Numerical and asymptotic aspects of parabolic cylinder functions

Transcription

1 Centrum voor Wiskunde en Informatica Numerical and asymptotic aspects of parabolic cylinder functions N.M. Temme Modelling, Analysis and Simulation (MAS) MAS-R005 May 3, 000

2 Report MAS-R005 ISSN CWI P.O. Box GB Amsterdam The Netherlands CWI is the National Research Institute for Mathematics and Computer Science. CWI is part of the Stichting Mathematisch Centrum (SMC), the Dutch foundation for promotion of mathematics and computer science and their applications. SMC is sponsored by the Netherlands Organization for Scientific Research (NWO). CWI is a member of ERCIM, the European Research Consortium for Informatics and Mathematics. Copyright Stichting Mathematisch Centrum P.O. Box 9079, 090 GB Amsterdam (NL) Kruislaan 3, 098 SJ Amsterdam (NL) Telephone Telefax

3 Numerical and Asymptotic Aspects of Parabolic Cylinder Functions Nico M. Temme CWI P.O. Box 9079, 090 GB Amsterdam, The Netherlands url: ABSTRACT Several uniform asymptotics expansions of the Weber parabolic cylinder functions are considered, one group in terms of elementary functions, another group in terms of Airy functions. Starting point for the discussion are asymptotic expansions given earlier by F.W.J. Olver. Some of his results are modied to improve the asymptotic properties and to enlarge the intervals for using the expansions in numerical algorithms. Olver's results are obtained from the dierential equation of the parabolic cylinder functions; we mention how modied expansions can be obtained from integral representations. Numerical tests are given for three expansions in terms of elementary functions. In this paper only real values of the parameters will be considered. 000 Mathematics Subject Classication: 33C5, A60, 65D0. Keywords and Phrases: parabolic cylinder functions, uniform asymptotic expansion, Airy-type expansions, numerical evaluation of special functions. Note: Work carried out under project MAS.3 Partial dierential equations in porous media research. This report has been accepted for publication in the Journal of Computational and Applied Mathematics. Introduction The solutions of the dierential equation d y dz? z + a y = 0 (.) are associated with the parabolic cylinder in harmonic analysis; see [0]. The solutions are called parabolic cylinder functions and are entire functions of z. Many properties in connection with physical applications are given in []. As in [], Chapter 9, we denote two standard solutions of (.) by U(a; z); V (a; z). These solutions are given by the representations?( 3 + a)? U(a; z) = p? a? y (a; z) V (a; z) = 3 y (a; z) 5 ;?( + a) p? a? tan?(? a) a + y (a; z)?( 3 + a) + cot a + 3 y (a; z) 5 ;?( + a) (.)

4 where y (a; z) = e? z F? a + ; ;? z = e z F a + ; ; z ; y (a; z) = ze? (.3) z F a + 3; 3; z = ze z F? a + 3; 3;? z and the conuent hypergeometric function is dened by F (a; c; z) = n=0 with (a) n =?(a + n)=?(a); n = 0; ; ; : : :. Another notation found in the literature is D (z) = U(?? ; z): (a) n (c) n z n n! ; (.) There is a relation with the Hermite polynomials. We have U?n? ; z =?n= e? z H n (z= p ); V n + ; z =?n= e z (?i) n H n (iz= p ): (.5) Other special cases are error functions and Fresnel integrals. The Wronskian relation between U(a; z) and V (a; z) reads: U(a; z)v 0 (a; z)? U 0 (a; z)v (a; z) = p =; (.6) which shows that U(a; z) and V (a; z) are independent solutions of (.) for all values of a. Other relations are U(a; z) = [V (a;?z)? sin a V (a; z)] ; cos a?( + a) V (a; z) =?( + a) [sin a U(a; z) + U(a;?z)] : The functions y (a; z) and y (a; z) are the simplest even and odd solutions of (.) and the Wronskian of this pair equals. From a numerical point of view, the pair fy ; y g is not a satisfactory pair [8], because they have almost the same asymptotic behavior at innity. The behavior of U(a; z) and V (a; z) is, for large positive z and z jaj: (.7) U(a; z) = e? z z?a? + O? z? ; V (a; z) = p =e z z a? + O? z? : (.8) Clearly, numerical computations of U(a; z) that are based on the representations in (.) should be done with great care, because of the loss of accuracy if z becomes large. Equation (.) has two turning points at p?a. For real parameters they become important if a is negative, and the asymptotic behavior of the solutions of (.) as a!? changes signicantly if z crosses the turning points. At these points Airy functions are needed. By

