Nield-Kuznetsov functions and Laplace transforms of parabolic cylinder functions
NNIELD-KUZNETSOV FUNCTIONS AND LAPLACE TRANSFORMSOF PARABOLIC CYLINDER FUNCTIONS
T. M. DUNSTER ∗ Abstract.
Nield-Kuznetsov functions of the first kind are studied, which are solutions of aninhomogeneous parabolic Weber equation, and have applications in fluid flow problems. Connec-tion formulas are constructed between them, numerically satisfactory solutions of the homogeneousversion of the differential equation, and a new complementary Nield-Kuznetsov function. Asymp-totic expansions are then derived that are uniformly valid for large values of the parameter andunbounded real and complex values of the argument. Laplace transforms of the parabolic cylin-der functions W ( a, x ) and U ( a, x ) are subsequently shown to be explicitly represented in terms ofthe complementary Nield-Kuznetsov function and closely related functions, and from these uniformasymptotic expansions are derived for the integrals. Key words.
Parabolic cylinder functions, WKB methods, Asymptotic expansions, Laplacetransforms
AMS subject classifications.
1. Introduction.
The parametric Nield-Kuznetsov function of the first kind isgiven by [1], [9](1.1) N W ( a, x ) = W ( a, x ) (cid:90) x W ( a, − t ) dt − W ( a, − x ) (cid:90) x W ( a, t ) dt, which is a solution of(1.2) d ydx + (cid:18) x − a (cid:19) y = − . In (1.1) the Weber parabolic cylinder functions W ( a, ± x ) are numerically satisfactorysolutions of the homogeneous form of (1.2), and for definitions and properties see [3,Sect. 12.4].This function N W ( a, x ) appears in the study of fluid flow through a variable per-meability porous layer, which involve Brinkman equations. These equations appearin the study of fast-moving fluids in porous media in which the kinetic potential fromfluid velocity, pressure, and gravity drives the flow. In the study of flow through cer-tain porous layers the Brinkman equations can be reduced to inhomogeneous forms ofvarious linear second order differential equations. When the distribution of the recip-rocal of the permeability is linear we get an inhomogeneous form of Airy’s equation[9]. We mention that this equation was studied in [4] where the inhomogeneous termwas either a polynomial or an exponential.For a quadratic distribution of the reciprocal of the permeability we get theparabolic cylinder equation, one form of which is (1.2) studied here; see [2] and [8].In [1] series solutions for N W ( a, x ) for small x were derived, and various other repre-sentations were derived in the above references.In [1] the authors state ”The current work involves real arguments of the We-ber functions. In the general analysis of Weber equation and associated functionswith complex arguments, the introduction of a Nield-Kuznetsov type function is still ∗ Department of Mathematics and Statistics, San Diego State University, 5500 Campanile Drive,San Diego, CA 92182-7720, USA. ([email protected], https://tmdunster.sdsu.edu).1 a r X i v : . [ m a t h . C A ] F e b T. M. DUNSTER challenging and requires further consideration.” In this paper we address this openproblem, as described below.From its definition the function N W ( a, x ) is readily seen to be even (and hence N (cid:48) W ( a,
0) = 0), and also vanishes at x = 0. Actually these two properties define N W ( a, x ) uniquely, since if y p ( a, x ) is another even particular solution of (1.2), thenfor some constant c ( a ) it must be related by(1.3) y p ( a, x ) = N W ( a, x ) + c ( a ) { W ( a, x ) + W ( a, − x ) } . But if also y p ( a,
0) = 0 then c ( a ) W ( a,
0) = 0 which implies c ( a ) = 0 since W ( a, (cid:54) = 0(see (1.7) below).We introduce and study the complementary parametric Nield-Kuznetsov function(1.4) ˆ N W ( a, x ) = W ( a, x ) (cid:90) ∞ x W ( a, − t ) dt − W ( a, − x ) (cid:90) ∞ x W ( a, t ) dt the negative of which is also a particular solution of (1.2). This has the advantageover N W ( a, x ) that it is uniquely defined by its recessive behavior at x = ∞ , andhence y ( a, x ) = − N W ( a, x ) is a numerically satisfactory particular solution of (1.2).Specifically, as we shall show, for large positive x this function decays monotonicallyand is O ( x − ), whereas all other solutions of (1.2) (including N W ( a, x )) are oscillatorywith an amplitude that decays more slowly, namely asymptotically proportional to x − / .Furthermore, we shall extend the definition of N W ( a, x ) and ˆ N W ( a, x ) to complexargument x = z (say). We shall demonstrate that ˆ N W ( a, z ) → z → ∞ for | arg( z ) ≤ π − δ ( δ > a → ∞ uniformly in part of the right halfplane that contains { z : (cid:60) ( z ) ≥ √ a } . In contrast, all other solutions of (1.2) areexponentially large as z → ∞ , and also as a → ∞ , in at least three of the fourquadrants.When x is real and small N W ( a, x ) is useful for solving (1.2) for the initial valueproblem y ( a,
0) = α , y (cid:48) ( a,
0) = β ; see [8]. However, a solution involving ˆ N W ( a, x )is more practicable computationally. This is because as x grows N W ( a, x ) becomesalmost indistinguishable from a homogeneous solution. In addition, as we shall showit grows exponentially as a → ∞ for all nonzero argument.The solution to the initial value problem in question is readily found to be(1.5) y ( a, x ) = C + ( a ) W ( a, x ) + C − ( a ) W ( a, − x ) + N W ( a, x ) , where(1.6) ˆ C ± ( a ) = ∓ βW ( a, − αW (cid:48) ( a, W ( a, W (cid:48) ( a,
0) = − , which follows from theWronskian of W ( a, ± x ) [3, Eq. 12.14.3], or alternatively from their values at x = 0[3, Eqs. 12.14.1, 12.14.2](1.7) W ( a,
0) = 2 − / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Γ (cid:0) + ia (cid:1) Γ (cid:0) + ia (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) / , and(1.8) W (cid:48) ( a,
0) = − − / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Γ (cid:0) + ia (cid:1) Γ (cid:0) + ia (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) / . IELD-KUZNETSOV FUNCTIONS W ( a,
0) = O ( a − / ) and W (cid:48) ( a,
0) = O ( a / ) as a → ∞ (see (2.58)and (2.59) below). Now (1.7) and (1.8) can of course can also be used in (1.6), andthen (1.5) can be used for small values of x . For all other values of x the Nield-Kuznetsov function N W ( a, x ) in (1.5) should be replaced by an expression involvingthe complementary function given by (1.4). This can be done using an appropriateconnection formula, which we derive in section 2 (see (2.20) and Theorem 2.7 below).In fact, ˆ N W ( a, x ) can be used for all x ; see (2.63) along with Lemmas 2.4 and 2.6 forthe case x small.For large real or complex values of z and/or large positive a we present uniformasymptotic expansions for ˆ N W ( a, z ) and related functions. These are given in sec-tion 3, employing recent results of [4] and [5]. They involve Scorer functions, whichare solutions of the inhomogeneous Airy equation, and slowly varying coefficient func-tions.In sections 4 and 5 we study the Laplace transforms of parabolic cylinder func-tions, which turn out to be related to the Nield-Kuznetsov functions. Laplace trans-forms for a wide range of other special functions are well documented. For example,see [3, Eqs. 9.10.14 and 10.22.49, Sect. 13.10(ii)] for Airy, Bessel and confluent hyper-geometric functions, respectively, and [7, pp. 170-176, 179–181] for orthogonal poly-nomials and associated Legendre functions, respectively. However for the paraboliccylinder functions there do not appear to be comparable results in the literature. Ex-isting formulas either involve an exponential factor with a quadratic exponent, and/orthe argument of the parabolic cylinder function having a square root.Consider the standard parabolic cylinder function U ( a, z ), which is a solution ofthe equation(1.9) d ydz − (cid:18) z + a (cid:19) y = 0 , having the integral representation [3, Eqs. 12.5.1 and 12.5.4](1.10) U ( a, z ) = e − z Γ (cid:0) + a (cid:1) (cid:90) ∞ t a − e − t − zt dt (cid:0) (cid:60) ( a ) > − (cid:1) . Its fundamental property is that it is the unique solution of (1.9) that is exponentiallysmall as z → ∞ for | arg( z ) | ≤ π − δ ( δ > z → ∞ (1.11) U ( a, z ) ∼ z − a − e − z (cid:0) | arg( z ) | ≤ π − δ < π (cid:1) . Then, as an example of a known Laplace transform involving this function butalso an exponential with quadratic exponent, from [7, Eq. 4.20.1] we find that for (cid:60) ( λ ) > (cid:60) ( a ) > − (1.12) (cid:90) ∞ e − λt e t U ( a, t ) dt = 1Γ (cid:0) + a (cid:1) (cid:90) ∞ x a − e − x λ + x dx. In sections 4 and 5 we fill the gap by obtaining explicit representations of theLaplace transforms of W ( a, ± t ) and U ( a, t ), respectively, without any extraneousmultiplicative terms. We find explicit expressions in terms of the complementaryNield-Kuznetsov function ˆ N W ( a, z ) and related functions. From these and the asymp-totics of section 3 we are then able to obtain powerful asymptotic expansions for the T. M. DUNSTER
Laplace transforms, as a → ∞ , that are uniformly valid for all real and complex val-ues of the Laplace variable λ . We also include asymptotic expansions for a boundedand | λ | large. We mention that an explicit representation for the Laplace transformfor W ( a, ± x ) for the case λ = 0 is also provided by (2.60) and Theorem 2.7 below,and likewise for U ( a, x ) by (2.29) (with R = 0 in that formula). Apart from (2.29) allthese results appear to be new.The general asymptotic expansions used from [4] and [5] for sections 3 to 5 arerigorously established via explicit error bounds, but for brevity we omit details of thebounds in this paper. To avoid phrase repetition, throughout this paper we denoteby δ an arbitrary small positive constant.
