. If we attempt to take either root as A, then condition (5.27) fails because of the presence of the other root.
One can proceed to cut the path into two parts, namely, the upper and lower halves of the unit circle. Thus let us take
where P1 is the upper half of the unit circle and P2 is the lower half.
On P1, we take
We can satisfy the condition (5.27) with a = −π/2, b = 0, and any reasonably small value of c, say c = 0.2. We get
so that
α = 3eπj/3
Then (5.29) takes the form
Substituting this into the integral (5.31) gives
From this we can get the first two terms of the desired asymptotic series, as well as some of the constituents of the third and fourth terms. To get all the constituents of the third term of the asymptotic series, we would have to carry the integration as far as the term in t12. In general, to get the (n + 1)st term of the asymptotic series, we would have to carry the integration as far as the term in t6n. We have verified, however, that
for
with
We can proceed similarly with I2(z), taking
Since we are interested only in real x, however, we can proceed more expeditiously by noting that in this case I1(z) and I2(z) are conjugate complexes of each other.
It will be noted that we were able to use the very simplest sort of path, but that the calculations were so complicated as to discourage us from attempting to compute even the third term of the series. We shall see shortly that this is not a serious matter, as the later coefficients can be obtained fairly readily by means of a differential equation. We digress temporarily, however, to show how one could simplify the calculations by using more subtlety in connection with the path of integration. Putting w = ejv in Eq. (5.33) and recalling that x is real gives
We can apply the principle stated earlier to the integral above, with
Then
Next, writing only those terms that will be needed to get three terms of the asymptotic series, we get
Substituting into the integral (5.31) and integrating term by term gives
with β as in (5.34).
We now show how to get more terms of the series by use of a differential equation. Starting from the series definition (5.32), we easily discover a differential equation
xF′″(x) + 2F″(x) + 2F(x) = 0
that F(x) satisfies. Unfortunately, F(x) equals not an explicit asymptotic series, but the real part of one, so that we have to work not with F(x) directly, but with a closely related function. We define
where L is the path in the w plane that starts at the origin, proceeds right along the real axis to w = 1, then counterclockwise halfway around the unit circle to w = −1, and finally left along the negative real axis to w = − ∞. Clearly, G(x) is closely related to
Indeed, an elementary estimate shows that the difference G − 2I1 is exponentially small compared with the asymptotic value of I1 that we discovered above. This lets us infer that G and 2I1 have the same asymptotic series, and so furnishes an instance of the case for which one has a path of infinite length, for example L, and justifies throwing away all but a finite portion in order to be able to apply the principle given above. We noted above that the real part of the asymptotic series for G and 2I1 is the asymptotic series for F(x).
By putting
w = (2x2)−⅓t
and deforming the contour of integration, we conclude that
Then
Thus we can proceed as in Sec. 5.5 to get the higher terms of the asymptotic series for G(x), with the result
5.12 Further Use of Integration by Parts
Suppose that we are considering the integral (5.13) and that the hypotheses of the theorem of Sec. 5.9 are satisfied. As we noted, most of the value of the integral (5.13) comes from the values of x near C. This suggests that
should be a good approximation for the integral (5.13) for large z. Indeed, it usually is. One commonly cannot get a closed form for the integral (5.35), so that this result is not particularly useful. When one does wish to use (5.35) as an approximation for (5.13), it is useful to have an estimate of the error, which is
Since our hypotheses ensure that g′(x) has a simple zero at x = C, we see that
is a well-behaved function for A ≤ x ≤ B. Thus we can integrate by parts in the right-hand integral above, getting
as an expression for the error incurred in using the integral (5.35) as an approximation for the integral (5.13). The integral on the right above has the same form as (5.13), so that one could repeat the process, usually getting an even better estimate for (5.13).
Instances of useful expansions obtained by repeated applications of the above process are given by Rosser.10 In the report by Franklin and Friedman,4 this process is studied for f and g of a special form, and it is shown that the process leads in this case to a convergent asymptotic series.
To give an example, let us consider again the function I(z) introduced in Sec. 5.9. We have
and so take A = 0, B = ∞, C = 1,
This gives
We integrate by parts on the right, getting
We can repeat the process on the right-hand integral, getting
One could repeat the process indefinitely. If we start the next repetition, but do not carry through the integration by parts, we get three terms of an approximate series for I(z) as follows:
The effectiveness of this series is illustrated in Table 5.2.