5 . Introduction 3 changing the parameters it is not dicult to verify that U(? ; t p ) and V (? ; t p ) satisfy the simple equation d y dt?? t? y = 0; (.9) with turning points at t =. For physical applications, negative a?values are most important (with special case the real Hermite polynomials, see (.5)). For positive a we can use the notation U( ; t p ) and V ( ; t p ), which satisfy the equation d y dt?? t + y = 0: (.0) The purpose of this paper is to give several asymptotic expansions of U(a; z) and V (a; z) that can be used for computing these functions for the case that at least one of the real parameters is large. In [0] an extensive collection of asymptotic expansions for the parabolic cylinder functions as jaj! has been derived from the dierential equation (.). The expansions are valid for complex values of the parameters and are given in terms of elementary functions and Airy functions. In Section we mention several expansions in terms of elementary functions derived by Olver and modify some his results in order to improve the asymptotic properties of the expansions, to enlarge the intervals for using the expansions in numerical algorithms, and to get new recursion relations for the coecients of the expansions. In Section 3 we give similar results for expansions in terms of Airy functions. In Section we give information on how to obtain the modied results by using integral representations of the parabolic cylinder functions. Finally we give numerical tests for three expansions in terms of elementary functions, with a few number of terms in the expansions. Only real parameters are considered in this paper.. Recent literature on numerical algorithms Recent papers on numerical algorithms for the parabolic cylinder functions are given in [] (Fortran; U(n; x) for natural n and positive x) and [3] (Fortran; U(a; x); V (a; x), a integer and half-integer and x 0). The methods are based on backward and forward recursion. [] gives programs in C for U(a; x); V (a; x), and uses representations in terms of the con- uent hypergeometric functions and asymptotic expressions, including those involving Airy functions. [3] gives Fortran programs for computing U(a; z); V (a; z) with real orders and real argument, and for half-integer order and complex argument, The methods are based on recursions, Maclaurin series and asymptotic expansions. They refer also to [3] for the evaluation of U(?ia; ze i ) for real a and z (this function is a solution of the dierential equation y 00 + ( z? a)y = 0). [9] uses series expansions and numerical quadrature; Fortran and C programs are given, and Mathematica cells to make graphical and numerical objects. Maple [5] has algorithms for hypergeometric functions, which can be used in (.) and (.3). Mathematica [] refers for the parabolic cylinder functions to their programs for the hypergeometric functions and the same advice is given in [].

6 . Expansions in terms of elementary functions. The case a 0; z > p?a;?a + z 0. Olver's expansions in terms of elementary functions are all based on the expansion O-(.3) g() e? U? ; t p (t? ) A s (t) ; (.) s as!, uniformly with respect to t [ + "; ); " is a small positive number and is given by = tp t?? ht p i ln + t? : (.) The expansion is valid for complex parameters in large domains of the? and t?planes; details on these domains are not given here. The coecients A s (t) are given by the recursion relation A s+ (t) = da p s (t) + t? dt 8 Z t c s+ 3u + (u? ) 5 A s (u) du; A 0 (t) = ; (.3) where the constants c s can be chosen in combination with the choice of g(). Olver chose the constants such that A s (t) = u s (t) ; (.) (t? ) 3s= where the u s (t) are polynomials in t of degree 3s, (s odd), 3s? (s even, s ). The rst few are u 0 (t) = ; u (t) = t(t? 6) and they satisfy the recurrence relation where ; u (t) =?9t + 9t + 5 ; 5 (t? )u 0 s(t)? 3stu s (t) = r s? (t); (.5) 8r s (t) = (3t + )u s (t)? (s + )tr s? (t) + (t? )r 0 s?(t): The quantity g() in (.) is only available in the form of an asymptotic expansion: where g() h()!? g s ; (.6) s h() =?? e?? ; (.7) g 0 = ; g = ; g 3 =? ; g s = 0 (s = ; ; : : : ); and in general g s = lim t! A s (t): We refer to Olver's equations by writing O-(.3), and so on. (.8)

