## Abstract

The integral considered here is a loop integral of the formwhere the integrand has branch points at *t*=±*θ*, *F*(*t*^{2}, *θ*^{2}) is an analytic function of its arguments and *N* is a large positive parameter. When *θ* is not small, its complete asymptotic expansion can be found by standard techniques. When *θ* is small, the branch points are nearly coincident, and it will be shown that there is a uniform asymptotic expansion involving Bessel functions of argument *Nθ*. An inequality will be established and will be used to show that the expansion is valid in a region including a disc |*θ*|≤*m* of the complex *θ*-plane, where *m* does not tend to 0, when *N* tends to ∞. The proof of this inequality uses the maximum-modulus principle of complex function theory.

## 1. Introduction

Gegenbauer polynomials of degree *n* and order *α* are defined by the generating function(1.1)(see Watson 1948, Section 3.32). It follows that(1.2)where the integral is taken along any circle |*σ*|=*ρ*<1. The principal contributions for large positive integers *n* come from the two branch points *σ*=e^{±iθ} in the integrand, and difficulties arise near *θ*=0, when the branch points are nearly coincident. We then look for asymptotic expansions involving appropriate special functions, in this case Bessel functions. Several versions have been given in the literature; these differ in the argument of the Bessel function and in the region of validity. Some versions are valid in an interval 0≤*θ*≤*ϵ*(*n*), where *ϵ*(*n*)→0, when *n*→∞, but here we shall be concerned with *uniform* expansions, in which the region of validity does not become small when *n* becomes large. The present work was inspired by the uniform expansion of Wong & Zhao (2005), in which *θ* is real and the argument of the Bessel functions is *nθ*. In the present work, we shall construct a uniform version in which the argument is *Nθ*=(*n*+*α*)*θ* and the region of validity includes a circle |*θ*|≤*m* of the complex *θ*-plane. Our method of proof differs significantly from their method and can probably also be used to establish their result in a disc. When the validity has been proved in a complex domain, independent of *n*, we know from experience (e.g. Ursell 1972) that it can often be extended with little difficulty to a larger complex domain. It will be seen in (3.10) that our uniform expansion involves Bessel functions of the orders *α*−1/2 and *α*−3/2.

We shall writethen it is not difficult to show that(1.3)where the contour of integration *C*(1) is an infinite loop starting at *t*=−*π*+i∞, passing below the branch points *t*=−*θ* and +*θ*, and ending at *t*=*π*+i∞. We shall initially be concerned with the asymptotic expansion of the integral(1.4)say, where the expression(1.5)differs from (1.3) by a constant phase factor. This modification is slightly more convenient for our purpose. Henceforth, we shall write(1.6)the condition that *n* is an integer is now no longer relevant.

## 2. The formal expansion

This is obtained by using a Bleistein sequence. We write(2.1)

Then,(2.2a)(2.2b)say, as can be readily verified. On repeating this process, we find that(2.3a)(2.3b)(2.3c)

As we have already noted, the integrals in (2.3*a*) and (2.3*b*) involve Bessel functions (see §3) and, in fact, we shall see that(2.4)where *N*=*n*+*α*, the functions *a*_{p}(*θ*^{2}) and *b*_{p}(*θ*^{2}) are analytic functions of *θ*^{2} and successive terms in the two asymptotic series decrease in powers of *N*^{−2}. In the expansion found by Wong & Zhao (2005), successive terms decrease in powers of *n*^{−1}, which is a slower rate of decrease. We find that(2.5)

So far, the calculations have been purely formal. To show that the Bleistein series represents an asymptotic expansion, we shall show that expression (2.3*c*) for the remainder is suitably small. Our fundamental integral inequality is the *remainder inequality*,(2.6)valid in a complex disc |*θ*|≤*m*, where *m* is independent of *N*. This condition is clearly sufficient. Its proof is the most difficult part of the mathematical argument, for the product |*Nθ*| and each of the integrals involving the two parameters *θ* and *N* may be small, moderate or large.

