Asymptotic Methods for Integrals

The following sections are included:

• Dedication
• Preface
• Acknowledgments
• Contents

The following sections are included:

• Symbols used in asymptotic estimates
• Asymptotic expansions
• A first example: Exponential integral
• Generalized asymptotic expansions
• Properties of asymptotic power series
• Optimal truncation of asymptotic expansions

We consider the large-z asymptotic expansions of Laplace-type integrals of the form

The standard form for Laplace's method is the integral

We have mentioned in Chapter 3 the connection between Laplace's method and the saddle point method. In this chapter we give several examples and details of the method. The integrals are presented as contour integrals in the complex plane, for which we select modifications of the contours before applying Laplace's method. Usually, these modified contours run through a saddle point of the integrand…

An analytic function f(z) may have different asymptotic approximations when |z| → ∞ in different domains of the complex plane. Usually these approximations are related and an approximation in one domain may also play a role in a different domain, although it may be exponentially small inside that domain. In the following section we explain this for the Airy function, together with the role of several approximations, and how they become important when going from one domain to another domain. All this is related to the Stokes phenomenon…

The Euler gamma function is usually defined by the integral representation

In Tricomi (1950) we find the remark “Seit einiger Zeit pflege ich die unvollständige Gammafunktion γ(α, x) das Aschenbrödel der Funktionen zu nennen”. In this and other chapters we take care of Tricomi's complaint…

Airy functions are special cases of Bessel functions of order and are named after G. B. Airy (1838), a British astronomer, who used them when studying rainbow phenomena. They occur in many other problems from physics, for example as solutions to boundary value problems in quantum mechanics and electromagnetic theory…

For Bessel functions with large argument detailed information is available about estimates for the remainders in the large-z expansions; see Olver (1997, pp. 266–270). These bounds are obtained by using the differential equation of the Bessel functions…

We consider asymptotic forms of this important class of functions, which are also known as confluent hypergeometric functions. The expansions in this chapter are for large argument z or for large parameters with other variables fixed, or in bounded domains. For other combinations of the parameters and argument we need uniform expansions. These will be given in other chapters as applications of general standard forms and as examples for certain methods; see Chapter 22 (in particular, §22.5) and Chapter 27…

Parabolic cylinder functions are associated with the differential equation

The Gauss hypergeometric function is defined by

Large parameter cases of the ₃F₂-functions arise frequently and there are no systematic methods to approach the problem. In Chapter 12 we have explained that we have suitable integral representations and connection formulas for the ₂F₁-functions for obtaining all kinds of expansions for large parameters. For the ₃F₂-functions these convenient starting points are missing, and even for the polynomial cases with argument ±1 the approach has to be based on ad hoc methods…

In the method of stationary phase integrals of the type

A generating function for a special function F_n(z) usually has the form of the convergent series

In §2.5 we have given examples in which the idea of Watson's lemma can be used for integrals of the form

In this chapter we mention expansions that can be viewed as alternatives or modifications of the expansions obtained by using Watson's lemma. Some of these expansions are convergent and have asymptotic properties.

In the derivation of uniform asymptotic expansions of integrals we often encounter the problem of expanding a function in two points. It seems that in Chester et al. (1957) this type of expansion has been used for the first time in asymptotic analysis. It was used to derive Airy-type expansions for integrals that have two nearby (or coalescing) saddle points. This reference does not give further details about two-point Taylor expansions…

Hermite polynomials show up in several problems of asymptotic analysis. We consider three different instances where these classical orthogonal polynomials can be used as main approximants…

It is clear from the examples in earlier chapters that an essential step in the saddle point method is to perform one or several substitutions after which the integral

When the function f in Laplace's method, see Chapter 3 and (3.0.1), has a singularity near the origin, the straightforward method, which is based on expanding this function in a power series, may fail. The coefficients of this expansion will show the effect of this singularity. In particular, when the singular point approaches the origin under the influence of a parameter, we need other methods. In this chapter we discuss the simple case of a single pole near the saddle point.

In this chapter we consider Case 4 of Table 20.1 in which the exponential function has a saddle point at t = α. When α ≤ α₀ < 0 (with α₀ fixed) we can use Laplace's method as we have explained in §10.4.2. When α ≥ α₁ > 0 we can use Watson's lemma. An extra point of attention is the factor t^β−1, an algebraic singularity at the origin…

In this chapter we consider Case 6 of Table 20.1 in which the exponential function has two saddle points at . When η ≠ 0 and fixed, we can use Laplace's method for one or both saddle points. When η → 0 we should take into account the contributions from both saddle points in one expansion. As we will see, we can use Airy functions to handle this case.