7 . Expansions in terms of elementary functions 5.. Modied expansions We modify the expansion in (.) by writing U? ; t p = h() e? s () F (t? ) (t); F (t) ; (.9) s where h() and are as before, and t = p t?? : (.0) The coecients s () are polynomials in, 0 () =, and are given by the recursion s+ () =? ( + ) d d s()? Z 0? 0u + 0u + 3 s (u) du: (.) This recursion follows from (.3) by substituting t = ( + )=p ( + ), which is the inverse of the relation in (.0). Explicitly, 0 () = ; () =? ( ); () = 88 ( ); 3 () =? ( ); (.) where is given in (.0). Observe that lim t! (t) = 0 and that all shown coecients s () vanish at innity for s > 0. These properties of s () follow by taking dierent constants c s than Olver did in (.3). In fact we have the relation g s s s () s u s (t) (t? ) 3 s s ; where the rst series appears in (.6). Explicitly, u s (t) =? t? 3 s sx j=0 g s?j j (): (.3) The relation (.3) can easily be veried for the early coecients, but it holds because of the unicity of Poincare-type asymptotic expansions. The expansion in (.9) has several advantages compared with (.). In the recursion relation (.5), both u s and u 0 s occur in the left-hand side. By using computer algebra it is not dicult to compute any number of coecients u s, but the relation for the polynomials s () is simpler than (.5), with this respect. The quantity h() in (.9) is dened as an exact relation, and not, as g() in (.), by an asymptotic expansion (cf. (.6)).

8 6 Most important, the expansion in (.9) has a double asymptotic property: it holds if one or both parameters t and are large, and not only if is large. For the function V (a; z) we have V? ; t p e = p P h()(t? ) (t); P (t) (?) s s() ; (.) s where the s () are the same as in (.9). This expansion is a modication of O-(.9) (see also O-(.)). For the derivatives we can use the identities e? d F dt (t? ) (t) =? (t? ) e? G (t); G (t) d e + P dt (t? ) (t) = + (t? ) e + Q (t); Q (t) The coecients s can be obtained from the relation s() s ; (?) s s() s : (.5) s(t) = s () + ( + )( + ) s? () + 8 ( + ) d s?() ; (.6) d s = 0; ; ; : : :. The rst few are 0(t) = ; (t) = ( ); v (t) =? 88 ( ); 3(t) = ( ): (.7) and This gives the modications (see O-(.3)) U 0? ; t p =? p h()(t? ) e? G (t); G (t) V 0? ; t p = (t? ) e p Q (t); Q (t) h() s() s (.8) (?) s s() : (.9) s Remark.. The functions F (t); G (t); P (t) and Q (t) introduced in the asymptotic representations satisfy the following exact relation: F (t) Q (t) + G (t) P (t) = : (.0) This follows from the Wronskian relation (.6). The relation in (.0) provides a convenient possibility for checking the accuracy in numerical algorithms that use the asymptotic expansions of F (t); G (t); P (t) and Q (t).

9 . Expansions in terms of elementary functions 7. The case a 0; z <? p?a;?a? z 0. For this case we mention the modication of O-(.6). That is, for t + " we have the representations = h() U? ;?t p sin (t? ) e? F (t) +?( + ) cos p h () 3 (.) e P (t) 5 ; where F (t) and P (t) have the expansions given in (.9) and (.), respectively. An expansion for V (? ;?t p ) follows from the second line in (.7), (.9) and (.). A few manipulations give V? ;?t p = h() (t? )?( + ) cos e? F (t)??( + ) sin p h () 3 (.) e P (t) 5 : Expansions for the derivatives follow from the identities in (.5). If a =? =?n? ; n = 0; ; ; : : :, the cosine in (.) vanishes, and, hence, the dominant part vanishes. This is the Hermite case, cf. (.5)..3 The case a 0;? p?a < z < p?a. For negative a and? < t < the expansions are essentially dierent, because now oscillations with respect to t occur. We have (O-(5.) and O-(5.3)) U? ; t p g() (? t " ) # (?) s u cos s (t) (?) s u s+ (t) ; (? t ) 3s+ 3 s+ X X? (? t ) 3s s? sin? with u s (t) dened in (.5) and g() in (.6), and U 0? ; t p " sin p g()(? t ) X? (?) s v s (t) X (? t ) 3s s + cos? (?) s v s+ (t) (? t ) 3s+ 3 s+ (.3) # ; (.) as!, uniformly with respect to jtj? ", where the coecients v s are given by (see O-(.5)) v s (t) = u s (t) + tu s?(t)? r s? (t); (.5)

