We obtain new uniform asymptotic approximations for integrals with a relatively exponentially small remainder. We illustrate how these results can be used to obtain remainder estimates in the Bleistein method. The method is created to deal with new types of integrals in which the usual methods for remainder estimates fail. As an application, we obtain an asymptotic expansion for as in |ph λ|≤π/2 uniformly for large |z|.
In this paper, we obtain new uniform asymptotic approximations for integrals with a relatively exponentially small remainder via a surprisingly simple method. We will also illustrate how these results can be used to obtain remainder estimates in the so-called Bleistein method. As an application of our results, we show in the final section that 1.1as in |ph λ|≤π/2 uniformly for large |z|. In this result, a and b are fixed complex constants and . For the notation of the hypergeometric function and the Kummer-U function, see ch. 13 and ch. 15 in [1,2]. Note that this result is a generalization of the well-known limit 1.2see 6.8(1) in .
In Temme , it was indicated how to obtain (1.1). The author even gave the integral representation for the left-hand side of (1.1) that we will use. It seemed that all we had to do is just apply the Bleistein method. That is exactly what we will do in §4. However, when we tried to show that the remainder was of the required order, we encountered a new problem: the approximant U(a,b,z) has a relatively complicated behaviour near z=0, as can be seen in §13.2(iii) in [1,2]. The usual methods did not work. Our new method is based on expanding the remainder in a new series with a relatively exponentially small remainder. After deriving this method, we realized that it can also be applied to the original integral itself. In this way, we obtain a uniform asymptotic approximation with a relatively exponentially small remainder. Note that the new approximations are not uniform asymptotic expansions. The expansions do not have an asymptotic property, but we will give estimates for the terms.
Let us consider integrals of the form 1.3For these integrals, the critical points are saddle points of p, the singularities of p and q, and possibly the endpoints of the contour of integration. We assume that the position of these critical points depends on a parameter ζ, that they coalesce at the origin when ζ=0, and that G(t) is analytic for at least |t|≤1. The functions p and q are chosen in such a way that the integral without the G is a good approximant. To obtain an asymptotic expansion for that holds uniformly for ζ near the origin, all the relevant critical points should contribute. The Bleistein method is a multi-point expansion of G(t) about the relevant critical points combined with a special integration by parts. For examples, see [5–12] and [13, ch. vii].
The Bleistein method is a very subtle process. We will show that one can also obtain a uniform asymptotic approximation by substituting the truncated Taylor series , where we insist that the number of terms depends on λ, i.e. M−γ|λ|∈[0,1), for some positive constant γ. In this way, we obtain an approximation with a remainder of the form (1.3), in which G(t) is replaced by tMS(t). By taking γ large enough, the integral representation for this remainder will have a dominant saddle point outside the region of coalescence, and simple estimates show that the remainder is exponentially small compared with the first approximants. Note that these new expansions are for |ζ| bounded, whereas in the normal Bleistein method, |ζ| is sometimes allowed to be unbounded.
This paper is organized as follows. In §2, we first illustrate the main ideas for probably the best-known example: two coalescing saddles. Even in this case, at the moment that we need an estimate for the integral representation of the remainder, one extra integration by parts would be required. We also give a few details on how one could obtain a uniform asymptotic approximation with a relatively exponentially small remainder. The main class of integrals is introduced in §3. These are integrals of the form (1.3) with p(t,ζ)=t, q(t,ζ)=tb−1(1+t/ζ)−a and . Theorem 3.1 contains the uniform asymptotic approximation with a relatively exponentially small remainder, and in lemma 3.2, we translate this result into an order estimate for the integral. This order estimate is exactly what is needed in §4, in which we apply the Bleistein method to obtain the uniform asymptotic expansion for this class of integrals. Finally, we apply these results in §5 to the hypergeometric function mentioned on the left-hand side of (1.1). We obtain the uniform asymptotic expansion in terms of the Kummer-U function, and illustrate the new uniform asymptotic approximation with exponentially small remainder.
2. Uniform asymptotics for integrals
We will illustrate some of the main steps in the process of obtaining uniform asymptotic expansions via the Bleistein method with probably the best-known example: two coalescing saddles. In the case that one encounters an integral in which two coalescing saddles dominate the asymptotics, one first converts the integral to its canonical form. We will omit that step and start with the canonical form. Let 2.1Often, in the literature, the function G(t) also depends on ζ. Without loss of generality and for simplicity of presentation, we write G(t) instead of G(t,ζ). Furthermore, we take λ>0 and assume that the function G(t) is analytic for |t|<1+ε, where ε is a positive constant, and also that G(t) is analytic and bounded along the path of integration. We are interested in the case that , for ζ in some neighbourhood of the origin, say . Note that the phase function has saddle points at . In the case that ζ>0, there exists a steepest descent path just passing through one saddle point, and in the case that ζ<0, the steepest descent path will pass through both saddle points (figure 1).
