1 Introduction

The particular Wright function under consideration (also known as a generalized Bessel function) is defined by

where λ is supposed real and μ is, in general, an arbitrary complex parameter. The series converges for all finite z provided λ > −1 and, when λ = 1, it reduces to the modified Bessel function z(1−μ)/2Iμ−1(2z). The asymptotics of this function were first studied by Wright [14, 15] using the method of steepest descents applied to the integral representation

The case corresponding to λ = −σ, 0 < σ < 1 arises in the analysis of time-fractional diffusion and diffusion-wave equations. The function with negative λ has been termed a Wright function of the second kind by Mainardi [3], with the function with λ > 0 being referred to as a Wright function of the first kind. In the former context, Mainardi [4, Appendix F] defined the auxiliary functions

These functions are interrelated by the following relation:

The case μ = 0 in (1.1) also finds application in probability theory and is discussed extensively in [13], where it is denoted by

and referred to as a ’reduced’ Wright function.

Plots of M_σ(x) for real x and varying σ are presented in [4, Appendix F] and [5]. These graphs illustrate the transition between the special values σ=0,12,1, where M_σ(x) has simple representations in terms of known functions. These are

where Ai is the Airy function. As σ → 1⁻, the function M_σ(x) tends to the Dirac delta function δ(x − 1).

In this paper we present the asymptotic expansions of F_σ(x) and M_σ(x) for x → ±∞ by exploiting the known asymptotics of the function ϕ(−σ, 0, x) discussed in [13]. The resulting expansions involve a combination of algebraic-type and exponential-type expansions, for which explicit representation of the coefficients in both types of expansion is given. In order to give a self-contained account, we describe the derivation of the expansion for M_σ(x) based on the asymptotics of integral functions of hypergeometric type described in [10] (see also [11, §4.2]). The asymptotic treatment of the function W_{λ, μ}(z) given by Wright [14], [15] did not give precise information about the coefficients appearing in the exponential expansions; see also [10] for a more detailed account.

2 The asymptotic expansions of F_σ(x) and M_σ(x) for x → ±∞

We define the quantities

The connection between F_σ(x) and the function ϕ defined in (1.6) is

The asymptotic expansions of ϕ(−σ, 0, x) for x → ±∞ when 0 < σ < 1 are given in [13, §5.2]. We therefore obtain the expansions stated in the following theorem:

The case σ=12 needs no special attention since

but see the comment at the end of Section 3 as this case is associated with a Stokes phenomenon.

The coefficients c_j(σ) appearing in the exponential expansions in Theorem 2.1 can be obtained ^[**] from [10, (4.6)] (when the parameter δ therein is replaced by σ). We have:

where the first few coefficients d_j(σ) are:

These polynomial coefficients are related to the so-called Zolotarev polynomials, see [13].

From the relation (1.5), we have M_σ(±x) = F_σ(±x)/(±πx) and after a little algebra we deduce the expansion of M_σ(x) given by:

For x → +∞, the function M_σ(x) is exponentially small for all values of σ in the interval 0 < σ < 1. The case of M_σ(−x), however, is seen to be more structured. When 0<σ<13, the factor cos πσ/κ > 0 and M_σ(−x) is exponentially large (with an oscillation) as x → +∞, with the algebraic expansion H^(x) being subdominant. When σ=13, this factor is zero and E^(x) is oscillatory with an algebraically controlled amplitude and H^(x)≡0. When 13<σ<12, the expansion E^(x) is exponentially small and the behaviour of M_σ(−x) is controlled by the algebraic expansion. Finally, when 12<σ<1 the expansion of M_σ(−x) is purely algebraic in character.

3 The asymptotic expansion of M_σ(x) for x → ±∞

In order to make this paper more self contained we present in this section an alternative derivation of the expansion of M_σ(x) as x → ±∞. Define the function

Then use of the reflection formula for the gamma function shows that the auxiliary Wright function M_σ(x) defined in (1.4) can be expressed in terms of ℱ(x) as

and in a similar manner

From the discussion in [10, Section 2], the Stokes lines for ℱ(z), where its exponential expansion is maximally subdominant relative to its algebraic expansion, are situated on the rays arg z = ±πκ. An important distinction between (3.2) and (3.3) when x > 0 is that for M_σ(−x) the arguments of the functions ℱ(xe±πiσ) are only situated on the Stokes lines arg z = ±πκ when σ=12, since κ=1−σ=12, whereas for M_σ(x) the arguments of ℱ(xe±πiκ) are situated on the Stokes lines for all values of σ in the range 0 < σ < 1.

From [10, §4.1] (see also [12, §2.3]), the asymptotic expansion of ℱ(z) is given by

as |z| → ∞. The upper or lower signs are chosen according as arg z > 0 or arg z < 0, respectively and ϵ denotes an arbitrarily small positive quantity. The formal exponential and algebraic expansions E(z) and H(z) are defined by

where the parameters κ, h, ϑ and A(σ) are defined in (2.1) and the coefficients c_j(σ) are those appearing in Theorem 1; see Appendix A for an algorithm for the calculation of these coefficients.

The exponential expansion E(z) is dominant in the sector |arg z|<12πκ and becomes exponentially small in the adjacent sectors 12πκ<|arg z|≤πκ. On arg z = ±πκ, E(z) is maximally subdominant relative to the algebraic expansion and switches off in a smooth manner (at fixed |z|) across these Stokes lines. The expansion in this case is given in Subsection 3.1.

4 Numerical results

We present some numerical results to verify the expansions in Theorems 1 and 2. In Table 1 the values (accurate to 10dp) of the coefficients c_j(σ) appearing in the exponential expansion are shown for two values of σ. Table 2 shows the absolute relative error in the computation of M_σ(x) as a function of the truncation index j with the expansion (2.5) in Theorem 2.2. Table 3 shows the same error in the computation of M_σ(−x) for different values of x with the expansion (2.6). Note that for σ = 1/4 and σ = 1/3 in Table 3 we have H^(x)≡0. For σ = 2/5, the algebraic expansion H^(x) has been optimally truncated, but for σ = 2/3 the truncation index was taken as k = 11.

The limit σ → 1⁻ in M_σ(x) can be obtained by setting σ = 1 − ϵ, ϵ → 0⁺ so that the parameters in (2.1) become

Then from Theorem 2.2 we obtain the leading behavior:

as x → +∞ and ϵ → 0. The above approximation for M_σ(x) agrees with that obtained in [6] by application of the saddle-point method applied to the integral (1.2). This argument is explained in Section 5.

Plots of M_σ(x) given by (4.1) are shown in Figs. 1, 2 and 3, and plots of M_σ(−x) given by (4.2) are shown in Fig. 4. These illustrate the transition to a Dirac delta function as ϵ → 0.

Figure 1

Plots of M_σ(x) for ϵ = 0.1 in linear (up) and semi-logarithmic scale (down)

5 The Kreis-Pipkin Method

This section focuses on the argument introduced as a variant of the saddle-point method by Kreis and Pipkin in [2] (revisited by Mainardi and Tomirotti in [7] for a wave problem in fractional viscoelasticity) to deal with sharply peaked functions around x ~ 1, in the limit where ϵ → 0⁺. The method is of interest from a numerical point of view, allowing us to deal with functions that are also physically relevant such as, in seismology, the pulse response in the nearly elastic limit.

Figure 2

Plots of M_σ(x) for ϵ = 0.01 in linear (up) and semi-logarithmic scale (down)

In this way it is possible, adapting the Kreis-Pipkin method to the M-Wright function, to study its asymmetric structure when it tends towards the Dirac delta function δ(x − 1).

We start by recalling the integral definition of the auxiliary Wright function F_σ(x) (compare (1.2))

related to the function M_σ(x) by (1.5). Taking into account the procedure described in [2], we have with σ = 1 − ϵ that the exponent is stationary at the point:

Figure 3

Plots of M_σ(x) for ϵ = 0.001 in linear (up) and semi-logarithmic scale (down)

Figure 4

Plots of M_σ(−x) for different values of ϵ in linear (up) and semi-logarithmic scale (down)

The next step is to expand t^−ϵ in powers of ϵln t/t₀, this being more accurate than expanding the exponent in powers of t − t₀, and using z = t/t₀. The final result is:

where we emphasize that this procedure is valid only in the limit ϵ → 0⁺. The relation (1.5) tells us that the expression of M_σ(x) can be simply obtained from knowledge of F_σ(x), and vice versa. The exponential factor appearing in (5.2) has a saddle point at z = 1 and the contour can be made to coincide with the steepest descent path, which is locally perpendicular to the real z-axis at the saddle. Then finally, by means of the steepest descent method, the function M_σ(x) as σ → 1⁻ can be expressed via a real integral.

The results are presented in Figs. 5, 6 and 7; each figure shows a comparison in linear and semi-logarithmic scale between three curves obtained using different methods. These are respectively the Kreis-Pipkin method, (4.1) of this work and the classical saddle-point method used by Mainardi and Tomirotti [6] (denoted by M-T 1995 in the figures). Note that the curves obtained via (4.1) and M-T 1995 are equivalent, and indeed can be simply shown to be analytically equivalent. The plots for 0 ≤ x ⋍ 1 in the Kreis-Pipkin method were obtained via an integral representation for M_σ(x) combined with matching to the leading asymptotic behavior. The method proposed by Kreis and Pipkin is thus seen to be a useful tool to reproduce the asymmetric structure of M_σ(x) that would be impossible with the standard saddle-point method.

6 Conclusions

We have given asymptotic expansions as x → ±∞ for the auxiliary Wright functions F_σ(x) and M_σ(x) defined in (1.3) and (1.4) when 0 < σ < 1. These expansions consist of series of an exponential and algebraic character whose relative dominance depends on the parameter σ. An algorithm for determining the coefficients in the exponential expansion is discussed and explicit representation of the first few coefficients has been given.

Numerical results are presented to confirm the accuracy of the expansions. Of particular interest is the the limit σ → 1⁻, where the function M_σ(x) approaches a Dirac delta function centered on x = 1. Graphical results based on the Kreis-Pipkin method are given that illustrate the leading asymptotic forms and the transition of M_σ(x) to a delta function.