A simplification of Laplace’s method: Applications to the Gamma function and Gauss hypergeometric function

The main difficulties in the Laplace’s method of asymptotic expansions of integrals are originated by a change of variables. We propose a variant of the method which avoids that change of variables and simplifies the computations. On the one hand, the calculation of the coefficients of the asymptotic expansion is remarkably simpler. On the other hand, the asymptotic sequence is as simple as in the standard Laplace’s method: inverse powers of the asymptotic variable. New asymptotic expansions of the Gamma function Γ(z) for large z and the Gauss hypergeometric function ₂F₁(a,b,c;z) for large b and c are given as illustrations. An explicit formula for the coefficients of the classical Stirling expansion of Γ(z) is also given.     

A Systematic “Saddle Point Near a Pole” Asymptotic Method with Application to the Gauss Hypergeometric Function

In recent works [ 1 ] and [ 2 ], we have proposed more systematic versions of the Laplace’s and saddle point methods for asymptotic expansions of integrals. Those variants of the standard methods avoid the classical change of variables and give closed algebraic formulas for the coefficients of the expansions. In this work we apply the ideas introduced in [ 1 ] and [ 2 ] to the uniform method “saddle point near a pole.” We obtain a computationally more systematic version of that uniform asymptotic method for integrals having a saddle point near a pole that, in many interesting examples, gives a closed algebraic formula for the coefficients. The asymptotic sequence is given, in general, in terms of exponential integrals of fractional order (or incomplete gamma functions). In particular, when the order of the saddle point is two, the basic approximant is given in terms of the error function (as in the standard method). As an application, we obtain new asymptotic expansions of the Gauss Hypergeometric function ₂F₁(a, b, c; z) for large b and c with c > b + 1 .     

Asymptotic Analysis of Numerical Steepest Descent with Path Approximations

We propose a variant of the numerical method of steepest descent for oscillatory integrals by using a low-cost explicit polynomial approximation of the paths of steepest descent. A loss of asymptotic order is observed, but in the most relevant cases the overall asymptotic order remains higher than a truncated asymptotic expansion at similar computational effort. Theoretical results based on number theory underpinning the mechanisms behind this effect are presented.     

An Explicit Formula for the Coefficients of the Saddle Point Method

Most standard textbooks about asymptotic approximations of integrals do not give explicit formulas for the coefficients of the asymptotic methods of Laplace and saddle point. In these techniques, those coefficients arise as the Taylor coefficients of a function defined in an implicit form, and the coefficients are not given by a closed algebraic formula. Despite this fact, we can extract from the literature some formulas of varying degrees of explicitness for those coefficients: Perron’s method (in Sitzungsber. Bayr. Akad. Wissensch. (Münch. Ber.), 191–219, 1917) offers an explicit computation in terms of the derivatives of an explicit function; in (de Bruijn, Asymptotic Methods in Analysis. Dover, New York, 1950) we can find a similar formula for the Laplace method which uses derivatives of an explicit function. Dingle (in Asymptotic Expansions: Their Derivation and Interpretation, Academic Press, New York, 1973) gives the coefficients of the saddle point method in terms of a contour integral. Perron’s method is rediscovered in (Campbell et al., Stud. Appl. Math. 77:151–172, 1987), but they also go farther and compute the above mentioned derivatives by means of a recurrence. The most recent contribution is (Wojdylo, SIAM Rev. 48(1):76–96, 2006), which rediscovers the Campbell, Fr?man and Walles’ formula and rewrites it in terms of Bell polynomials (in the Laplace method) using new ideas of combinatorial analysis which efficiently simplify and systematize the computations. In this paper we continue the research line of these authors. We combine the more systematic version of the saddle point method introduced in (López et al., J. Math. Anal. Appl. 354(1):347–359, 2009) with Wojdylo’s idea to derive a new and more explicit formula for the coefficients of the saddle point method, similar to Wojdylo’s formula for the coefficients of the Laplace method. As an example, we show the application of this formula to the Bessel function.     

An Explicit Formula for the Coefficients in Laplace’s Method

Laplace’s method is one of the fundamental techniques in the asymptotic approximation of integrals. The coefficients appearing in the resulting asymptotic expansion arise as the coefficients of a convergent or asymptotic series of a function defined in an implicit form. Due to the tedious computation of these coefficients, most standard textbooks on asymptotic approximations of integrals do not give explicit formulas for them. Nevertheless, we can find some more or less explicit representations for the coefficients in the literature: Perron’s formula gives them in terms of derivatives of an explicit function; Campbell, Fröman and Walles simplified Perron’s method by computing these derivatives using an explicit recurrence relation. The most recent contribution is due to Wojdylo, who rediscovered the Campbell, Fröman and Walles formula and rewrote it in terms of partial ordinary Bell polynomials. In this paper, we provide an alternative representation for the coefficients that contains ordinary potential polynomials. The proof is based on Perron’s formula and a theorem of Comtet. The asymptotic expansions of the gamma function and the incomplete gamma function are given as illustrations.     