It follows that in the case where we want an asymptotic approximation for that holds for ζ close to the origin, then both saddle points should contribute. One option would be to use a multi-point expansion such as the one that is discussed in , but we will use the Bleistein method. For more details, see Olde Daalhuis . We take G0(t)=G(t), and define Hn(t), Gn+1(t), n=0,1,2,…, by writing 2.2with pn, qn following from the substitution of , 2.3It is important to note that we do not introduce any new singularities, and the growth of Gn(t) at infinity is similar to the growth of G0(t). If we use (2.2) in (2.1) and integrate N times by parts, we obtain 2.4where 2.5Ai(λ) is the Airy function and Ai′(λ) its derivative. For the Airy function, see [1,2], ch. 9. To show that the expansion in (2.4) has an asymptotic property, one has to show that there exists a positive constant KN such that 2.6for say and λ>λ0. Usually , p. 371, one splits the proof in two parts: for the case |ζλ2/3|≥ρ, one can check the contributions of the saddle points, and for the case |ζλ2/3|≤ρ, one makes the following observation. Note that in this case, the Airy function in (2.6) and its derivative are just bounded functions. It is easy to show that in this case , as , but this is not sufficient because the Airy function in (2.6) could be zero. One extra integration by parts is needed, 2.7and because , as , bound (2.6) holds. Here, we also use the fact that the Airy function and its derivative cannot be zero at the same time.
Let us investigate what would happen when we use some ideas from exponential asymptotics, and in (2.1) expand 2.8where for the moment is a contour that encircles t and the origin once. We obtain 2.9with 2.10and 2.11Note that functions very similar to um(λ,ζ) were introduced in , §10.4.
By taking a suitable contour for , we can guarantee that there exists a constant K such that |S(t)|≤K for all t on our path of integration. We also take M=λ/2+ρ with ρ∈[0,1). Hence, the phase function in (2.11) becomes . Because p′(t)=t2−ζ+1/(2t), this phase function has three saddle points, say at t=t1, t2, t3, and because we restrict , we can choose them such that ph t1∈(0,π/2), ph t2∈(−π/2,0) and . Note that now two saddle points are active. In the lower half plane, the steepest descent path emanates from , passes through t2 and ends at the origin, and in the upper half plane, it starts at the origin, passes through t1 and ends at (figure 1).
The main difference between the integrals in (2.1) and (2.11) is the factor tM. For the integral in (2.11), the saddle points are located at t=t1,t2, and we have |tj|<1, j=1,2. It follows that is clearly exponentially small compared with f(λ,ζ). Hence, the finite sum in (2.9) is an asymptotic approximation for f(λ,ζ), with a relatively exponentially small error. Via the saddle-point method and using the identity , we obtain the estimate 2.12
We do not claim that expansion (2.9) has an asymptotic property. Let us investigate this expansion. We identify u0=λ−1/3Ai(ζλ2/3) and u1=−λ−2/3Ai′(ζλ2/3). Via integration by parts, we obtain the recurrence relation 2.13where in the case m=0 the third term vanishes. Hence, it is possible to express um in terms of the first two: um=amu0+bmu1, where a0=b1=1, a1=b0=b2=0, a2=ζ, and am and bm satisfy recurrence relation (2.13). am and bm are polynomials in ζ and 1/λ. Because we assume and take m≤M−2, i.e. , it is easy to show via induction that 2.14Hence, we can rewrite (2.9) as 2.15Note that because we assume that G(t) is analytic for |t|<1+ε, it follows that the gm are bounded. Thus, the sums in (2.15) are clearly bounded, and it follows that there exists a constant K such that |f(λ,ζ)|≤K(λ−1/3|Ai(ζλ2/3)|+λ−2/3|Ai′(ζλ2/3)|) for say and λ>λ0. Hence, this is an alternative way of obtaining order estimates for these integrals. To obtain (2.6), one extra integration by parts was needed, whereas for the new method, we expand in many terms.
Note that the expansions (2.4) and (2.15) are very similar. The two series in (2.4) clearly have an asymptotic property, whereas the two series in (2.15) behave like convergent geometric series. Often the calculation of the Taylor coefficients gm is much easier than the calculation of the pn and qn. The big advantage of (2.4) is that even taking N=1 gives a one-term approximation for the left-hand side, whereas in (2.15), we have to take many terms.