In this chapter we consider what happens when for certain polynomials the degree n is large, together with other parameters. We will see that Hermite polynomials can be used in uniform expansions. The main examples considered in this chapter are , the Gegenbauer or ultraspherical polynomials (with as special case the relativistic Hermite polynomials) and , the Tricomi–Carlitz polynomials for large values of the degree n and the orders γ or α. The Laguerre polynomials for large α are considered in §32.4. For all these cases we show how to derive approximations of the zeros of the polynomials in terms of the zeros of the Hermite polynomials, and we compare these with numerical values. In §24.4 we mention several other examples considered in the literature…

We consider Laplace-type integrals of the form

In this chapter we consider Case 8 of Table 20.1. The integrand has the form t^λ−1 e^−zt f(t), which is of a simple Laplace-type. However, we consider an incomplete Laplace integral with interval [α,∞), where α ≥ 0, and we consider both α and λ as uniformity parameters that can range through all nonnegative values, whether or not they are large in comparison with z…

In this chapter we consider Laplace-type integrals in which the exponential function exp(−z(t + β²/t) has an essential singularity at the origin and two saddle points at ±β that approach the origin as β → 0. As indicated in Case 9 and Case 10 of Table 20.1, we can use modified Bessel functions to handle this asymptotic feature…

The Kummer function ₁F₁, or confluent hypergeometric function (see Chapter 10), can be defined as the limit

Legendre functions are special cases of the Gauss hypergeometric functions, and we can use the many connection formulas for these functions. For certain combinations of the parameters the hypergeometric functions satisfy quadratic transformations, and Legendre functions are those special cases of the hypergeometric functions for which a quadratic transformation exists. This property gives special relations that we will use for asymptotic representations…

In Chapter 11 we have given a few properties of parabolic cylinder functions and we have derived the expansions for large argument z. When the parameter is large, we need expansions in the form of uniform expansions, which are in terms of elementary functions and in terms of Airy functions, and have been obtained in Olver (1959) by using the governing differential equation. In Temme (2000) we have indicated that some of Olver's expansions can be modified in the sense that they become valid with a double asymptotic property: valid when either the argument or the order, or both, are large. Jones (2006) has also given similar expansions, by using the differential equation in the complex plane, and he provided error bounds for remainders…

The differential equation

Laguerre polynomials have been considered in §15.5, where we have given expansions for large n and bounded values of z and α. In this chapter we give more details on the large degree asymptotics. Frst we consider bounded values of z, with two cases: α bounded and α depending on n. When z is allowed to become large as well, we need other uniform expansions. For this we summarize the results given in Frenzen and Wong (1988), with expansions in terms of Airy functions and Bessel functions, and those from Temme (1986b), with expansions in terms of Hermite polynomials…

Generalized Bessel polynomials of degree n, complex order μ and complex argument z, denoted by , have been introduced in Krall and Frink (1949), and can be defined by their generating function. We have (Grosswald, 1978):

We describe a method for obtaining an asymptotic expansion of both Stirling numbers and for large values of n. The expansions are obtained by using a modification of the saddle point method, and are valid uniformly for 0 ≤ m ≤ n. Short tables are given to show the results for n = 10. These experiments confirm the uniform character of our estimates.

We consider the integral

In this chapter we show how to transform a selection of well-known distribution functions, such as the gamma and beta distributions, into a standard form. We derive asymptotic expansions with respect to one parameter, and the expansion is uniformly valid with respect to a second parameter. The standard form is a convenient starting point in several cases, however, we will see that for some examples it is better to use contour integral representations. This will be explained in Chapter 37 for the incomplete gamma functions, in Chapter 38 for the incomplete beta functions, and in Chapter 39 for the non-central chi-square distribution functions (or Marcum functions). In a final chapter we consider the problem of inverting the cumulative distribution functions by using asymptotic methods.

We recall the definitions of the incomplete gamma functions given in Chapter 7:

The incomplete beta function is considered in §26.3, where a uniform expansion is given by using a certain real incomplete Laplace integral and a loop integral. In this chapter we give uniform expansions in terms of the complementary error function, but we start with simpler expansions…

We consider for positive x, y, μ the functions

We consider the sum

We consider the asymptotic behavior of the function

The inversion of cumulative distribution functions is an important topic in statistics, probability theory, and econometrics, in particular for computing percentage points of chi-square, F, and Student's t-distributions. In the tails of these distributions the numerical inversion is not very easy, and for the standard distributions asymptotic formulas are available…

The following sections are included:

• Bibliography
• Index