2. Fundamental solutions and connection formulas.
For z ∈ C the homo-geneous Weber differential equation(2.1) d ydz + (cid:18) z − a (cid:19) y = 0 , has numerically satisfactory solutions(2.2) W j ( a, z ) = U (cid:16) ( − j ia, ( − i ) j ze − πi/ (cid:17) ( j = 0 , , , . These solutions are important because each W j ( a, z ) is recessive (exponentially small)at z = e πi/ i j ∞ , whereas all other linearly independent solutions are exponentiallylarge at that singularity. Note also that W ( a, z ) = W ( a, − z ) and W ( a, z ) = W ( a, − z ), and so we can primarily focus on W ( a, z ) and W ( a, z ).As z → ∞ for j = 0 and j = 3 we have from (1.12) and (2.2)(2.3) W j ( a, z ) ∼ e − πa ± πi z ∓ ia − e ± iz (cid:16)(cid:12)(cid:12)(cid:12) arg (cid:16) ze ∓ πi/ (cid:17)(cid:12)(cid:12)(cid:12) ≤ π − δ (cid:17) , where the upper signs are taken for j = 0 and lower signs for j = 3. From this it isclear that these two form a numerically satisfactory pair of solutions of (2.1) in theright half plane. The same is true for W ( a, z ) and W ( a, z ) in the left half plane.From (2.2) and [3, Eq. 12.2.19] we note the connection formulas(2.4) W ( a, − z ) = − ie πa W ( a, z ) + √ πe πa + πi Γ (cid:0) + ia (cid:1) W ( a, z ) , and(2.5) W ( a, − z ) = √ πe πa − πi Γ (cid:0) − ia (cid:1) W ( a, z ) + ie πa W ( a, z ) . With these definitions, the Weber function W ( a, x ) can then be extended tocomplex argument x = z via the relation [3, Eq. 12.14.4](2.6) W ( a, z ) = (cid:113) k e πa/ (cid:8) e iρ W ( a, z ) + e − iρ W ( a, z ) (cid:9) , where(2.7) k = (cid:112) e πa − e πa , (2.8) ρ = φ + π, IELD-KUZNETSOV FUNCTIONS φ = arg (cid:8) Γ (cid:0) + ia (cid:1)(cid:9) . In (2.9) the branch of arg is taken to be zero when a = 0 and then defined by continuityfor a > d ydz + (cid:18) z − a (cid:19) y = z R ( R = 0 , , , · · · ) . From Eq. (4.31) of this reference we have the solution that interests us the most,namely(2.11) W (0 , R ( a, z ) = ie πa/ (cid:20) W ( a, z ) (cid:90) ze πi/ ∞ t R W ( a, t ) dt − W ( a, z ) (cid:90) ze − πi/ ∞ t R W ( a, t ) dt (cid:21) . This is defined and converges for all nonnegative integers R , and is significant becauseit is the unique solution that does not grow exponentially in the right half plane. Inparticular it is O ( z R − ) as z → ∞ for | arg( z ) ≤ π − δ , uniformly for all real valuesof a . We shall primarily focus on the cases R = 0 and R = 1.Firstly, for R = 0 the following identification allows us to extend ˆ N W ( a, x ) tocomplex argument z , and later extract its asymptotic behavior for large parameterand unbounded argument. Theorem
For the analytic continuation of (1.4) via (2.6) we have (2.12) ˆ N W ( a, z ) = W (0 , ( a, z ) = ie πa/ (cid:20) W ( a, z ) (cid:90) ze πi/ ∞ W ( a, t ) dt − W ( a, z ) (cid:90) ze − πi/ ∞ W ( a, t ) dt (cid:21) . Proof. As x → ∞ it follows from [3, Eqs. 12.14.17 and 12.14.18] that(2.13) W ( a, x ) ∼ (cid:114) kx cos( ω ) , and(2.14) W ( a, − x ) ∼ (cid:114) kx sin( ω ) , where k is given by (2.7),(2.15) ω = x − a ln x + π + φ , and φ is given by (2.9). Now from (2.13), (2.14), and (2.15) one finds that(2.16) (cid:90) ∞ x W ( a, t ) dt ∼ −√ k (cid:18) x (cid:19) / sin( ω ) , T. M. DUNSTER and(2.17) (cid:90) ∞ x W ( a, − t ) dt ∼ √ k (cid:18) x (cid:19) / cos( ω ) . So from (1.4), (2.13), (2.14), (2.16) and (2.17) we deduce that(2.18) ˆ N W ( a, x ) ∼ /x . Now any solution y ( a, x ) of (1.2) can be expressed as(2.19) y ( a, x ) = c ( a ) W ( a, x ) + c ( a ) W ( a, − x ) − ˆ N W ( a, x ) , for some constants c ( a ) and c ( a ). But from (2.13), (2.14), (2.18) and (2.19) it isclear that y ( a, x ) ∼ − x − as x → ∞ iff c ( a ) = c ( a ) = 0. Therefore ˆ N W ( a, z )and W (0 , ( a, z ) must be equal, since the latter shares the unique behavior (2.18) as z = x → ∞ , and the negative of both are solutions of (1.2).Our primary focus now is the important connection formula, given as follows. Lemma
The Nield-Kuznetsov function (1.1) and the complementary version(1.4) are related by (2.20) N W ( a, x ) = c +0 ( a ) W ( a, x ) + c − ( a ) W ( a, − x ) − ˆ N W ( a, x ) , where (2.21) c ± ( a ) = ∓ ˆ N (cid:48) W ( a, W ( a, − ˆ N W ( a, W (cid:48) ( a, . Proof.
The function y ( a, x ) = − ˆ N W ( a, x ) is a solution of (1.2) satisfying theinitial conditions y ( a,
0) = α = − ˆ N W ( a,
0) and y (cid:48) ( a,
0) = β = − ˆ N (cid:48) W ( a, N W ( a,
0) and ˆ N (cid:48) W ( a, Lemma N W ( a,
0) = − πe πa/ (cid:61) (cid:40) e − πi/ Γ (cid:0) + ia (cid:1) F (cid:0) , − ia ; (cid:1)(cid:41) , where F is Olver’s scaled hypergeometric function [3, Eq. 15.2.2] (2.23) F ( a, b ; c, z ) = F ( a, b ; c, z )Γ( c ) = ∞ (cid:88) s =0 ( a ) s ( b ) s z s Γ( c + s ) s ! . Proof.
From (2.12) we have(2.24) ˆ N W ( a,
0) = ie πa/ (cid:34) W ( a, (cid:90) e − πi/ ∞ W ( a, t ) dt − W ( a, (cid:90) e πi/ ∞ W ( a, t ) dt (cid:35) . IELD-KUZNETSOV FUNCTIONS W functions in terms of U given by (2.2) then allows us to recastthis in the form(2.25) ˆ N W ( a,
0) = ie πa/ (cid:34) U ( ia, (cid:90) e − πi/ ∞ U (cid:16) − ia, te πi/ (cid:17) dt − U ( − ia, (cid:90) e πi/ ∞ U (cid:16) ia, te − πi/ (cid:17) dt (cid:35) . We next make change of integration variables t → te ± πi/ to arrive at(2.26) ˆ N W ( a,
0) = ie πa/ (cid:20) e − πi/ U ( ia, (cid:90) ∞ U ( − ia, t ) dt − e πi/ U ( − ia, (cid:90) ∞ U ( ia, t ) dt (cid:21) , which yields(2.27) ˆ N W ( a,
0) = − e πa/ (cid:61) (cid:26) e − πi/ U ( ia, (cid:90) ∞ U ( − ia, t ) dt (cid:27) . Next from [3, Eq. 12.2.6](2.28) U ( ia,
0) = √ π ia + Γ (cid:0) + ia (cid:1) . We then put (2.28) into (2.27), along with the very useful but little-known identity[11, Eq. 2.11.2.1](2.29) (cid:90) ∞ t R U ( a, t ) dt = 2 − a − R − √ πR ! × F (cid:0) R + , R + 1; a + R + ; (cid:1) ( R = 0 , , , · · · ) , with R = 0 and a replaced by − ia , and as a result we arrive at (2.22).Although we now have an explicit representation for ˆ N W ( a, a is large, since the imaginary part we need to extract isexponentially small relative to the real part of the function in question. For exampleif a = 20, then to 4 decimal places(2.30) e − πi/ Γ (cid:0) + ia (cid:1) F (cid:0) , − ia ; (cid:1) = 7 . × − . i. We overcome this with the following numerically stable representation.
Lemma N W ( a,
0) = 2 e πa X ( a ) − Y ( a ) , where (2.32) X ( a ) = (cid:12)(cid:12) Γ (cid:0) + ia (cid:1)(cid:12)(cid:12) √ π , and (2.33) Y ( a ) = (cid:61) (cid:8) (1 − ia ) − F (cid:0) , − ia ; (cid:1)(cid:9) . T. M. DUNSTER
Proof.
From (2.23) and well-known functional relations for the Gamma functionΓ( z + 1) = z Γ( z ) and(2.34) 1Γ( z ) = sin( πz )Γ(1 − z ) π , we obtain(2.35) e − πi/ Γ (cid:0) + ia (cid:1) F (cid:0) , − ia ; (cid:1) = e − πi/ Γ (cid:0) + ia (cid:1) Γ (cid:0) − ia (cid:1) F (cid:0) , − ia ; (cid:1) = 2 (cid:0) ie πa/ − e − πa/ (cid:1) (1 − ia ) π F (cid:0) , − ia ; (cid:1) . Thus from (2.22) we get (2.31) where(2.36) X ( a ) = (cid:60) (cid:8) (1 − ia ) − F (cid:0) , − ia ; (cid:1)(cid:9) , and Y ( a ) is given by (2.33).Finally, from [3, Eqs. 5.4.6, 5.5.1, 15.4.6, 15.10.21] we obtain the relation(2.37) F (cid:0) , − ia ; (cid:1) = − − ia ia F (cid:0) , + ia ; (cid:1) + (1 − ia ) (cid:12)(cid:12) Γ (cid:0) + ia (cid:1)(cid:12)(cid:12) √ π . We then divide both sides by 1 − ia , equate the real parts of both sides, and refer to(2.36), and as a result we verify (2.32).We shall frequently use Stirling’s formula [3, Eq. 5.11.3](2.38) Γ ( z ) ∼ e − z z z (cid:18) πz (cid:19) / ∞ (cid:88) k =0 g k z k , as z → ∞ for | arg( z ) | ≤ π − δ , where g = 1, g = and subsequent coefficientsgiven by [3, Eqs. 5.11.5 and 5.11.6]. From this we have: Lemma As a → ∞ (2.39) ˆ N W ( a,
0) = (cid:114) πa e πa/ (cid:26) O (cid:18) a (cid:19)(cid:27) . Proof.