3. Main example: an exponentially small accurate uniform asymptotic approximation
For fixed parameters, we consider the following integral: 3.1where and |ph (ζ)|<π. The integrand in (3.1) has possible branch points at the origin and at t=−ζ, which coalesce in the case that ζ=0. Let ε be a positive constant. We assume that G(t) is analytic in a domain , which is the union of the disc |t|≤1+ε and the sector |ph t|≤θ0, where (figure 2). We also assume that G(t) is of at most polynomial growth as in .
It is well known that to obtain an asymptotic expansion for an integral of the form (3.1) for that is supposed to hold uniformly for ζ close to the origin, both the endpoint t=0 and the branch point t=−ζ should contribute. The Bleistein method will do that for us, and will give us the uniform asymptotic approximation in terms of the Kummer-U function. We will show the details in §4. However, the proof of that result is not straightforward, and one new step is needed in which we use the lemma mentioned at the end of this section. The surprising fact is that we can also obtain a uniform asymptotic approximation by just focusing on the endpoint t=0 and letting the number of terms depend on |λ|.
The main theorem of this paper is the following.
Let and G(t) be analytic in region as shown in figure 2 and of at most polynomial growth as in . Then, 3.2with , such that γ1∈[0,1), and 3.3as in |ph λ|≤π/2 uniformly for and |ph ζ|≤π.
Here, gm,m=0,1,2,…, are the Taylor coefficients of G(t) about t=0 and they are bounded. We define α0=1,α1=0,β0=0,β1=1 and the other αm,βm satisfy the recurrence relation, 3.4Hence, αm,βm are polynomials in a, b, ζ and 1/λ. Below we show that in the case that we take |λ| large enough, we have the bounds |αm|≤Pm, |βm|≤Pm−1, for m=0,1,…,M, with P=0.9101397…
Let us start with the contour of integration in (3.1). In the case that ph (λ)≠0, we would like to rotate the contour to the line ph (t)=−ph (λ). There are two possible restrictions.
(1) Because we assume that G(t) is analytic in , we will, for , always take |ph t|≤π/8. To be more precise, we take for the straight line ph t=θ, with 3.5
(2) The integrand has a branch point at t=−ζ, and we will allow this branch point to approach the positive real axis. Hence, we have to allow for indents in our new contour of integration . The worst case is illustrated in figure 3, which shows the case ph (λ)<0 and ph (ζ)=−π, that is, the branch point has approached the positive real axis from the upper half plane. The indent will be a circular arc, in which we choose the radius to be δ|ζ|, where we choose .
We use (2.8) in (3.1) and obtain 3.6where 3.7and 3.8where contour encircles contour . For |t|>1, the distance between the two contours is ε, and contour contains also a big part of the unit circle (figure 3).
We will use the standard notation for the Kummer-U function given in [1,2], ch. 13, and also integral representation (13.4.4) in that reference. We identify 3.9The reader can verify via integrating by parts that the functions um(λ,ζ) satisfy recurrence relation (3.4). It follows that um(λ,ζ) can be written as a linear combination of u0(λ,ζ) and u1(λ,ζ), i.e. 3.10where the αm,βm are defined above. Let us write M=γ0|λ|+γ1, such that γ1∈[0,1). We are going to show that there is a P∈(0,1) such that |αm|≤Pm for m=0,1,…,M. This inequality clearly holds for m=0,1. Suppose that |αj|≤Pj for j=m,m+1, then we obtain from recurrence relation (3.4) 3.11Recall that we already assumed , and we take |λ| large enough to guarantee that |b−a+1|+|ζb|≤|λ|/10. Then, we have 3.12where we use for the second inequality that m≤M−2<γ0|λ|. Thus, we want to take P such that . We can take for example , then P=0.9101397… With these choices for the constant, we also have |βm|≤Pm−1 for m=0,1,…,M.
Now, we want to show that the remainder term defined in (3.8) is relatively exponentially small. Because the growth of G(τ) is at most polynomial for large τ, we can assume that as . With our choice for contour , it follows that 3.13as , uniformly with respect to . Hence, there is a constant K such that 3.14We divide the contour , where is the finite path that starts at the origin and ends at , and is the path t=reiθ, . For the finite path, we will derive below that there is a constant K1 such that 3.15Along the contour , we note that |(1+t/ζ)−a| is bounded. Typically, the exponential in the integrand is bounded by 1. However, for example, in the case that ph λ=−π/2, and −ζ has approached the positive real axis from above, we have to make a small indent in the contour of integration, an arc with radius δ|ζ| (figure 3). In that case, the exponential along is bounded by e|λζ|δ. Note that for all t along , we have .
For the path , we let t=eiθτ, where |θ|≤π/8. We obtain 3.16where, again, we have used M=γ0|λ|+γ1. Here, the phase function has a minimum at . We take . Application of the saddle-point method (see §2.4(iv) in [1,2]) shows that 3.17as . When we take, as before, and use the fact that with our choice for θ (see (3.5)), we have |T|≤1, we can write this result as 3.18as .