## 3. Integral representations of Bessel functions

We have(3.1)(3.2)(3.3)(3.4)when the phases of (*τ*^{2}−1)^{ν−1/2} are suitably chosen (see Watson 1948, Section 6.1). Here, the contour *C*(2) is a figure-of-eight encircling the branch points *τ*=−1 and +1 in the negative and positive directions, respectively.

We also have (see Watson 1948, Sections 3.6 (2) and 3.62 (5), (6))(3.5)(3.6)(3.7)(3.8)Now we write *τ*=*t*/*θ*, *z*=*Nθ*, *ν*=(1/2)−*α* where for the moment it is assumed that *θ* is real and positive. Then,(3.9)(3.10)(3.11)(3.12)

We also introduce the modified functions(3.13)

(3.14)

(3.15)

(3.16)

## 4. The functions *A*_{p}(*θ*, *N*) and *B*_{p}(*θ*, *N*) and their analytic continuation

In Wong & Zhao (2005), an inequality of form (2.6) was obtained for real *θ* by an elaborate argument; in the present work, we shall obtain such a bound more simply by treating *θ* as a complex variable, and using the following ideas put forward in earlier work by the author in Ursell (1972, 1990) and other papers.

In the preceding argument, the contour of integration was *C*(1), but the argument remains valid for several other contours, in particular for the contour *C*(2), a figure-of-eight encircling the branch points *t*=−*θ* and +*θ* in the negative and positive directions, respectively; also for the two contours *C*(±) starting respectively at ±*θ*+i∞, encircling the branch point ±*θ* in the positive direction and returning to ±*θ*+i∞. It is easy to see that on these contours, the integrated terms in (2.2*a*) vanish. Initially, we shall choose *C*(2) as our second contour. We note that the finite asymptotic series in (2.3*a*) and (2.3*b*) are analytic functions of *θ*, but that the corresponding infinite series do not necessarily converge and therefore do not necessarily define analytic functions. We shall try to find analytic functions corresponding to these series. (The functions defined by the integrals are multivalued functions.) We therefore consider functions *A*_{p}(*θ*, *N*) and *B*_{p}(*θ*, *N*), satisfying the equations(4.1)(4.2)(Henceforth the suffixes *p* will usually be omitted, and the dependence of *Δ*(*θ*), *A*(*θ*) and *B*(*θ*) on *N* and *F* will not always be noted explicitly.) Then,(4.3)(4.4)where(4.5)and we see that *A*(*θ*, *N*) and *B*(*θ*, *N*) are uniquely defined, provided that *Δ*(*θ*, *N*)≠0, as we shall for the moment assume. (Evidently, *A*(*θ*, *N*) and *B*(*θ*, *N*) also satisfy the corresponding equations in which *C*(1) and *C*(2) are replaced by any pair of distinct contours.) The argument now proceeds as follows. We wish to find bounds for the functions *A*(*θ*) and *B*(*θ*) in a finite neighbourhood |*θ*|≤*m*, independent of *N*. We begin by showing that *A*(*θ*) and *B*(*θ*) are single-valued near *θ*=0. The function *Δ*(*θ*) has a branch point at *θ*=0 and is known explicitly from properties of Bessel functions (4.6). Analytic continuation of *Δ*(*θ*)*A*(*θ*) and *Δ*(*θ*)*B*(*θ*) now shows that *A*(*θ*) and *B*(*θ*) are single-valued (perhaps unbounded) functions of the complex variable *θ* in some neighbourhood |*θ*|≤*m*. We next show that *A*(*θ*) and *B*(*θ*) are bounded near *θ*=0; they are therefore analytic in the neighbourhood |*θ*|≤*m* of *θ*=0. It remains to find the bounds.