10 8 and = arccos t? tp? t : (.6) For the function V (a; z) we have (O-(.0) and O(.)) V? ; t p g()?( + )(? t ) " cos + (?) s u s (t) (? t ) 3s s? sin V 0? ; t p p g()(? t )?( + ) " sin + (?) s v s (t) (? t ) 3s s + cos + + (?) s u s+ (t) (? t ) 3s+ 3 s+ (?) s v s+ (t) (? t ) 3s+ 3 s+ # # ; ; (.7) By using the Wronskian relation (.6) it follows that we have the following asymptotic identity (?) s u s (t) (? t ) 3s s (?) s u s+ (t) (? t ) 3s+ 3 s+ (?) s v s (t) (? t ) 3s s + (?) s v s+ (t) (? t ) 3s+ 3 s+?( + ) p g ()? : : : : (.8).3. Modied expansions We can give modied versions based on our earlier modications, with g() replaced with h(), and so on. Because in the present case t belongs to a nite domain, the modied expansions don't have the double asymptotic property. We prefer Olver's versions for this case. This completes the description of U(a; z); U 0 (a; z); V (a; z); V 0 (a; z) in terms of elementary functions for negative values of a.. The case a 0; z 0; a + z 0. For positive values of a the asymptotic behavior is rather simple because no oscillations occur now. The results follow from Olver's expansions O-(.0) and O-(.). The modied forms are where U ; t p e h() e? e = (t + ) e = ef (t); e F (t) (?) s s(e) ; (.9) s h p t + t + ln t p i + + t ; (.30)

11 . Expansions in terms of elementary functions 9 e h() = e??? : (.3) The coecients s in (.9) are the same as in (.9), with replaced by t e = p? : (.3) + t For the derivative we have U 0 ; t p =? p e h()( + t ) e? e Ga e (z); Ga e (z) where s (e) is given in (.6), with e dened in (.3). (?) s s(e) ; (.33) s.5 The case a 0; z 0; a? z 0. Olver's expansion O-(.0) and O(.) cover both cases z 0 and z 0. We have the modied expansions U ;?t p U 0 ;?t p p h()e e =?( + ) ( + t ) ep (t); =? p p?( + ) h()e e ( + t ) e Q (t); (.3) where ep (t) s (e) s ; e Q (t) s(e) s : In Section.. we give details on the derivation of these expansions. Remark.. By using the second relation in (.7), the representations for V (a; z) and V 0 (a; z) for positive a can be obtained from the results for U(a; z) and U 0 (a; z) in (.9), (.33) and (.3). Remark.3. The functions F e (t); G e (t); P e (t) and Q e (t) introduced in (.9), (.3) and (.3) satisfy the following exact relation: ef (t) e Q (t) + e G (t) e P (t) = : (.35) This follows from the Wronskian relation p U(a; z)u 0 (a;?z) + U 0 (a; z)u(a;?z) =??(a + ): See also Remark.. Remark.. The expansions of Sections. and.5 have the double asymptotic property: they are valid if the a + jzj!. In Sections. and.5 we consider the cases z 0 and z 0, respectively, as two separate cases. Olver's corresponding expansions O-(.0) and O-(.) cover both cases and are valid for? < t <. As always, in Olver's expansions large values of are needed, whatever the size of t.