From (2.32) and (2.38) we find that(2.40) X ( a ) = √ πe − πa/ √ a (cid:26) O (cid:18) a (cid:19)(cid:27) . We also have directly from (2.23) and (2.33)(2.41) Y ( a ) ∼ ∞ (cid:88) k =0 y k +1 a k +1 , where the first four coefficients are y = , y = , y = and y = . Thus from(2.31) we get (2.39) IELD-KUZNETSOV FUNCTIONS Remark A more accurate approximation comes from retaining the Gammafunction in the leading term, so that (2.42) ˆ N W ( a,
0) = e πa (cid:12)(cid:12) Γ (cid:0) + ia (cid:1)(cid:12)(cid:12) √ π (cid:26) O (cid:18) e − πa/ √ a (cid:19)(cid:27) ( a → ∞ ) . Next we establish a simple explicit value for the derivative at the origin.
Lemma N (cid:48) W ( a,
0) = −√ πe πa/ . Proof.
From differentiating (2.11) we obtain similarly to (2.25)(2.44) ˆ N (cid:48) W ( a,
0) = ie πa/ (cid:34) e − πi/ U (cid:48) ( ia, (cid:90) e − πi/ ∞ U (cid:16) − ia, te πi/ (cid:17) dt − e πi/ U (cid:48) ( − ia, (cid:90) e πi/ ∞ U (cid:16) ia, te − πi/ (cid:17) dt (cid:35) , and hence following the same steps leading to (2.27) yields(2.45) ˆ N (cid:48) W ( a,
0) = 2 e πa/ (cid:60) (cid:26) U (cid:48) ( ia, (cid:90) ∞ U ( − ia, t ) dt (cid:27) . Next from [3, Eq. 12.2.7](2.46) U (cid:48) ( ia,
0) = − √ π ia − Γ (cid:0) + ia (cid:1) , and so from (2.29), (2.45) and (2.46) we have(2.47) ˆ N (cid:48) W ( a,
0) = −√ πe πa/ (cid:60) (cid:40) (cid:0) + ia (cid:1) F (cid:0) , − ia ; (cid:1)(cid:41) . Now from (2.23), (2.33) and (2.36) and the Gamma function recurrence relation wefind that(2.48) 1Γ (cid:0) + ia (cid:1) F (cid:0) , − ia ; (cid:1) = 1Γ (cid:0) + ia (cid:1) Γ (cid:0) − ia (cid:1) F (cid:0) , − ia ; (cid:1) = 4Γ (cid:0) + ia (cid:1) Γ (cid:0) − ia (cid:1) F (cid:0) , − ia ; (cid:1) − ia = 4 { X ( a ) + iY ( a ) } (cid:12)(cid:12) Γ (cid:0) + ia (cid:1)(cid:12)(cid:12) . Hence from (2.47) we deduce that(2.49) ˆ N (cid:48) W ( a,
0) = − √ πe πa/ X ( a ) (cid:12)(cid:12) Γ (cid:0) + ia (cid:1)(cid:12)(cid:12) , and consequently from (2.32) we arrive at (2.43), as asserted.0 T. M. DUNSTER
Bringing everything together, we arrive at our desired representations for theconnection coefficients.
Theorem
In (2.20) we have (2.50) c +0 ( a ) = W ( a, (cid:104) √ πe πa/ (cid:110) (cid:0) e − πa (cid:1) − / (cid:111) − Y ( a ) { W (cid:48) ( a, } (cid:105) , (2.51) c − ( a ) = − W ( a, (cid:20) Y ( a ) { W (cid:48) ( a, } + √ πe − πa/ (cid:110) e − πa + (cid:0) e − πa (cid:1) / (cid:111) − (cid:21) , and (2.52) c ± ( − a ) = W ( a, (cid:104) Y ( a ) { W (cid:48) ( a, } ± √ πe − πa/ (cid:110) ± (cid:0) e πa (cid:1) − / (cid:111)(cid:105) , where W ( a, , W (cid:48) ( a, and Y ( a ) are given by (1.7), (1.8) and (2.33), respectively,and in the latter F is the (unscaled) hypergeometric function defined by the rapidlyconverging series in (2.23).Proof. From (1.7), (1.8), (2.31), and (2.32)(2.53) ˆ N W ( a, W (cid:48) ( a, W ( a,
0) = 4 Y ( a ) { W (cid:48) ( a, } − e πa √ π (cid:12)(cid:12) Γ (cid:0) + ia (cid:1) Γ (cid:0) + ia (cid:1)(cid:12)(cid:12) . On using (2.34) we have(2.54) e πa √ π (cid:12)(cid:12) Γ (cid:0) + ia (cid:1) Γ (cid:0) + ia (cid:1)(cid:12)(cid:12) = e πa/ (cid:18) π e − πa (cid:19) / , and thus from (2.21), (2.43), (2.53), and (2.54) we get (2.50) and (2.51).Next, for a replaced by − a we have that (1.7) and (1.8) are unchanged, and from(2.33) Y ( − a ) = − Y ( a ). Hence from (2.50) and (2.51) we arrive at (2.52), which onthe assumption that a > a that c +0 ( a ) is exponentially large, whereas c − ( a )and c ± ( − a ) are O ( a − / ). Corollary As a → ∞ (2.55) c +0 ( a ) = √ πa − / e πa/ (cid:8) O ( a − ) (cid:9) , (2.56) c − ( a ) = − − / a − / (cid:8) O ( a − ) (cid:9) , and (2.57) c ± ( − a ) = 2 − / a − / (cid:8) O ( a − ) (cid:9) . Proof.
From (1.7), (1.8) and (2.38) one can show that(2.58) W ( a,
0) = 2 − / a − / (cid:8) O ( a − ) (cid:9) , and(2.59) W (cid:48) ( a,
0) = − − / a / (cid:8) O ( a − ) (cid:9) , and hence using these along with (2.41), (2.50), (2.51), and (2.52) we obtain the givenapproximations. IELD-KUZNETSOV FUNCTIONS Remark From (1.4) and (2.20), and their differentiated forms, we obtain theinteresting integrals (2.60) (cid:90) ∞ W ( a, ± t ) dt = ∓ c ∓ ( a ) , where c ∓ ( a ) are given by Theorem .Also, from (1.7), (2.21), (2.43) and (2.60), we note that (2.61) (cid:90) ∞−∞ W ( a, t ) dt = − N (cid:48) W ( a, W ( a,
0) = 2 / √ πe πa/ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Γ (cid:0) + ia (cid:1) Γ (cid:0) + ia (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) / . An interesting wrinkle in (2.20) is that while it is numerically satisfactory forlarge a and moderate to unbounded x , there is large cancellation for large a when x is very small. This is on account of N W ( a, x ) and its first derivative vanishing at x = 0, whereas the three terms on the RHS are exponentially large at x = 0 for large a . To overcome this we wish to replace the three functions on the RHS of (2.20) withalternative ones that mimic the behaviour N W ( a, x ) at x = 0. To do so we note bysetting x = 0 in this equation and its derivative that(2.62) c +0 ( a ) W ( a,
0) + c − ( a ) W ( a, − ˆ N W ( a, c +0 ( a ) W (cid:48) ( a, − c − ( a ) W (cid:48) ( a, − ˆ N (cid:48) W ( a, . Thus our desired modification of (2.20) is(2.63) N W ( a, x ) = c +0 ( a ) { W ( a, x ) − W ( a, − W (cid:48) ( a, x } + c − ( a ) { W ( a, − x ) − W ( a,
0) + W (cid:48) ( a, x }− (cid:110) ˆ N W ( a, x ) − ˆ N W ( a, − ˆ N (cid:48) W ( a, x (cid:111) , which in conjunction with (1.7), (1.8), (2.31) and (2.43) can be used if a is large and x is small. For all other values (2.20) is stable.We complete this section by obtaining a connection formula that can be used forcomplex argument z , and we only need to consider (cid:60) ( z ) ≥ N W ( a, z ) is even.Instead of W ( a, ± x ) we need the numerically satisfactory pair W ( a, z ) and W ( a, z )as companions to ˆ N W ( a, z ). Our result is as follows. Theorem
For z ∈ C let W j ( a, z ) ( j = 0 , ) be defined by (2.2), and ˆ N W ( a, z ) be given by (2.12). For x = z let N W ( a, z ) be given by (1.1) and (2.6). Then thesefunctions are related by (2.64) N W ( a, z ) = c (0)0 ( a ) W ( a, z ) + c (3)0 ( a ) W ( a, z ) − ˆ N W ( a, z ) , where (2.65) c (0)0 ( a ) = √ π + ia e πa + πi Y ( a )Γ (cid:0) − ia (cid:1) + 2 − + ia e πa + πi Γ (cid:0) + ia (cid:1) ,c (3)0 ( a ) = c (0)0 ( a ) , and Y ( a ) is defined by (2.33). T. M. DUNSTER
Proof.
Firstly from (2.64) we see that c (3)0 ( a ) = c (0)0 ( a ), since when z = x ∈ R both N W ( a, x ) and ˆ N W ( a, x ) are real, and W ( a, x ) = W ( a, x ). Next, on recallingthat N W ( a,
0) = N (cid:48) W ( a,
0) = 0, we have from (2.64) and its derivative(2.66) c (0)0 ( a ) = ˆ N W ( a, W (cid:48) ( a, − ˆ N (cid:48) W ( a, W ( a, W { W , W } ( a, . Now from (2.2) and [3, Eq. 12.2.12] the required Wronskian is given by(2.67) W { W ( a, z ) , W ( a, z ) } = − ie − πa/ , and from (2.2) and [3, Eqs. 12.2.6, 12.2.7] we have(2.68) W ( a,
0) = W ( a,
0) = √ π − ia Γ (cid:0) − ia (cid:1) , and(2.69) W (cid:48) ( a,
0) = W (cid:48) ( a,
0) = √ πe − πi/ + ia Γ (cid:0) − ia (cid:1) . Therefore using Lemmas 2.4 and 2.6, (2.66) - (2.69), along with the Gamma functionreflection formula (2.34), we arrive at (2.65).In (2.65) the first term dominates when a is large since it is exponentially largein comparison to the second term, and from (2.38) and (2.41) we then get(2.70) c (0)0 ( a ) = 2 − / a − + ia e πa + πi − ia (cid:8) O ( a − ) (cid:9) , with the corresponding approximation for c (3)0 ( a ) being the complex conjugate of this.If a is large and | z | is small then similarly to (2.63) one should use in place of(2.64) the alternative form(2.71) N W ( a, z ) = c (0)0 ( a ) { W ( a, z ) − W ( a, − W (cid:48) ( a, z } + c (3)0 ( a ) { W ( a, z ) − W ( a, − W (cid:48) ( a, z }− (cid:110) ˆ N W ( a, z ) − ˆ N W ( a, − ˆ N (cid:48) W ( a, z (cid:111) .