When |*θ*|=*m* and *N* is large, we can readily find bounds for the integrals defining these analytic functions, because the branch points *t*=±*θ* are now well separated. The same bounds are valid in the whole neighbourhood |*θ*|≤*m* by the maximum-modulus principle, which states that in a closed domain of the *z*-plane, the modulus |*w*(*z*)| of an analytic function *w*(*z*) attains its maximum value on the boundary of the domain, not at an interior point.

We now carry out the steps outlined above. We had equations (4.3)–(4.5). We continue each term in these equations analytically along a circle |*θ*|=const. in the complex *θ*-plane. This is not difficult for *Δ*(*θ*), because the relevant integrals are multiples of Bessel functions; in fact, it is known from (3.9) and (3.10) that(4.6)(see Watson 1948, Section 3.2(7)). To find the variation of *Δ*(*θ*)*A*(*θ*) and *Δ*(*θ*)*B*(*θ*), we need to consider the same Bessel functions and also integrals of the form

To illustrate our method, let us first consider *C*(1) and put *F*(*t*^{2}, *θ*^{2})≡1. Let the contour *C*(1) be deformed to lie outside the circle |*t*|=*π*, and take |*θ*|<*m*. (A possible value is *m*=(3/4)*π*.) Then,(4.7)andan analytic function of *θ*^{2} when |*θ*|<*m*, in fact for all *θ* as we already know from (3.14). The treatment of the integrals(4.8)is similar. For the functions *F*_{p}(*t*^{2}, *θ*^{2}) that have branch points in the *t*-plane, the branch points nearest to *t*=0 are readily seen to be at *t*=±2*π*±*θ*. Suppose that |*θ*| is chosen to be sufficiently small, say |*θ*|<*m*, then the contour *C*(1) can evidently be chosen to lie inside these branch points, and analogous choices can be made for all the functions *F*_{p}(*t*^{2}, *θ*^{2}). We see, as before, that the integrals (4.8) are analytic functions of *θ*^{2}, when |*θ*|<*m*.

We next consider the integrals(4.9)in which the contour is a figure-of-eight with its centre at *t*=0. When we rotate *t* and *θ* in small steps through an angle *π*, we see that the value of the integral is multiplied by a phase factor, actually e^{(1−2α)iπ}, since it is evidently the same factor as when *F*_{p} is replaced by 1. Thus, we see that(4.10)

We also see from (4.6) that(4.11)and it follows that *B*(*θ*e^{iπ})=*B*(*θ*) and similarly that *A*(*θ*e^{iπ})=*A*(*θ*), when |*θ*|<(3/4)*π*. Hence, also *A*(*θ*e^{2iπ})=*A*(*θ*) and *B*(*θ*e^{2iπ})=*B*(*θ*), and so the functions *A*(*θ*) and *B*(*θ*) are single-valued in this domain. The functions *A*(*θ*) and *B*(*θ*) can therefore be expanded in Laurent series in *θ*^{2} near *θ*=0.

To show that they are analytic in the same domain, it is sufficient to show that they are bounded near *θ*=0. For this purpose, it is sufficient to suppose that *Nθ* is small. Let us examine the integrals on the right-hand side of (4.3). The integrals not involving *F* are given explicitly by (3.9) and (3.10) in terms of Bessel functions. To estimate(4.12)for small *θ*, we put *θ*=0 and obtain(4.13)a finite integral. To estimate(4.14)we write *t*=*Tθ* and obtain(4.15)which in the limit of small *θ* varies as *θ*^{1−2α}. Thus, we see that the order of magnitude of the integrals in the expression for *Δ*(*θ*)*A*(*θ*) near *θ*=0 is *O*(1).*O*(*θ*^{3−2α})+*O*(*θ*^{1−2α}).*O*(1)=*O*(*θ*^{1−2α}), also *Δ*(*θ*)=const.*θ*^{1−2α}, thus *A*(*θ*) is bounded near *θ*=0. It follows that *A*(*θ*) is analytic in *θ*, when |*θ*|<*m*. To show that *B*(*θ*) is analytic, we recall that(4.16)and it follows by a similar argument that *B*(*θ*) is analytic. In fact, the order of magnitude near *θ*=0 of the integrals on the right-hand side is *O*(1).*O*(*θ*^{1−2α})+*O*(*θ*^{1−2α}).*O*(1)=*O*(*θ*^{1−2α}), the same as for *A*(*θ*). We recall that *A*(*θ*) and *B*(*θ*) depend on *N* and *F*.