12 0 In Figure we show the domains in the t; a?plane where the various expansions of U(a; z) of this section are valid. (.33) Ο (.0) a (.8) Ο (.0) t (.) Ο (.6) (.3) Ο (5.) (.9) Ο (.3) (3.6) Ο (9.7) (3.) Ο (8.) Figure. Regions for the modied asymptotic expansions of U(a; z) given in Section and the Airy-type expansions of Section 3 (which are valid in much larger domains than those indicated by the arrows). 3. Expansions in terms of Airy functions The Airy-type expansions are needed if z runs through an interval containing one of the turning points p?a, that is, t =. 3. The case a 0; z 0. We summarize the basic results O-(8.), O-(8.5) and O-(.) (see also O-(.)): U? ; t p " # = 3 g()() Ai( 3 ) A () + Ai0 ( 3 ) B 8 () ; (3.) 3 U 0? ; t p " # = () 3 g() Ai( 3 ) C () () + Ai 0 ( 3 ) D () ; (3.) 3 V? ; t p " # = 3 g()()?( + Bi( 3 ) A () + Bi0 ( 3 ) B ) 8 () ; (3.3) 3 V 0? ; t p " # = () 3 g() Bi( 3 ) ()?( + C ) () + Bi 0 ( 3 ) D () : (3.) 3 The coecient functions A (); B (); C () and D () have the following asymptotic expansions A () a s () s ; B () b s () ; (3.5) s

13 3. Expansions in terms of Airy functions C () c s () s ; D () d s () ; (3.6) s as!, uniformly with respect to t? +, where is a small xed positive number. The quantity is dened by 3 (?) 3 = (t);? < t ; ( 0) 3 3 = (t); t; ( 0) (3.7) where ; follow from (.6), (.), respectively, and () = : (3.8) t? The function (t) is real for t >? and analytic at t =. We can invert (t) into t(), and obtain t = +?=3? 0?= : : : : The function g() has the expansion given in (.6) and the coecients a s (); b s () are given by a s () = Xs m=0 m? 3 m A s?m (t) where A s (t) are used in (.), 0 = and s+ p X bs () =? m=0 m? 3 m A s?m+ (t); (3.9) m = (m + ) (m + 3) (6m? ) m! () m ; m =? 6m + 6m? a m: (3.0) A recursion for m reads: (6m + 5)(6m + 3)(6m + ) m+ = m ; m = 0; ; ; : : : : (m + )(m + ) The numbers m ; m occur in the asymptotic expansions of the Airy functions, and the relations in (3.9) follow from solving (3.) and (3.3) for A () and B (), expanding the Airy functions (assuming that is bounded away from 0) and by using (.) and a similar result for V (a; z) (O-(.6) and O-(.)). For negative values of (i.e.,? < t < ) we can use (O-(3.)) where P a s () = (?) s s m=0 m(?)? 3 m As?m e (t); p P?bs () = (?) s? s+ m=0 m(?)? 3 m As?m+ e (t); ea s (t) = u s(t) (? t ) 3 s : (3.)

14 The functions C () and D () of (3.) and (3.) are given by C () = ()A () + A 0 () + B (); D () = A () + ()B () + B 0 () : The coecients c s () and d s () in (3.6) are given by where (3.) c s () = ()a s () + a 0 s() + b s (); d s () = a s () + ()b s? () + b 0 s?(); (3.3) () = 0 () () =? t[()]6 ; (3.) with () given in (3.8). Explicitly, X s+ p c s () =? m=0 m? 3 m B s?m+ () d s () =? Xs m=0 m? 3 m B s?m (); (3.5) where B s () = v s (t)=(t? ) 3 s, with v s (t) dened in (.5). needed for negative values of,i.e., if? < t < ; see (3.). Other versions of (3.5) are 3. The case a 0; z 0. Near the other turning point t =? we can use the representations (O-(9.7)) U? ;?t p Ai 0 ( 3 ) B () 8 3 ) = 3 g()() + cos " ( sin Ai( 3 )A ()+ ( )# Bi( 3 )A () + Bi0 ( 3 ) 8 3 B () ; (3.6) as!, uniformly with respect to t? +, where is a small xed positive number. Expansions for V (a; z) follow from (3.) and (3.6) and the second relation in (.7). Results for the derivatives of U(a; z) and V (a; z) follow easily from the earlier results. 3.3 Modied forms of Olver's Airy-type expansions Modied versions of the Airy-type expansions (3.) { (3.) can also be given. In the case of the expansions in terms of elementary functions our main motivation for introducing modi- ed expansions was the double asymptotic property of these expansions. In the case of the Airy-type expansions the interesting domains for the parameter t, from a numerical point of view, are nite domains that contain the turning points. So, considering the expansions given so far, there is no need to have Airy-type expansions with the double asymptotic property; if remains nite and jtj we can use the expansions in terms of elementary functions. However, we have another interest in modied expansions in the case of Airy-type expansions. We explain this by rst discussing a few properties of the coecient functions A (); B (); C () and D (). By using the Wronskian relation (.6) we can verify the relation A ()D ()? B ()C () =?( + ) p g () ; (3.7)