Let and G(t) be analytic in region and of at most polynomial growth as in . Then, 3.19as in |ph λ|≤π/2 uniformly for and |ph ζ|≤π.
Let for a moment w(λ)=ζbU(b,b−a+1,λζ). Then, it follows from (13.3.22) in [1,2] that w′(λ)=−bζb+1U(b+1,b−a+2,λζ). Because w(λ) satisfies a second-order linear differential equation, it follows that the two terms on the right-hand side of (3.19) cannot vanish at the same time. Hence, we can absorb the relatively exponentially small remainder RM(λ,ζ) into the right-hand side of (3.19). Because G(t) is analytic in it follows that there is a constant K such that |gm|≤K for all m. Hence, , and similarly is also bounded. Thus, the right-hand side of (3.2) can be absorbed into the right-hand side of (3.19).
4. Main example: a uniform asymptotic expansion
Now, we will obtain the uniform asymptotic expansion for (3.1) via Bleistein's method. We take G0(t)=G(t) and define Hn(t), Gn+1(t), n=0,1,2,…, by writing 4.1and 4.2with pn, qn following from the substitution of t=0 and t=−ζ, 4.3
Let and G(t) be analytic in region and of at most polynomial growth as in . Then, 4.4with 4.5as in |ph λ|≤π/2 uniformly for and |ph ζ|≤π.
To obtain the uniform asymptotic expansion, we apply the Bleistein method, i.e. we substitute (4.1) (with n=0) in (3.1), 4.6Integration by parts and using (4.2) results in 4.7Because the integral on the right-hand side of (4.7) is of the same form as (3.1), we can repeat this process and obtain (4.4) with 4.8We note that the process defined in (4.1) and (4.2) does not introduce new singularities for Gn(t). Thus, the Gn(t) are analytic in and of at most polynomial growth as in .
At this stage, one would split the proof into two cases. The case |λζ| is bounded is much harder than usual because the behaviour of U(α,β,x) is complicated for x near the origin. See §13.2(iii) in [1,2]. Hence, it is not easy to just use integral representation (4.8) and obtain order estimate (4.5). Even in the much simpler case of two coalescing saddles, discussed in §2, one extra integration by parts was needed to obtain the required order estimate. Here, that one extra step would not be sufficient. However, using many extra steps works. The result is lemma 3.2. Note that the integral in (4.8) is the same as the one in the definition of fa,b(λ,ζ), with G(t) replaced by Gn(t). Hence, we can use lemma 3.2 and obtain 4.9as in |ph λ|≤π/2 uniformly for and |ph ζ|≤π. This is not exactly (4.5), but because recurrence relation (13.3.9) in [1,2] can be presented as 4.10order estimate (4.5) follows.
5. The main application
The large λ asymptotics of the hypergeometric function in the case that a, b are fixed complex parameters and z is a bounded variable, is well understood. For large parts of the bounded complex z-plane, the Gauss series itself 5.1is an asymptotic expansion. For more details and restrictions, see Wagner . In this section, we will consider unbounded |z|.
The following substitution and expansion were suggested in . We consider 5.2where and and |ph (1+z)|<π. Substituting 1−τ=e−t in (5.2), we obtain 5.3which we write as 5.4where 5.5and 5.6To guarantee that , we take |z|≥(e1/4−1)−1 and keep on assuming that |ph (1+z)|<π. Note that when then ζ→0.
The singularities of G(t) are at the points 5.7As in sector we have . Hence, we can use theorem 4.1, 5.8with order estimate (4.5). The coefficients pn and qn can be found using the process defined in (4.1) and (4.2). For the coefficients p0 and q0, we need 5.9Taking n=1 in (5.8) gives us the approximation (1.1).
As a final numerical example, we apply theorem 3.1 to the hypergeometric function (5.4), 5.10where the Taylor coefficients gm of G(t) about t=0 can be computed via (5.5). However, this process is not straightforward. It is more convenient to consider the logarithmic derivative of G(t), 5.11The computation of hk is much easier, and to compute gm, we use g0=1 and 5.12
If we take , , λ=10i and z=4. Then, M=3, ζ=0.22314355 and the ‘exact’ value of the left-hand side of (5.10) is −0.0028078130−0.0011238098i, whereas taking M=3 on the right-hand side gives us the approximation −0.0028078267−0.0011237993i. Thus, the relative error is 5.71×10−6. In the case that we take λ=20i, we need M=7 and the relative error is 6.46×10−12.
- Received January 4, 2013.
- Accepted February 6, 2013.
- © 2013 The Author(s) Published by the Royal Society. All rights reserved.