## 5. Bounds for *A*_{p}(*θ*, *N*) and *B*_{p}(*θ*, *N*)

To find uniform bounds for small and moderate *θ* (independent of *N*), say inside the circle |*θ*|<*m*, from the expressions for *A*(*θ*, *N*) and *B*(*θ*, *N*), it is sufficient to find such bounds on the circumference |*θ*|=*m* of this circle, and then to apply the maximum-modulus principle. The leading terms come from the branch points *t*=±*θ*, which are well separated and to which the regular method of steepest descents (in fact, Watson's lemma for loop integrals) is therefore applicable. We express the integrals along the double loops *C*(1) and *C*(2) in terms of integrals involving single loops. We note that(5.1)(5.2)(see (3.5) and (3.6)). These identities express each of the double loops (3.9) and (3.10) along *C*(2) and *C*(1), respectively, as the sum of two single loops (3.11) and (3.12) about *θ* and −*θ*. It is readily seen that analogous expressions, containing the same coefficients, are valid when the additional analytic factor *F*(*t*^{2}, *θ*^{2}) is contained in each of the loop integrals (3.13)–(3.16). We now express each of the double-loop integrals in the expression for *Δ*(*θ*, *N*)*A*(*θ*, *N*) as the sum of the appropriate two single-loop integrals.

We have

We now use equations (5.1) and (5.2) for *J*_{1/2−α} and *J*_{α−1/2}, respectively, and the corresponding expressions for and Then, we find thatan expression involving only single loops. We recall from (4.6) that(see Watson 1948, Section 3.2(7)). It follows that(5.3)

Since |*θ*| is bounded away from 0, say |*θ*|=*m*, the distance |2*θ*| between the branch points ±*θ* is bounded away from 0. Each term on the right-hand side of (5.3) can therefore be estimated for large *N* by Watson's lemma. Thus, from (3.12), we havewhere we should note that here and in the following equations the coefficients *C*_{j}(*α*) and *C*_{j}(*α*−1) are constants, which are not infinite. We do not need to determine these constants at this stage. Similarly,

The treatment of the other terms in expression (5.3) for *A*(*N*, *θ*) is similar. We thus see thatand

We must still examine the region of validity of the preceding calculations based on Watson's lemma. It is not difficult to see that the single loops do not interfere with each other when the phase of *θ* is sufficiently small, in particular when −(1/2)*π*≤ph(*θ*)≤(1/2)*π*. When the asymptotic expressions are substituted in (5.3), we see thatwhere *M*(*F*) is independent of *N* and *θ* lies on the semicircle (|*θ*|=*m*, −(1/2)*π*≤ph(*θ*)≤(1/2)*π*) The same bound is actually valid on the whole circle, because it has already been shown that *A*(*θ*)=*A*(*θ*e^{iπ}). The maximum-modulus principle applied to the interior of the circle |*θ*|=*m* now shows that(5.4)

A similar bound can be found for *B*(*θ*, *N*; *F*). We hadsay, whereThe calculations for *B*^{†}(*θ*, *N*; *F*^{†}) now follows the same lines as for *A*(*θ*, *N*; *F*). As before, we find that(5.5)When these bounds are substituted in (4.1)(5.6)we obtain the desired inequality (2.6)(5.7)