High-precision computation of the confluent hypergeometric functions via Franklin-Friedman expansion

We present a method of high-precision computation of the confluent hypergeometric functions using an effective computational approach of what we termed Franklin-Friedman expansions. These expansions are convergent under mild conditions of the involved amplitude function and for some interesting cases the coefficients can be rapidly computed, thus providing a viable alternative to the conventional dichotomy between series expansion and asymptotic expansion. The present method has been extensively tested in different regimes of the parameters and compared with recently investigated convergent and uniform asymptotic expansions.     

The Riemann-Hilbert approach to strong asymptotics for orthogonal polynomials on [−1,1]

We consider polynomials that are orthogonal on [−1,1] with respect to a modified Jacobi weight (1−x)^α(1+x)^βh(x), with α,β>−1 and h real analytic and strictly positive on [−1,1]. We obtain full asymptotic expansions for the monic and orthonormal polynomials outside the interval [−1,1], for the recurrence coefficients and for the leading coefficients of the orthonormal polynomials. We also deduce asymptotic behavior for the Hankel determinants and for the monic orthogonal polynomials on the interval [−1,1]. For the asymptotic analysis we use the steepest descent technique for Riemann-Hilbert problems developed by Deift and Zhou, and applied to orthogonal polynomials on the real line by Deift, Kriecherbauer, McLaughlin, Venakides, and Zhou. In the steepest descent method we will use the Szeg? function associated with the weight and for the local analysis around the endpoints ±1 we use Bessel functions of appropriate order, whereas Deift et al. use Airy functions.     

Error bounds for the uniform asymptotic expansion of the incomplete gamma function

We derive simple, explicit error bounds for the uniform asymptotic expansion of the incomplete gamma function Γ(a,z) valid for complex values of a and z as |a|→∞. Their evaluation depends on numerically pre-computed bounds for the coefficients c_k(η) in the expansion of Γ(a,z) taken along rays in the complex η plane, where η is a variable related to z/a. The bounds are compared with numerical computations of the remainder in the truncated expansion.     

Asymptotic expansions for the distributions of functions of a correlation matrix

This paper deals with asymptotic expansions for the non-null distributions of certain test statistics concerning a correlation matrix in a multivariate normal distribution. For this purpose an asymptotic expansion is given for the distribution of a function of the sample correlation matrix. As special cases of the resulting expansion, asymptotic expansions for the distributions of the sample correlation coefficient, Fisher's z-transformation and arcsine transformation are also given.     

Strong asymptotics of the recurrence coefficients of orthogonal polynomials associated to the generalized Jacobi weight

We study asymptotics of the recurrence coefficients of orthogonal polynomials associated to the generalized Jacobi weight, which is a weight function with a finite number of algebraic singularities on [−1,1]. The recurrence coefficients can be written in terms of the solution of the corresponding Riemann–Hilbert (RH) problem for orthogonal polynomials. Using the steepest descent method of Deift and Zhou, we analyze the RH problem, and obtain complete asymptotic expansions of the recurrence coefficients. We will determine explicitly the order 1/n terms in the expansions. A critical step in the analysis of the RH problem will be the local analysis around the algebraic singularities, for which we use Bessel functions of appropriate order. In addition, the RH approach gives us also strong asymptotics of the orthogonal polynomials near the algebraic singularities in terms of Bessel functions.     

Derivation of transitional asymptotic expansions from integral representations

A method for deriving transitional asymptotic expansions from integral representations is described and applied to Anger function and modified Hankel function. The method consists in deriving asymptotic expansions of the function considered as well as its first derivativeat the transition point using conventional methods such as Laplace’s method or the method of steepest descents. Since both the functions considered satisfy a second order linear differential equation, it is possible to obtain asymptotic expansions of higher order derivatives of the functions from the first two expansions. Thus asymptotic expressions for all the derivatives at the transition point are known and a Taylor expansion of the function in the neighbourhood of the transition point can be written. The method is also applicable to the generalized exponential integral, Weber’s parabolic cylinder function and Poiseuille function.     

New series expansions of the Gauss hypergeometric function

The Gauss hypergeometric function ₂ F ₁(a,b,c;z) can be computed by using the power series in powers of $z, z/(z-1), 1-z, 1/z, 1/(1-z),~\textrm{and}~(z-1)/z$ . With these expansions, ₂ F ₁(a,b,c;z) is not completely computable for all complex values of z. As pointed out in Gil et al. (2007, §2.3), the points z?=?e ^±iπ/3 are always excluded from the domains of convergence of these expansions. Bühring (SIAM J Math Anal 18:884–889, 1987) has given a power series expansion that allows computation at and near these points. But, when b???a is an integer, the coefficients of that expansion become indeterminate and its computation requires a nontrivial limiting process. Moreover, the convergence becomes slower and slower in that case. In this paper, we obtain new expansions of the Gauss hypergeometric function in terms of rational functions of z for which the points z?=?e ^±iπ/3 are well inside their domains of convergence. In addition, these expansions are well defined when b???a is an integer and no limits are needed in that case. Numerical computations show that these expansions converge faster than Bühring’s expansion for z in the neighborhood of the points e ^±iπ/3, especially when b???a is close to an integer number.     