3. Uniform asymptotic expansions.
Here we record asymptotic expansionsfor the Nield-Kuznetsov functions for large a that are uniformly valid for all real andcomplex values of the argument z . We extract the required results from [5, Sect. 4.1],and refer the reader for proofs of the foregoing results. We first must define termsthat are used, and the notation here differs slightly from the above reference.It is convenient to write a = u and rescale the argument from z to √ u z . Thuswe shall focus primarily on the function W (0 , R (cid:0) u, √ u z (cid:1) . Although we have so farconsidered R = 0 (which is the complementary Nield-Kuznetsov function) we alsoshall include R = 1, since both of these will be used in the proceeding section onLaplace transforms.Firstly, with z rescaled as above, let(3.1) β = z √ − z , IELD-KUZNETSOV FUNCTIONS − < z < −∞ , −
1] and [1 , ∞ ). Note β → ± i as z → ∞ in theupper and lower half planes, respectively.Next define(3.2) ξ = ζ / = arccos( z ) − z (cid:112) − z . The branch in (3.2) is chosen so that ξ ≥ ζ ≥ − ≤ z ≤
1, and bycontinuity elsewhere in the z plane having a cut along ( −∞ , −
1] (as well as [1 , ∞ ) for ξ ). Hence ζ ≤ ≤ z < ∞ , and is given by(3.3) ( − ζ ) / = z (cid:112) z − − ln (cid:16) z + (cid:112) z − (cid:17) . We remark that ζ (unlike ξ ) is an analytic function of z for (cid:60) ( z ) ≥
0, in particular at z = 1.We now define an easily computed sequence of polynomials via(3.4) E ( β ) = β (cid:0) β + 6 (cid:1) , (3.5) E ( β ) = (cid:0) β + 1 (cid:1) (cid:0) β + 2 (cid:1) , and for s = 2 , , · · · (3.6) E s +1 ( β ) = 12 (cid:0) β + 1 (cid:1) E (cid:48) s ( β ) + 12 (cid:90) βiσ ( s ) (cid:0) p + 1 (cid:1) s − (cid:88) j =1 E (cid:48) j ( p )E (cid:48) s − j ( p ) dp. Here σ ( s ) = 1 for s odd and σ ( s ) = 0 for s even, so that the even and odd coefficientsare even and odd functions of β , respectively, with E s ( ± i ) = 0 ( s = 1 , , , · · · ).We further define two sequences { a s } ∞ s =1 and { ˜ a s } ∞ s =1 by a = a = , ˜ a = ˜ a = − , with subsequent terms a s and ˜ a s ( s = 2 , , · · · ) satisfying the same recursionformula(3.7) b s +1 = 12 ( s + 1) b s + 12 s − (cid:88) j =1 b j b s − j . With the above definitions we then introduce a sequence of coefficient functions(3.8) E s ( z ) = E s ( β ) + ( − s a s s − ξ − s , and(3.9) ˜ E s ( z ) = E s ( β ) + ( − s ˜ a s s − ξ − s . Next A ( u, z ) and B ( u, z ) are two coefficient functions which are analytic in thecut plane C \ ( −∞ , − u → ∞ , uniformly for z bounded away by a distance δ from this cut, they possess the asymptotic expansions(3.10) A ( u, z ) ∼ φ ( z ) exp (cid:40) ∞ (cid:88) s =1 ˜ E s ( z ) u s (cid:41) cosh (cid:40) ∞ (cid:88) s =0 ˜ E s +1 ( z ) u s +1 (cid:41) , T. M. DUNSTER and(3.11) B ( u, z ) ∼ φ ( z ) u / √ ζ exp (cid:40) ∞ (cid:88) s =1 E s ( z ) u s (cid:41) sinh (cid:40) ∞ (cid:88) s =0 E s +1 ( z ) u s +1 (cid:41) , where(3.12) φ ( z ) = (cid:18) ζ − z (cid:19) / . Note that φ ( z ) has a removable singularity at z = 1 since ζ , as defined by (3.2), hasa simple zero at this point. We also remark that z = 1 is a turning point from thedifferential equation from which these expansions were derived.The coefficients E s ( z ) and ˜ E s ( z ) are not analytic at z = 1. Thus these expansions(or rather a truncated series in (3.10) and (3.11)) cannot be used directly near thissingularity. However both A ( u, z ) and B ( u, z ) are themselves analytic at this point,and there are two ways to compute them near and at this point, described next.If many terms are required for high accuracy, Cauchy’s integral formula can beused, as given in [6, Thm. 4.2]. Basically the expansions for A ( u, z ) and B ( u, z ) areinserting in the Cauchy integral representation of these functions, and being validon the contour of integration that encloses z = 1 this allows accurate and stablecomputation of each function.If only a few terms are required we can expand these functions into a traditionalasymptotic expansion involving inverse powers of u . Specifically, for A ( u, z ) we findfrom (3.10) as u → ∞ (3.13) A ( u, z ) ∼ φ ( z ) ∞ (cid:88) s =0 A s ( z ) u s , where A ( z ) = 1,(3.14) A ( z ) = (cid:110) ˜ E ( z ) + 2 ˜ E ( z ) (cid:111) , (3.15) A ( z ) = (cid:110) ˜ E ( z ) + 12 ˜ E ( z ) ˜ E ( z ) + 24 ˜ E ( z ) ˜ E ( z ) + 12 ˜ E ( z ) + 24 ˜ E ( z ) (cid:111) , and so on.Similarly from (3.11) we have(3.16) B ( u, z ) ∼ φ ( z ) u / ∞ (cid:88) s =0 B s ( z ) u s , with the first two terms being(3.17) B ( z ) = ζ − / E ( z ) , and(3.18) B ( z ) = ζ − / (cid:8) E ( z ) + E ( z ) E ( z ) + E ( z ) (cid:9) . From general turning point theory of differential equations [10, Chap. 11] weknow that these coefficients must be defined and analytic at z = 1, even though ˜ E s ( z ) IELD-KUZNETSOV FUNCTIONS z = 1 cancel out in each of A s ( z ) and B s ( z ), renderingthem analytic at the turning point. Thus a Taylor series can be employed very closeto the removable singularity. For example, as z → from (3.1) - (3.9), (3.14), (3.15), (3.17) and (3.18)(3.19) A ( z ) = 7900 − z −
1) + O (cid:8) ( z − (cid:9) , (3.20) A ( z ) = 7284626913970880000 − z −
1) + O (cid:8) ( z − (cid:9) , (3.21) B ( z ) = − / + 7450 2 / ( z −
1) + O (cid:8) ( z − (cid:9) , and(3.22) B ( z ) = − / + 5549641517440000 2 / ( z −
1) + O (cid:8) ( z − (cid:9) . In a more general setting, analyticity of asymptotic coefficients in this and similarsituations follows from the following result, and we will use this later.
Theorem
Let < ρ < ρ , u > , z ∈ C , H ( u, z ) be an analytic function of z in the open disk D = { z : | z − z | < ρ } , and h s ( z ) ( s = 0 , , , · · · ) be a sequence offunctions that are analytic in D except possibly for an isolated singularity at z = z .If H ( u, z ) is known to possess the asymptotic expansion (3.23) H ( u, z ) ∼ ∞ (cid:88) s =0 h s ( z ) u s ( u → ∞ ) , in the annulus ρ < | z − z | < ρ , then z is an ordinary point or a removablesingularity for each h s ( z ) , and the expansion (3.23) actually holds for all z ∈ D (with h s ( z ) defined by the limit of this function at z if it is a removable singularity).Proof. To establish both assertions we shall use certain Cauchy integrals. Firstly,since z is an isolated singularity of h s ( z ) for each s it is expressible as a Laurentseries at this point, of the form(3.24) h s ( z ) = ∞ (cid:88) j = −∞ a s,j ( z − z ) j (0 < | z − z | < ρ ) , for some coefficients a s,j .Now choose any ρ (cid:48) such that ρ < ρ (cid:48) < ρ , and let Γ be a positively orientatedcircle centered at z = z of radius ρ (cid:48) . By hypothesis (3.23) certainly holds for allpoints on this contour, and since Γ ∈ D we have for arbitrary positive integer n andnonnegative integer r , from the Cauchy-Goursat theorem and analyticity of H ( u, z )(3.25) 0 = (cid:73) Γ ( z − z ) r H ( u, z ) dz = n − (cid:88) s =0 h r,s u s + O (cid:18) u n (cid:19) , Maple 2020. Maplesoft, a division of Waterloo Maple Inc., Waterloo, Ontario T. M. DUNSTER where(3.26) h r,s = (cid:73) Γ ( z − z ) r h s ( z ) dz. Now by uniqueness of asymptotic series it follows from (3.25) for each s that h r,s = 0for all nonnegative r . But from (3.24), (3.26) and the residue theorem we have h r,s =2 πia s, − r − ( r = 0 , , , · · · ), and hence all negative powers of ( z − z ) in the Laurentexpansion (3.24) vanish. Thus each function h s ( z ) is either analytic at z = z , or hasa removable singularity there, as asserted; if the latter, define h s ( z ) = a s, .Having established each h s ( z ) is analytic in D we can now employ Cauchy’sintegral formula, and for any point z interior to Γ, again using (3.23) for points onthe contour, we have(3.27) H ( u, z ) = 12 πi (cid:73) Γ H ( u, t ) t − z dt = n − (cid:88) s =0 πiu s (cid:73) Γ h s ( t ) t − z dt + O (cid:18) u n (cid:19) = n − (cid:88) s =0 h s ( z ) u s + O (cid:18) u n (cid:19) . This verifies that (3.23) holds for all z ∈ D .Next, there are some constants and other coefficients we require. Firstly, we have(3.28) γ ( u ) = 2 − + iu π − / u − / e πu + i ( χ ( u )+ π ) Γ (cid:0) + iu (cid:1) , and(3.29) γ ( u ) = 2 − + iu π − / u − / e πu + i ( χ ( u ) − π ) Γ (cid:0) + iu (cid:1) . In these χ ( u ) is another constant that is defined by a certain asymptotic series anderror term (which has an explicit bound). For brevity we omit details of the errorbound here, and instead note that this constant possesses the following asymptoticexpansion as u → ∞ (3.30) χ ( u ) ∼ u (cid:18) eu (cid:19) + 1 u ∞ (cid:88) s =0 e s u s , where e s are the coefficients in the following relation (which can be derived using(2.38))(3.31) 12 (cid:61) (cid:8) ln (cid:0) Γ (cid:0) − iu (cid:1)(cid:1)(cid:9) ∼ u (cid:18) eu (cid:19) + 1 u ∞ (cid:88) s =0 e s u s . The first two coefficients are e = − and e = − .Incidentally, one can show from (3.30) and (3.31) that for large u and arbitrarypositive integer n (3.32) e iχ ( u ) = (cid:40) Γ (cid:0) − iu (cid:1) Γ (cid:0) + iu (cid:1) (cid:41) / (cid:26) O (cid:18) u n (cid:19)(cid:27) , IELD-KUZNETSOV FUNCTIONS γ ( u ) = 1 √ u / (cid:26) u + 41128 u + O (cid:18) u (cid:19)(cid:27) , and(3.34) γ ( u ) = 1 √ u / (cid:26) − u − u + O (cid:18) u (cid:19)(cid:27) , as u → ∞ Next for R = 0 , G ,R ( z ) = z R / (cid:0) z − (cid:1) , and(3.36) G s +1 ,R ( z ) = G (cid:48)(cid:48) s,R ( z ) / (cid:0) − z (cid:1) ( s = 0 , , , · · · ) . For R = 0 and R = 1 a third slowly-varying coefficient function G R ( u, z ), companionto (3.10) and (3.11) and valid for the same values of z , is then defined such that for u → ∞ (3.37) G R ( u, z ) ∼ u ∞ (cid:88) s =0 G s,R ( z ) u s − γ R ( u ) J ( u, z ) u / ζ (cid:18) ζ − z (cid:19) / , where(3.38) J ( u, z ) ∼ − exp (cid:40) ∞ (cid:88) s =1 ˜ E s ( z ) u s (cid:41) cosh (cid:40) ∞ (cid:88) s =0 ˜ E s +1 ( z ) u s +1 (cid:41) ∞ (cid:88) k =0 (3 k )! k ! (3 u ζ ) k + 1 uζ / exp (cid:40) ∞ (cid:88) s =1 E s ( z ) u s (cid:41) sinh (cid:40) ∞ (cid:88) s =0 E s +1 ( z ) u s +1 (cid:41) ∞ (cid:88) k =0 (3 k + 1)! k ! (3 u ζ ) k . The function G R ( u, z ) is analytic at z = 1, but the truncated series (3.37) and(3.38) are not and so break down near this point. An asymptotic expansion for G R ( u, z ) can be computed near this point in similar way to A ( u, z ) and B ( u, z ), asdescribed above. That is, either by Cauchy’s integral formula, or by re-expanding theseries in inverse powers of u for a few terms, similarly to (3.13) and (3.16). In thelatter case the coefficients are analytic at z = 1 from Theorem 3.1.Having defined all the terms required, from [5, Sect. 4.1] we then have the fol-lowing main result, which in conjunction with (2.11) provides a uniform asymptoticexpansion for ˆ N W (cid:0) u, √ u z (cid:1) for the case R = 0 (and likewise (3.43) below). Theorem
For R = 0 , W (0 , R (cid:16) u, √ u z (cid:17) = (2 u ) R +1 (cid:104) G R ( u, z )+ πγ R ( u ) (cid:110) Hi (cid:16) u / ζ (cid:17) A ( u, z ) + Hi (cid:48) (cid:16) u / ζ (cid:17) B ( u, z ) (cid:111)(cid:105) , where Hi( z ) is the Scorer function defined by (3.40) Hi( z ) = 1 π (cid:90) ∞ exp (cid:0) − t + zt (cid:1) dt, T. M. DUNSTER ζ is defined by (3.2), and A ( u, z ) , B ( u, z ) , and G R ( u, z ) possess the asymptotic expan-sions (3.10), (3.11) and (3.37), respectively, as u → ∞ uniformly for z bounded awayby a distance δ from the cut ( −∞ , − (suitably modified as described after (3.12) if z is close to or equal to 1). Remark
3. Hi( z ) is the uniquely defined particular solution of the inhomogeneousAiry equation (3.41) d wdz − zw = 1 π , having the behavior (3.42) Hi( z ) ∼ − πz (cid:0) z → ∞ , | arg( − z ) | ≤ π − δ (cid:1) . This function grows exponentially as z → ∞ in | arg( z ) | ≤ π − δ (see [3, Eq. 9.12.29],and also [3, Sect. 9.12] for further properties.) As z → ∞ it follows from Theorem 3.2 that W (0 , R ( a, z ) = O ( z − R ) for | arg( z ) |≤ π − δ , and is exponentially large as z → ∞ for δ < | arg( − z ) | ≤ π − δ . No othersolution of (2.10) shares this recessive property in parts of both the first and fourthquadrants.Moreover we have that for large u (3.43) W (0 , R (cid:16) u, √ u z (cid:17) ∼ (2 u ) R +1 u ∞ (cid:88) s =0 G s,R ( z ) u s = O (cid:18) u − R z − R (cid:19) , uniformly z lying in ”most” of the right half plane. On the other hand this functionis exponentially large in u in a part of the right half plane where (cid:60) ( z ) <
1, as well asin the whole of left half plane. More precisely, the boundary that separates these tworegions is depicted in Figure 1, and consists of two level curves (cid:60) ( ξ ) = 0 in the firstand fourth quadrants, emanating from z = 1, where ξ is given by (3.2). The simpleexpansion (3.43) can be considered as a special case of (3.39) that is uniformly validin a more restricted region, namely for all unbounded z to the right of the curves (cid:60) ( ξ ) = 0 and that lie at least a distance δ from them.We finish with expansions for homogeneous solutions. We use the standard nota-tion for Airy functions of complex argument Ai l ( z ) = Ai( ze − πil/ ) ( l = 0 , ± w l ( u, z ) = Ai l (cid:16) u / ζ (cid:17) A ( u, z ) + Ai (cid:48) l (cid:16) u / ζ (cid:17) B ( u, z ) . Then we have for unbounded z as described in Theorem 3.2(3.45) W j (cid:16) u, √ uz (cid:17) = 2 / √ πu − / e − πu/ e ± i ( χ ( u )+ π ) w ∓ ( u, z ) , where upper signs for j = 0 and lower sign for j = 3.Recall for large z that W ( u, √ uz ), W ( u, √ uz ) and W (0 , R ( u, √ uz ) forma numerically satisfactory set of functions in the subdominant region depicted inFigure 1, since W ( u, √ uz ) is recessive (exponentially small at infinity) in the firstquadrant, W ( u, √ uz ) is recessive (exponentially small at infinity) in the fourthquadrant, and W (0 , R ( u, √ uz ) is the unique particular solution that is subdominantin the above described region. IELD-KUZNETSOV FUNCTIONS (0) ( (cid:1) (1) Exponentially LargeExponentially Large O ! u − R z − R " O ! u − R z − R " Fig. 1 . Regions of subdominance and dominance for W (0 , R ( u, √ u z ) With these three fundamental solutions, an asymptotic expansion for N W ( a, z )follows from Theorems 2.9 and 3.2, (3.44) and (3.45). This is certainly uniformlyvalid for (cid:60) ( z ) ≥
0, with of course the left half plane immediately being covered bythe evenness of N W ( a, z ).We also remark that an asymptotic expansion for ˆ N W ( u, √ u z ) valid in the0 T. M. DUNSTER whole left half plane, and in particular including the cut −∞ < z ≤ − N W ( a, − z ) = ˆ N W ( a, z ) − / πie πa (cid:26) ia/ Γ( − ia ) W ( a, z ) − − ia/ Γ( + ia ) W ( a, z ) (cid:27) , which can be deduced from [5, Eq.(5.33)].
4. Laplace Transforms of W ( a, ± t ) . We consider the Laplace transform of theWeber functions defined by(4.1) L ± W ( a, λ ) = (cid:90) ∞ e − λt W ( a, ± t ) dt (cid:0) | arg( λ ) | ≤ π (cid:1) , and by analytic continuation for other values of arg( λ ). As we shall show, both areentire functions of λ .For bounded a and large λ the asymptotics are simple to obtain, and read asfollows. Theorem
For fixed a and λ → ∞ with | arg( λ ) | ≤ π − δ (4.2) L ± W ( a, λ ) ∼ W ( a, λ ∞ (cid:88) s =0 α s ( a ) λ s ± W (cid:48) ( a, λ ∞ (cid:88) s =0 β s ( a ) λ s , where α ( a ) = β ( a ) = 1 , α ( a ) = β ( a ) = a , and for s = 0 , , , · · · (4.3) α s +2 = aα s +1 − ( s + 1)(2 s + 1) α s , (4.4) β s +2 = aβ s +1 − ( s + 1)(2 s + 3) β s , where W ( a, and W (cid:48) ( a, are given by (1.7) and (1.8).Proof. We have by Watson’s lemma [10, Chap. 4, Thm. 3.2] applied to (4.1)(4.5) L ± W ( a, λ ) ∼ ∞ (cid:88) s =0 ( ± s a s s ! λ s +1 , for | arg( λ ) | ≤ π − δ , where a s are the coefficients in the asymptotic expansion(4.6) W ( a, t ) ∼ ∞ (cid:88) s =0 a s t s ( t → . These coefficients are given by [3, Eqs. 12.14.8 - 12.14.12] (actually (4.6) is a conver-gent series), and inserting them into (4.5) yields (4.2) - (4.4).We now consider the more difficult problem of obtaining asymptotic expansionsfor large a that are uniformly valid for all real and complex λ , including the analyticcontinuation of (4.1) to the left half λ plane where the integral diverges. To this endwe begin with: Lemma
Each L ± W ( a, λ ) is an entire function of λ and satisfies the differentialequation (4.7) ∂ L ± W ( a, λ ) ∂λ = 4 (cid:0) a − λ (cid:1) L ± W ( a, λ ) ± W (cid:48) ( a,
0) + 4 λW ( a, . IELD-KUZNETSOV FUNCTIONS Proof.