Table 5.2 Efficiency of the Asymptotic Series J(z) for the Integral I(z)
EXERCISES
Most of the exercises given below are intended to illustrate points raised in this chapter and to afford the reader an opportunity to test his mastery of these points. The final exercise (No. 8) is a problem that arose in the theory of boundary layers in aerodynamics, and it is included as an illustration of the kind of problem that can arise in practical applications.
1. For each of the following integrals, find an asymptotic formula or series as requested and state in what angle in the z plane it is valid.
a. Asymptotic series for
b. Asymptotic formula for
d. Asymptotic formula for
e. Asymptotic formula for
f. Asymptotic formula for
2. Derive an asymptotic formula for
Hint: First perform an integration by parts in the manner suggested for handling the integral (5.13) when g′(x) ≠ 0; note that, since the factor x − 1 is present, this integration by parts will succeed even though g′(x) = 0 in the present case.
3. Note that the integral (5.42) plus twice the integral (5.40) equals the integral (5.39), but the same does not hold for their asymptotic formulas. Explain this.
4. Find an asymptotic formula for
where C is the path given by
5. By using the principle set forth in Sec. 5.11, get two nonvanishing terms of the asymptotic series for
6. Compute a four-decimal value of Γ(j). Hint: For some positive integer N, compute a sufficiently accurate value for Γ(N + 1 + j) by the series (5.7). Then use
7. Prove that
Hint: Define
By the result Γ(w + 1) = wΓ(w), prove that G(z) = G(z + 1), and so conclude that G(z) = G(z + N), where N is a large integer. Now use the series (5.7) to show that
8. Determine the two leading terms of a series that gives the asymptotic behavior of
for a and b both large and real. Hint: Clearly the integral (5.43) is the real part of
Now proceed as suggested under the integral (5.13).
REFERENCES
1. Chester, C., and B. Friedman, “Uniform Asymptotic Expansions,” Report No. IMM-NYU 219, New York University Institute of Mathematical Sciences, July, 1955.
2. de Bruijn, N. G., “Asymptotic Methods in Analysis,” Interscience Publishers, Inc., New York, 1958.
3. Erdélyi, A., “Asymptotic Expansions,” Dover Publications, New York, 1956.
4. Franklin, J., and B. Friedman, “A Convergent Asymptotic Representation for Integrals,” Research Report No. BR-9, New York University Institute of Mathematical Sciences, December, 1954.
5. Ince, E. L., “Ordinary Differential Equations,” Longmans, Green & Co., Ltd., London, 1926. Reprinted by Dover Publications, New York, 1956.
6. Jeffreys, H., and B. S. Jeffreys, “Methods of Mathematical Physics,” 2d ed., Cambridge University Press, New York, 1950.
7. Morrey, Charles B., Nonlinear Methods, chap. 16 in “Modern Mathematics for the Engineer,” First Series, edited by E. F. Beckenbach, McGraw-Hill Book Company, Inc., New York, 1956.
8. Morse, P. M., and H. Feshbach, “Methods of Theoretical Physics,” McGraw-Hill Book Company, Inc., New York, 1953.
9. Rosser, J. B., Transformations to Speed the Convergence of Series, J. Res. Natl. Bur. Standards, vol. 46, pp. 56–64, 1951.
10. ——, Explicit Remainder Terms for Some Asymptotic Series, J. Rational Mech. Anal., vol. 4, pp. 595–626, 1955.
11. ——, Some Sufficient Conditions for the Existence of an Asymptotic Formula or an Asymptotic Expansion, a chapter (pp. 371–387) in “On Numerical Approxition—Proceedings of a Symposium Conducted by the Mathematics Research Center, U.S. Army, at the University of Wisconsin, Madison, April 21–23, 1958,” University of Wisconsin Press, Madison, Wis., 1959.
12. Wall, H. S., “Analytic Theory of Continued Fractions,” D. Van Nostrand Company, Inc., Princeton, N.J., 1948.
13. Whittaker, E. T., and G. N. Watson, “A Course of Modern Analysis,” American ed., The Macmillan Company, New York, 1946.