Uniform asymptotic methods for integrals

We give an overview of basic methods that can be used for obtaining asymptotic expansions of integrals: Watson’s lemma, Laplace’s method, the saddle point method, and the method of stationary phase. Certain developments in the field of asymptotic analysis will be compared with De Bruijn’s book Asymptotic Methods in Analysis. The classical methods can be modified for obtaining expansions that hold uniformly with respect to additional parameters. We give an overview of examples in which special functions, such as the complementary error function, Airy functions, and Bessel functions, are used as approximations in uniform asymptotic expansions.     

Global asymptotic expansions of the Laguerre polynomials—a Riemann–Hilbert approach

By using the steepest descent method for Riemann–Hilbert problems introduced by Deift–Zhou (Ann Math 137:295–370, 1993), we derive two asymptotic expansions for the scaled Laguerre polynomial $L^{(\alpha)}_n(\nu z)$ as n→∞, where ν=4n+2α+2. One expansion holds uniformly in a right half-plane $\text{Re}\; z\geq \delta_1, 0<\delta_1<1$ , which contains the critical point z=1; the other expansion holds uniformly in a left half-plane $\text{Re}\; z\leq 1-\delta_2, 0<\delta_2<1-\delta_1$ , which contains the other critical point z=0. The two half-planes together cover the entire complex z-plane. The critical points z=1 and z=0 correspond, respectively, to the turning point and the singularity of the differential equation satisfied by $L^{(\alpha)}_n(\nu z)$ .     

Stokes line width

The concept of Stokes line width is introduced for the asymptotic expansions of functions near an essential singularity. Explicit expressions are found for functions (switching functions) that switch on the exponentially small terms for the Dawson integral, Airy function, and the gamma function. A different, more natural representation of a function, not associated with expansion in an asymptotic series, in the form of dominant and recessive terms is obtained by a special division of the contour integral which represents the function into contributions of higher and lower saddle points. This division leads to a narrower, natural Stokes line width and a switching function of an argument that depends on the topology of the lines of steepest descent from the saddle point.P. N. Lebedev Physics Institute, Rossiskoi Adademii Nauk. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 92, No. 1, pp. 24–40, July, 1992.     

Asymptotic expansions of the Hurwitz-Lerch zeta function

The Hurwitz-Lerch zeta function Φ(z,s,a) is considered for large and small values of a∈C, and for large values of z∈C, with |Arg(a)|<π, z∉[1,∞) and s∈C. This function is originally defined as a power series in z, convergent for |z|<1, s∈C and 1−a∉N. An integral representation is obtained for Φ(z,s,a) which define the analytical continuation of the Hurwitz-Lerch zeta function to the cut complex z-plane C?[1,∞). From this integral we derive three complete asymptotic expansions for either large or small a and large z. These expansions are accompanied by error bounds at any order of the approximation. Numerical experiments show that these bounds are very accurate for real values of the asymptotic variables.     

Uniform asymptotic expansion for a class of polynomials biorthogonal on the unit circle

An asymptotic expansion including error bounds is given for polynomials {P _n, Q_n} that are biorthogonal on the unit circle with respect to the weight function (1?e^iθ)^α+β(1?e^?iθ)^α?β. The asymptotic parameter isn; the expansion is uniform with respect toz in compact subsets ofC{0}. The pointz=1 is an interesting point, where the asymptotic behavior of the polynomials strongly changes. The approximants in the expansions are confluent hyper-geometric functions. The polynomials are special cases of the Gauss hyper-geometric functions. In fact, with the results of the paper it follows how (in a uniform way) the confluent hypergeometric function is obtained as the limit of the hypergeometric function₂ F ₁(a, b; c; z/b), asb→±∞,z≠b, withz=0 as “transition” point in the uniform expansion.     

A uniform asymptotic expansion for the incomplete gamma function

We describe a new uniform asymptotic expansion for the incomplete gamma function Γ(a,z) valid for large values of z. This expansion contains a complementary error function of an argument measuring transition across the point z=a (which is different from that in the well-known uniform expansion for large a of Temme), with easily computable coefficients that do not involve a removable singularity at z=a. Our expansion is, however, valid in a smaller domain of the parameters than that of Temme. Numerical examples are given to illustrate the accuracy of the expansion.     

Generalized Fresnel integrals

A general class of (finite dimensional) oscillatory integrals with polynomially growing phase functions is studied. A representation formula of the Parseval type is proven as well as a formula giving the integrals in terms of analytically continued absolutely convergent integrals. Their asymptotic expansion for “strong oscillations” is given. The expansion is in powers of ?1/2M, where ? is a small parameters and 2M is the order of growth of the phase function. Additional assumptions on the integrands are found which are sufficient to yield convergent, resp. Borel summable, expansions.     

Exponentially small expansions in the asymptotics of the Wright function

We consider exponentially small expansions present in the asymptotics of the generalised hypergeometric function, or Wright function, _pΨ_q(z) for large |z| that have not been considered in the existing theory. Our interest is principally with those functions of this class that possess either a finite algebraic expansion or no such expansion and with parameter values that produce exponentially small expansions in the neighbourhood of the negative real z axis. Numerical examples are presented to demonstrate the presence of these exponentially small expansions.