Assume temporarily that | arg( λ ) | ≤ π − δ . Differentiation of (4.1) twicewith respect to λ gives(4.8) ∂ L ± W ( a, λ ) ∂λ = (cid:90) ∞ t e − λt W ( a, ± t ) dt. Then from the Weber differential equation (2.1) we find that(4.9) ∂ L ± W ( a, λ ) ∂λ = 4 a (cid:90) ∞ e − λt W ( a, ± t ) dt − (cid:90) ∞ e − λt W (cid:48)(cid:48) ( a, ± t ) dt. Next from integration by parts twice of the second integral on the RHS of (4.9), andreferring to (4.1), we then get (4.7). Finally, by analytic continuation the restrictionon arg( λ ) can be relaxed and L ± W ( a, λ ) are seen to be entire functions of λ since thedifferential equations (4.7) they satisfy have no finite singularities.As a consequence the following explicit representation can be inferred. Theorem
The Laplace transforms (4.1) are expressible in terms of the gen-eral Nield-Kuznetsov functions (2.11) via (4.10) L ± W ( a, λ ) = ± W (cid:48) ( a, W (0 , ( a, λ ) + W ( a, W (0 , ( a, λ ) . Proof.
The differential equation (4.7) is an inhomogeneous form of (2.1) with z replaced 2 λ . Thus(4.11) L ± W ( a, λ ) = ± W (cid:48) ( a, W (0 , ( a, λ ) + W ( a, W (0 , ( a, λ )+ A ( a ) W ( a, λ ) + A ( a ) W ( a, λ ) , for some constants A ( a ) and A ( a ). Now as λ → ∞ with δ ≤ arg( λ ) ≤ π − δ all terms on the RHS of (4.11) vanish, with the exception of W ( a, λ ) which isunbounded. But from (4.2) the LHS vanishes in this limit, and hence A ( a ) = 0.Similarly by letting λ → ∞ with − π + δ ≤ arg( λ ) ≤ − δ < A ( a ) = 0.We now proceed to obtain uniform asymptotic expansions using these represen-tations. If we let u = 2 a and define z by(4.12) 2 λ = √ u z, then from using the asymptotic expansions of Theorem 3.2 in the RHS of (4.10) wehave under the conditions of that theorem(4.13) L ± W (cid:18) u, (cid:113) u z (cid:19) ∼ ± u W (cid:48) (cid:0) u, (cid:1) ∞ (cid:88) s =0 G s, ( z ) u s + (cid:114) u W (cid:0) u, (cid:1) ∞ (cid:88) s =0 G s, ( z ) u s + ν ± ( u ) (cid:40) π Hi (cid:16) u / ζ (cid:17) A ( u, z ) + π Hi (cid:48) (cid:16) u / ζ (cid:17) B ( u, z ) − J ( u, z ) u / ζ (cid:18) ζ − z (cid:19) / (cid:41) , where(4.14) ν ± ( u ) = ± uγ ( u ) W (cid:48) (cid:0) u, (cid:1) + √ u / γ ( u ) W (cid:0) u, (cid:1) . T. M. DUNSTER
Consider first L + W ( u, (cid:113) u z ). We find from (1.7), (1.8), (2.38), (3.28), (3.29),and (3.32) for large u and arbitrary positive integer n that(4.15) 2 uγ ( u ) W (cid:48) (cid:0) u, (cid:1) = − − / u − / (cid:8) O (cid:0) u − n (cid:1)(cid:9) , and(4.16) √ u / γ ( u ) W (cid:0) u, (cid:1) = 2 − / u − / (cid:8) O (cid:0) u − n (cid:1)(cid:9) . Now from these two equations and (4.14) we see that ν + ( u ) = O ( u − n ), and hencefrom (1.7), (1.8), (2.38), (3.35), (3.36), and (4.13) we find that(4.17) L + W (cid:18) u, (cid:113) u z (cid:19) ∼ / u / ∞ (cid:88) s =0 G + s ( z ) u s , where G + s ( z ) are coefficients in the formal expansion(4.18) 2 u W (cid:48) (cid:0) u, (cid:1) ∞ (cid:88) s =0 G s, ( z ) u s + (cid:114) u W (cid:0) u, (cid:1) ∞ (cid:88) s =0 G s, ( z ) u s ∼ / u / ∞ (cid:88) s =0 G + s ( z ) u s . These can be computed by using (1.7), (1.8), (2.38), (3.35), and (3.36). For the firsttwo we find(4.19) G +0 ( z ) = 1 z + 1 , and(4.20) G +1 ( z ) = ( z + 3) (cid:0) z + 2 z + 5 (cid:1) z + 1) . The expansion (4.17) is certainly valid to the right of the curves (cid:60) ( ξ ) = 0 shownin Figure 1, but it is not clear if the term in braces in (4.13) is negligible elsewhere,or indeed if the all the coefficients are defined at z = 1. To verify this, and improveaccuracy, we obtain an improved version of the expansion. We do this by using theconnection formula [5, Eqs. (5.26) and (5.27)](4.21) W (0 , R (cid:16) u, √ u z (cid:17) = W (0 , R (cid:16) u, √ u z (cid:17) + µ R ( u ) W (cid:16) u, √ u z (cid:17) , where(4.22) µ ( u ) = 2 − + iu e πi/ e uπ/ Γ (cid:0) + iu (cid:1) , and(4.23) µ ( u ) = 2 + iu e uπ/ Γ (cid:0) + iu (cid:1) . Thus from (4.10) and (4.21) we have the exact expression(4.24) L + W (cid:18) u, (cid:113) uz (cid:19) = W (cid:48) (cid:0) u, (cid:1) W (0 , (cid:16) u, √ u z (cid:17) + W (cid:0) u, (cid:1) W (0 , (cid:16) u, √ u z (cid:17) + µ ( u ) W (cid:16) u, √ u z (cid:17) , IELD-KUZNETSOV FUNCTIONS µ ( u ) = µ ( u ) W (cid:48) (cid:0) u, (cid:1) + µ ( u ) W (cid:0) u, (cid:1) . Similarly, in terms of W (1 , ( u, √ u z ) instead of W (0 , ( u, √ u z ), we can showthat(4.26) L + W (cid:18) u, (cid:113) uz (cid:19) = W (cid:48) (cid:0) u, (cid:1) W (1 , (cid:16) u, √ u z (cid:17) + W (cid:0) u, (cid:1) W (1 , (cid:16) u, √ u z (cid:17) + µ ( u ) W (cid:16) u, √ u z (cid:17) . At this stage it is worth emphasizing for large z and either R = 0 or R = 1 that W (0 , R ( u, √ u z ) is characterized as being bounded in the first and third quadrants,and W (1 , R ( u, √ u z ) is characterized as being bounded in the second and fourthquadrants. While (4.24) and (4.26) can both in theory be used for all z , we only usethem in the quadrants in which these functions are bounded. Taken together, bothof them cover the whole complex plane.Thus for example, for the right half plane (4.24) (respectively (4.26)) is only nu-merically useful in the first (respectively fourth) quadrant, since in the other quadrantall functions of the RHS of (4.24) and (4.26) are exponentially large, with large can-cellations due to the fact L + W ( u, (cid:113) uz ) is bounded in the right half plane to theright of the curves depicted in Figure 1 emanating from z = 1.In summary, since W (0 , R ( u, √ u z ) is bounded in the first and third quadrantswe use (4.24) in these quadrants (excluding points on or near the cut −∞ < z ≤ − W (0 , R ( u, √ u z ) is the same as (3.39) except withthe Scorer function Hi( z ) replaced by its rotated form e πi/ Hi( ze πi/ ). We thus findsimilarly to the derivation of the expansion (4.17) that(4.27) W (cid:48) (cid:0) u, (cid:1) W (0 , (cid:16) u, √ u z (cid:17) + W (cid:0) u, (cid:1) W (0 , (cid:16) u, √ u z (cid:17) ∼ / u / ∞ (cid:88) s =0 G + s ( z ) u s . For 0 ≤ arg( ± z ) ≤ π/ u → ∞ we then have from (4.24) and (4.27)(4.28) L + W (cid:18) u, (cid:113) u z (cid:19) ∼ / u / ∞ (cid:88) s =0 G + s ( z ) u s + µ ( u ) W (cid:16) u, √ u z (cid:17) , and similarly from (4.26) for − π/ ≤ arg( ± z ) ≤ L + W (cid:18) u, (cid:113) u z (cid:19) ∼ / u / ∞ (cid:88) s =0 G + s ( z ) u s + µ ( u ) W (cid:16) u, √ u z (cid:17) , in both cases excluding points on or near the cut −∞ < z ≤ − µ ( u ) = (cid:18) πe πu (cid:19) / (cid:8) Γ (cid:0) + iu (cid:1)(cid:9) / (cid:104) T / ( u ) − e πi/ T − / ( u ) (cid:105) , T. M. DUNSTER where(4.31) T ( u ) = 1 + i tanh (cid:0) πu (cid:1) − i tanh (cid:0) πu (cid:1) . From this it follows that(4.32) µ ( u ) = e πi/ (cid:16) π e πu (cid:17) / (cid:8) Γ (cid:0) + iu (cid:1)(cid:9) / (cid:8) O (cid:0) e − πu (cid:1)(cid:9) , which incidentally (with the aid of (2.38)) shows that µ ( u ) = O ( e − πu/ ).Our first desired uniform asymptotic expansion is then given as follows. Theorem
Let Ai ∓ ( z ) = Ai( ze ± πi/ ) . Then as u → ∞ we have uniformly (4.33) L + W (cid:18) u, (cid:113) u z (cid:19) ∼ / u / ∞ (cid:88) s =0 G + s ( z ) u s + √ π / u − / e − πu ± πi (cid:12)(cid:12) Γ (cid:0) + iu (cid:1)(cid:12)(cid:12) / × (cid:110) Ai ∓ (cid:16) u / ζ (cid:17) A ( u, z ) + Ai (cid:48)∓ (cid:16) u / ζ (cid:17) B ( u, z ) (cid:111) , where upper signs are taken for z in first and third quadrants, and lower signs for z in second and fourth quadrants. In both cases points within a distance of δ of thecut −∞ < z ≤ − must be excluded, but otherwise z is unrestricted. The coefficients G + s ( z ) are rational functions whose only poles are at z = − , and are given by (4.18)using (1.7), (1.8), (2.38), (3.35), (3.36). Also, ζ is given by (3.2), and A ( u, z ) and B ( u, z ) are functions (analytic at z = 1 ) having the expansions (3.10) and (3.11),respectively.Proof. The expansions (4.33) follow from (3.44), (3.45), (4.28), (4.29), and (4.32).Next one can show for | ζ | < ρ , where ρ > u . Now ζ = 0 corresponds to z = 1, and therefore (4.17) holds for0 < ρ < | z − | < ρ for certain constants ρ and ρ . We then apply Theorem 3.1to establish the analyticity of G + s ( z ) at z = 1, as well as the validity of (4.33) in aneighborhood of z = 1.Let us now turn our attention to L − W ( u, (cid:113) u z ). Then, similarly to (4.18) wedefine coefficients G − s ( z ) via the formal expansion(4.34) − u W (cid:48) (cid:0) u, (cid:1) ∞ (cid:88) s =0 G s, ( z ) u s + (cid:114) u W (cid:0) u, (cid:1) ∞ (cid:88) s =0 G s, ( z ) u s − J ( u, z ) u / ζ (cid:18) ζ − z (cid:19) / ∼ / u / ∞ (cid:88) s =0 G − s ( z ) u s , where J ( u, z ) is a function (analytic at z = 1) having the asymptotic expansion (3.38).The first two coefficients are found to be(4.35) G − ( z ) = 1 z − √ φ ( z ) ζ , IELD-KUZNETSOV FUNCTIONS G − ( z ) = ( z − (cid:0) z − z + 5 (cid:1) z − + φ ( z ) (cid:110) ˜ E ( z ) + 2 ˜ E ( z ) (cid:111) √ ζ − √ φ ( z ) E ( z ) ζ / + 2 / φ ( z ) ζ . Unlike G + s ( z ) these are not rational functions of z , however from Theorem 3.1 onecan show that they also have removable singularities at z = 1 ( ζ = 0), and so can beconsidered analytic at z = 1. For example, using Maple we find as z → G − ( z ) = 15 − z −
1) + O (cid:8) ( z − (cid:9) , and(4.38) G − ( z ) = 5863498624000 − z −
1) + O (cid:8) ( z − (cid:9) . From (4.13), (4.14), (4.15), (4.16), and (4.34) we now obtain our desired result.