We have now established the desired asymptotic expansion in terms of loop integrals. It remains to express this in the form of Bessel functions (2.4), and this can be done by a detailed study of the neglected external factors multiplying the loop integrals defining the Bessel functions. This troublesome calculation (cf. Watson 1948, ch. 6) can, however, be avoided. As we know that, for real *θ*, the asymptotic expansion must be real, of the formTo determine *P*(*α*) and *Q*(*α*), it is sufficient to consider the value *θ*=0. We know from (2.5) that *a*_{0}(0)=1, *b*_{0}(0)=(1/12)*α*, and it can be shown by considering *F*_{1}(*t*^{2}, 0) that *a*_{1}(0)=0. It follows that(5.8)

We also see from the generating function (1.1) thatand therefore we see thatIf we now write *u*=e^{−v}, then

By referring to (5.8), we see thatand that(5.9)where *N*=*n*+*α*. An equivalent expression was given in (2.4). We note again that the terms in the two asymptotic series decrease in powers of *N*^{−2}. We recall that we have provisionally excluded values of *α*, for which *Δ*(*θ*, *N*)=0.

## 6. Discussion and conclusion

We have now derived the asymptotic expansion (5.9) for the integralIt is not difficult to see that this expansion is valid inside a circle |*θ*|=*m* in the complex *θ*-plane, where *m* is independent of *N*. By deforming the contour of integration, the validity can be extended in a simple manner to the circle |*θ*|<*π*, excluding small circles |*θ*±*π*|<*m*_{1}. (When *θ*=±*π*, the branch points are coincident.) Our result should be compared with that of Wong & Zhao (2005), which uses different ideas, contains Bessel functions of argument *nθ* and is valid for real *θ* in an interval 0≤*θ*≤*π*−*ϵ*. Successive terms in our asymptotic series decrease by a factor *N*^{−2}; in their series by a factor *n*^{−1}, which is a slower rate of decrease. There can be little doubt that our method can be used to extend their result into the complex *θ*-plane.

It will be recalled that the functions *A*_{p}(*θ*, *N*) and *B*_{p}(*θ*, *N*) were originally introduced as the formal sums of asymptotic series and were then shown to be analytic functions. It is therefore of interest to show that, conversely, the same asymptotic series are asymptotic expansions of the functions *A*_{p}(*θ*, *N*) and *B*_{p}(*θ*, *N*). In fact, we have the identity(6.1)which is obtained by substituting expansion (2.3*c*) in (4.3) and noting expression (4.5) for *Δ*(*θ*). It follows that

We have incidentally derived an asymptotic method for integral (3.13) along the loop *C*(2), but this will not be pursued further here. It may be observed that we have worked initially with double loops, but that single loops were introduced in §5. An attempt was made to work initially with single loops, but this was found to be less convenient.

Our method is applicable for general values of *α*, but we still need to re-examine the critical values of *α* at which *Δ*(*θ*)=0, i.e. the values *α*=1/2 and *α*=1, 0, −1, −2, …, (4.6). This difficulty has arisen because we have chosen *C*(2) as our second contour. In fact, it can be seen from equation (5.3) that our argument for *A*(*θ*) is valid for all values of *α*, with a corresponding result for *B*(*θ*). We may also note that our earlier argument in fact shows that near these critical values, the functions *A*(*θ*) and *B*(*θ*) continue to be well defined and to behave smoothly, even though |*Δ*(*θ*)| is large. When *α*=1/2 and *α*=1, the functions *A*(*θ*) and *B*(*θ*) are not defined by (4.1) and (4.2), but can be defined by continuity. At the remaining critical values *α*=0, −1, −2, …, we note that the generating function is a polynomial; it follows that , when *n*≥−2*α*+1.

The expansion derived by Wong & Zhao (2005) is expressed in terms of Bessel functions of argument *nθ*. Evidently, we would expect to obtain an expansion of this form if we replace our Bleistein sequence (2.1), with first memberby a Bleistein sequence with first memberand continue by following our line of argument. It would be interesting to carry out this calculation.

## Acknowledgments

I am grateful to Adri Olde Daalhuis and Roderick Wong for their helpful comments.

## Footnotes

- Received July 28, 2006.
- Accepted October 10, 2006.

- © 2006 The Royal Society