Theorem As u → ∞ (4.39) L − W (cid:18) u, (cid:113) u z (cid:19) ∼ / u / ∞ (cid:88) s =0 G − s ( z ) u s + 2 / πu / (cid:110) Hi (cid:16) u / ζ (cid:17) A ( u, z ) + Hi (cid:48) (cid:16) u / ζ (cid:17) B ( u, z ) (cid:111) , uniformly for all z except those within a distance of δ of the cut −∞ < z ≤ − . Here Hi( z ) is the Scorer function defined by (3.40). The following theorem provides asymptotic expansions uniformly valid in a regioncontaining the interval −∞ < z ≤ −
1, which was excluded from Theorems 4.4 and 4.5.
Theorem
Let z lie on, or to the right of the level curves (cid:60) ( ξ ) = 0 in thefirst and fourth quadrants, emanating from z = 1 , as depicted in Figure , where ξ is given by (3.2). Then as u → ∞ we have uniformly for z in this unbounded closedregion (4.40) L + W (cid:18) u, − (cid:113) u z (cid:19) ∼ − L − W (cid:18) u, (cid:113) u z (cid:19) + √ π / u − / e πu/ (cid:12)(cid:12) Γ (cid:0) + iu (cid:1)(cid:12)(cid:12) / × (cid:110) Bi (cid:16) u / ζ (cid:17) A ( u, z ) + Bi (cid:48) (cid:16) u / ζ (cid:17) B ( u, z ) (cid:111) , and (4.41) L − W (cid:18) u, − (cid:113) u z (cid:19) ∼ − L + W (cid:18) u, (cid:113) u z (cid:19) + 2 √ π / u − / e πu/ (cid:12)(cid:12) Γ (cid:0) + iu (cid:1)(cid:12)(cid:12) / × (cid:110) Ai (cid:16) u / ζ (cid:17) A ( u, z ) + Ai (cid:48) (cid:16) u / ζ (cid:17) B ( u, z ) (cid:111) . T. M. DUNSTER
Remark In these the expansions of L ± W ( u, (cid:113) u z ) given by Theorems and can be used. Also, the Airy functions in both these expansions are exponen-tially large as u → ∞ in the interior of the region of validity, except on the real axiswhere they are oscillatory for ≤ z < ∞ .Proof. From [5, Eq. (5.33)] we have for R = 0 , W (0 , R (cid:16) u, −√ uz (cid:17) = ( − R W (0 , R (cid:16) u, √ uz (cid:17) + (cid:110) ( − R +1 − ie πu/ (cid:111) µ R ( u ) W (cid:16) u, √ uz (cid:17) + (cid:110) ( − R +1 + ie πu/ (cid:111) µ R ( u ) W (cid:16) u, √ uz (cid:17) where µ ( u ) and µ ( u ) are given by (4.22) and (4.23), respectively. Thus inserting(4.42) into (4.10) and using (4.25) we obtain(4.43) L ± W (cid:18) u, − (cid:113) u z (cid:19) = − L ∓ W (cid:18) u, (cid:113) u z (cid:19) + ˆ µ ± ( u ) W (cid:16) u, √ uz (cid:17) + ˆ µ ± ( u ) W (cid:16) u, √ uz (cid:17) , where(4.44) ˆ µ + ( u ) = µ ( u ) W (cid:0) u, (cid:1) − µ ( u ) W (cid:48) (cid:0) u, (cid:1) − ie πu/ µ ( u ) , and(4.45) ˆ µ − ( u ) = µ ( u ) − ie πu/ (cid:8) µ ( u ) W (cid:0) u, (cid:1) − µ ( u ) W (cid:48) (cid:0) u, (cid:1)(cid:9) . We find from (1.7), (1.8), (2.34), (4.22), (4.23) and (4.30), for large u ,(4.46) µ + ( u ) = e πi/ (cid:18) πe πu (cid:19) / (cid:8) Γ (cid:0) + iu (cid:1)(cid:9) / (cid:8) O (cid:0) e − πu (cid:1)(cid:9) , and(4.47) µ − ( u ) = 2 / π / e − πi/ e πu/ (cid:8) Γ (cid:0) + iu (cid:1)(cid:9) / (cid:8) O (cid:0) e − πu (cid:1)(cid:9) . From (2.38), (4.46) and (4.47) it can be shown that µ + ( u ) = O ( e πu/ ) and µ − ( u ) = O ( e πu/ ) as u → ∞ . Now from [3, Eq. 9.2.10](4.48) Bi( z ) = e − πi/ Ai ( z ) + e πi/ Ai − ( z ) , and so from this and (3.32), (3.44), (3.45), (4.43), (4.44) and (4.46) we obtain (4.40),provided z lies in the stated closed region. To the left of the boundary curves thisexpansion cannot be used, since for large u there is a severe cancellation of exponen-tially large terms on the RHS, whereas the LHS is bounded. However, this region iscovered by Theorem 4.4.Similarly, using [3, Eq. 9.2.12](4.49) Ai( z ) = e πi/ Ai ( z ) + e − πi/ Ai − ( z ) , along with (3.32), (3.44), (3.45), (4.43), (4.45) and (4.47) we obtain (4.41). Unlike(4.40) this is actually still valid in part of the region to the left of the boundary, butagain we can use Theorem 4.5 in this case. IELD-KUZNETSOV FUNCTIONS
5. Laplace Transform of U ( a, t ) . Let us now consider(5.1) L U ( a, λ ) = (cid:90) ∞ e − λt U ( a, t ) dt (cid:0) | arg( λ ) | ≤ π (cid:1) . When defined by analytic continuation for other values of arg( λ ) we again will findthat it is an entire function of λ .The procedures here are similar to those section 4, although a major difference isthat we obtain a simpler slowly varying expansion for this Laplace transform that isvalid in the whole right half λ plane, and also in part of the left half plane. To avoidintroducing new terms we shall use the same notation in this section as the previousone, noting any differences of the functions and parameters involved.Beginning with large λ with a fixed, similarly to Theorem 4.1 we obtain an asymp-totic expansion for L U ( a, λ ). The main difference here is that, unlike W ( a, ± t ), U ( a, t )is exponentially small in a sector containing the real t axis: see (1.11). This allowsus to apply a more general version of Watson’s lemma for analytic integrands whichare exponentially small in sectors [10, Chap. 4, Thm. 3.3]. So from this and from [3,Eqs. 12.2.6 and 12.2.7] we then obtain the following. Theorem
For fixed a and λ → ∞ with | arg( λ ) | ≤ π − δ (5.2) L U ( a, λ ) ∼ U ( a, λ ∞ (cid:88) s =0 α s ( a ) λ s + U (cid:48) ( a, λ ∞ (cid:88) s =0 β s ( a ) λ s , where α ( a ) = β ( a ) = 1 , α ( a ) = β ( a ) = a , and for s = 0 , , , · · · (5.3) α s +2 = aα s +1 + ( s + 1)(2 s + 1) α s , (5.4) β s +2 = aβ s +1 + ( s + 1)(2 s + 3) β s , (5.5) U ( a,
0) = √ π a + Γ (cid:0) + a (cid:1) , and (5.6) U (cid:48) ( a,
0) = − √ π a − Γ (cid:0) + a (cid:1) . We omit the proof since it is very similar to that of Theorem 4.1, although wenote that the λ sector of asymptotic validity in Theorem 5.1 is larger.Next we express L U ( a, λ ) in terms of certain functions U ( j,k ) R ( a, z ) (( j, k ) = (0 , ,
3) and (1 , z R ( R = 0 , z = ∞ that U (1 , R ( a, z ) is bounded when π ≤ | arg( ± z ) | ≤ π , U (0 , R ( a, z ) is bounded when − π ≤ arg( z ) ≤ π , and U (0 , R ( a, z ) is bounded when − π ≤ arg( z ) ≤ π .In [5] they are defined similarly to (2.11), namely(5.7) U (1 , R ( a, z ) = i Γ (cid:0) − a (cid:1) √ π (cid:20) U ( − a, iz ) (cid:90) zi ∞ t R U ( − a, − it ) dt − U ( − a, − iz ) (cid:90) z − i ∞ t R U ( − a, it ) dt (cid:21) , T. M. DUNSTER (5.8) U (0 , R ( a, z ) = e πi ( a − ) (cid:20) U ( − a, − iz ) (cid:90) z ∞ t R U ( a, t ) dt − U ( a, z ) (cid:90) zi ∞ t R U ( − a, − it ) dt (cid:21) , and U (0 , R ( a, z ) be given by (5.8) with i replaced by − i .It turns out L U ( a, λ ) can be expressed in terms of any of these three functions(with a replaced by − a ): Theorem
For ( j, k ) = (0 , , (0 , or (1 , L U ( a, λ ) = − U (cid:48) ( a, U ( j,k )0 ( − a, λ ) − U ( a, U ( j,k )1 ( − a, λ ) . Proof.
Similarly to Lemma 4.2 we find from (5.1)(5.10) ∂ L U ( a, λ ) ∂λ = 4 (cid:0) λ − a (cid:1) L U ( a, λ ) − U (cid:48) ( a, − λU ( a, . This is an inhomogeneous version of (1.9) with z replaced 2 λ and a replaced by − a .Now for ( j, k ) = (0 , , (0 ,
3) or (1 , A j ( a ) and A k ( a ), the solution L U ( a, λ ) can be expressed in the form(5.11) L U ( a, λ ) = − U (cid:48) ( a, U ( j,k )0 ( − a, λ ) − U ( a, U ( j,k )1 ( − a, λ )+ A j ( a ) U j ( − a, λ ) + A k ( a ) U k ( − a, λ ) , where (analogous to (2.2))(5.12) U j ( a, z ) = U (cid:0) ( − j a, ( − i ) j z (cid:1) ( j = 0 , , , are solutions of (1.9) that are exponentially small at infinity in the sectors | arg(( − i ) j z ) | < π (see (1.11)). Then by letting λ → ∞ in various appropriate direc-tions, and noting that from Theorem 5.1 L U ( a, λ ) is bounded for | arg( λ ) | ≤ π − δ ,we deduce that A j ( a ) = A k ( a ) = 0 in all three cases, and the result follows.We again define z by (4.12), and with this introduce closed regions Z j ( j =0 , , ,
3) as depicted in Figure 2. In this the boundaries (thick lines) comprise theinterval − ≤ z ≤ z = ± (cid:60) ( ξ ) = constant,where instead of (3.2) we have(5.13) ξ = (cid:90) z (cid:0) t − (cid:1) / dt = z (cid:112) z − − ln (cid:16) z + (cid:112) z − (cid:17) , which is positive for 1 < z < ∞ and a continuous function of z for z ∈ C \ ( −∞ , Z ( j,k ) to be the interior of Z j ∪ Z k ( j (cid:54) = k ).Next, in place of (3.35) and (3.36), define coefficients(5.14) G ,R ( z ) = − z R / (cid:0) z − (cid:1) , and(5.15) G s +1 ,R ( z ) = G (cid:48)(cid:48) s,R ( z ) / (cid:0) z − (cid:1) ( s = 0 , , , · · · ) . IELD-KUZNETSOV FUNCTIONS (0) ( (cid:1) (1) Z Z Z Z Fig. 2 . Domains Z j ( j = 0 , , , ) Then, with the aid of (2.38), (5.5), (5.6), (5.14), and (5.15) we define a sequence of0
T. M. DUNSTER coefficients ˆ G s ( z ) ( s = 0 , , , · · · ) via the formal expansion (cf. (4.18))(5.16) − u U (cid:48) (cid:0) u, (cid:1) ∞ (cid:88) s =0 G s, ( z ) u s − (cid:114) u U (cid:0) u, (cid:1) ∞ (cid:88) s =0 G s, ( z ) u s ∼ / u / (cid:18) eu (cid:19) u/ ∞ (cid:88) s =0 ˆ G s ( z ) u s . The first three are found to be(5.17) ˆ G ( z ) = 11 + z , (5.18) ˆ G ( z ) = 124(1 + z ) , and(5.19) ˆ G ( z ) = − z + 717 z + 1581 z + 21591152(1 + z ) . Our main result then reads as follows.
Theorem
The coefficients ˆ G s ( z ) are rational functions whose only poles areat z = − , and as u → ∞ (5.20) L U (cid:18) u, (cid:113) u z (cid:19) ∼ / u / (cid:18) eu (cid:19) u/ ∞ (cid:88) s =0 ˆ G s ( z ) u s , for z in the interior of Z ∪ Z ∪ Z , uniformly for all points in this set whose distancefrom its boundary is greater than or equal to δ .Proof. It is evident from (5.14) - (5.16) that ˆ G s ( z ) are rational functions, whoseonly possible poles are at z = ±
1. Next, by applying [4, Thm. 4] we obtain the simpleasymptotic expansions for large u (5.21) U ( j,k ) R (cid:16) − u, √ u z (cid:17) ∼ u ) R u ∞ (cid:88) s =0 G s,R ( z ) u s ( R = 0 , , where ( j, k ) = (0 , , (0 ,
3) or (1 , z ∈ Z ( j,k ) , and uniformly forall points in this set whose distance from its boundary is greater than or equal to δ . Note that z = ± Z ∪ Z ∪ Z , except possibly in a closed neighborhood of z = 1. However, from Theorem 3.1 we see that the expansion is valid there too, andmoreover the coefficients ˆ G s ( z ) are analytic at z = 1. Finally, the error bounds for(5.21) supplied by [4, Thm. 4] establish that the expansion is uniformly valid for all(unbounded) z that are at least a distance δ from the boundary of Z ∪ Z ∪ Z .It remains to provide an alternative expansion valid for a domain containingpoints excluded from Theorem 5.3. This is more complicated, since like Theorem 4.5it will involve the Scorer function. IELD-KUZNETSOV FUNCTIONS β = z √ z − , where the branch of the square root is positive for z > − , β → z → ∞ in any direction. Furtherdefine(5.23) E ( β ) = β (cid:0) β − (cid:1) , (5.24) E ( β ) = (cid:0) β − (cid:1) (cid:0) β − (cid:1) , and for s = 2 , , · · · (5.25) E s +1 ( β ) = 12 (cid:0) β − (cid:1) E (cid:48) s ( β ) + 12 (cid:90) βσ ( s ) (cid:0) p − (cid:1) s − (cid:88) j =1 E (cid:48) j ( p )E (cid:48) s − j ( p ) dp, where again σ ( s ) = 1 for s odd and σ ( s ) = 0 for s even, so that the even andodd coefficients are even and odd functions of β , respectively, with E s (1) = 0 ( s =1 , , , · · · ).Next, in place of (3.28), (3.29), (4.14) let(5.26) ν ( u ) = −√ uγ ( u ) U (cid:48) (cid:0) u, (cid:1) + u / γ ( u ) U (cid:0) u, (cid:1) , where γ R ( u ) ( R = 0 ,
1) are constants possessing the asymptotic expansions(5.27) γ R ( u ) ∼ √ πu u − R − u − R + Γ (cid:0) u − R + (cid:1) exp (cid:40) − u + ∞ (cid:88) s =0 E s +1 (1) u s +1 (cid:41) , as u → ∞ . Then if ζ = (3 ξ/ / (where ξ is given by (5.13)) we define coefficients˜ G s ( z ) which, like those of (4.34), are analytic at z = 1 and are given by(5.28) − u U (cid:48) (cid:0) u, (cid:1) ∞ (cid:88) s =0 G s, ( z ) u s + (cid:114) u U (cid:0) u, (cid:1) ∞ (cid:88) s =0 G s, ( z ) u s − √ ν ( u ) J ( u, z ) u / ζ (cid:18) ζ − z (cid:19) / ∼ / u / (cid:18) eu (cid:19) u/ ∞ (cid:88) s =0 ˜ G s ( z ) u s . The desired expansion is given as follows.
Theorem As u → ∞ (5.29) L U (cid:18) u, − (cid:113) u z (cid:19) ∼ / u / (cid:18) eu (cid:19) u/ (cid:34) ∞ (cid:88) s =0 ˜ G s ( z ) u s + √ πu / exp (cid:40) − ∞ (cid:88) s =0 E s +1 (1) u s +1 (cid:41) (cid:110) Hi (cid:16) u / ζ (cid:17) A ( u, z ) + Hi (cid:48) (cid:16) u / ζ (cid:17) B ( u, z ) (cid:111) (cid:35) , for z in the interior of Z ∪ Z ∪ Z , uniformly for all points in this set whose distancefrom its boundary is no less than δ . T. M. DUNSTER
Proof.
Firstly we note that U (1 , R ( − a, − λ ) = ( − R U (1 , R ( − a, λ ) since both areparticular solutions of the same differential equation with the same unique subdomi-nant behavior at λ = ± i ∞ . Thus from (5.9) with ( j, k ) = (1 ,
3) we have(5.30) L U ( a, − λ ) = − U (cid:48) ( a, U (1 , ( − a, − λ ) − U ( a, U (1 , ( − a, − λ )= − U (cid:48) ( a, U (1 , ( − a, λ ) + U ( a, U (1 , ( − a, λ ) . Now for R = 0 and R = 1 the functions U (1 , R ( u, √ u z ) possess the sameexpansions as W (0 , R ( u, √ u z ) given by Theorem 3.2, but with coefficients G s,R ( z )and E s ( β ) given by (5.14), (5.15), (5.22) - (5.25) and γ R ( u ) given by (5.27). We theninsert these into (5.30) (with a = u and 2 λ = √ u z ) and use (5.28), and this yieldsin the stated region(5.31) L U (cid:18) u, − (cid:113) u z (cid:19) ∼ / u / (cid:18) eu (cid:19) u/ ∞ (cid:88) s =0 ˆ G s ( z ) u s + √ πν ( u ) (cid:110) Hi (cid:16) u / ζ (cid:17) A ( u, z ) + Hi (cid:48) (cid:16) u / ζ (cid:17) B ( u, z ) (cid:111) . Next from (2.38), (5.5), (5.6), (5.26) and (5.27) we find that for large u (5.32) ν ( u ) ∼ √ π / u / Γ (cid:0) + u (cid:1) (cid:16) u e (cid:17) u/ exp (cid:40) ∞ (cid:88) s =0 E s +1 (1) u s +1 (cid:41) . Now from (4.49), [6, Eqs. (3.15), (3.35), (3.36)] and [3, Eq. 12.2.18] one can showthat for large u (5.33) √ π Γ (cid:0) + u (cid:1) (cid:16) u e (cid:17) u/ ∼ exp (cid:40) − ∞ (cid:88) s =0 E s +1 (1) u s +1 (cid:41) , and thus from (5.32)(5.34) ν ( u ) ∼ / u / (cid:18) eu (cid:19) u/ exp (cid:40) − ∞ (cid:88) s =0 E s +1 (1) u s +1 (cid:41) . Plugging this into (5.31) then yields (5.29).
Acknowledgments.
Financial support from Ministerio de Ciencia e Innovaci´on,Spain, project PGC2018-098279-B-I00 (MCIU/AEI/FEDER, UE) is acknowledged.
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