15 3. Expansions in terms of Airy functions 3 where g() is dened by means of an asymptotic expansion given in (.6). dierential equation (O-(7.)) By using the d W d = + () W; (3.8) where () = 5 6? (3t + ) h i (t? ) = 3 =3? ?=3? ?= : : : ; 85 (3.9) we can derive the following system of equations for the functions A (); B (): A 00 + B 0 + B? ()A = 0; B 00 + A 0? ()B = 0; (3.0) where primes denote dierentiation with respect to. A Wronskian for this system follows by eliminating the terms with (). This gives A 0 A + AB 00? A 00 B? B 0 B? B = 0; which can be integrated: A () + A ()B 0 ()? A 0 ()B ()? B () =?( + ) p g () ; (3.) where the quantity on the right-hand side follows from (3.7) and (3.). It has the expansion? : : : ; (3.) as follows from O-(.) and O-(5.). As mentioned before, the interesting domain of the Airy-type expansions given in this section is the domain that contains the turning point t =, or = 0. The representations of the coecients of the expansions given in (3.9) cannot be used in numerical algorithms when jj is small, unless we expand all relevant coecients in powers of. This is one way how to handle this problem numerically; see [8]. In that paper we have discussed another method that is based on a system like (3.0), with applications to Bessel functions. In that method the functions A () and B () are expanded in powers of, for suciently small values of jj, say jj, and the Maclaurin coecients are computed from (3.0) by recursion. A normalizing relation (the analogue of (3.)) plays a crucial role in that algorithm. The method works quite well for relatively small values of a parameter (the order of the Bessel functions) that is the analogue of. When we want to use this algorithm for the present case only large values of are allowed because the function g() that is used in (3.){(3.) and (3.) is only dened for large values of. For this reason we give the modied versions of Olver's Airy-type expansions. The modied versions are more complicated than the Olver's expansions, because the analogues of the series in (3.5) and (3.6) are in powers of?, and not in powers of?. Hence, when we use these series for numerical computations we need more coecients in the modied expansions,

16 which is certainly not desirable from a numerical point of view, given the complexity of the coecients in Airy-type expansions. However, in the algorithm based on Maclaurin expansions of the analogues of the coecient functions A (); B (); C () and D () this point is of minor concern. The modied expansions are the following: U V? ; t p " # =?( + )() Ai( 3 ) F () + Ai0 ( 3 ) G 3 h() 8 () ; (3.3) 3? ; t p " # = () Bi( 3 ) F () + Bi0 ( 3 ) G 3 h() 8 () : (3.) 3 The functions F () and G () have the following asymptotic expansions F () f s () s ; G () g s () : (3.5) s The quantity and the functions () and h() are as in Section 3.. Comparing (3.3), (3.) with (3.), (3.3) we conclude that F () = H()A (); G () = H()B (); H() = p g()h()?( + : (3.6) ) The function H() can be expanded (see O-(.), O-(.7), O-(6.) and (.6)) H() + s= (?) s s ( ) ; (3.7) s where s are the coecients in the gamma function expansions?( + z) p e?z z z X The rst few coecients are 0 = ; =? ; = 5 ; 3 = : s z ; s?( + z) ez z?z X p (?) s s z : (3.8) s The second expansion in (3.8) can be used in (3.6) to nd relations between the coecients a s () and b s () of (3.5) and of f s () and g s () of (3.5). That is, f 0 () = ; f () = ; f () = a () ; f 3() = a ()? ; g 0 () = b 0 (); g () = b 0(); g () = b () b 0(); g 3 () = b ()? b 0(): The coecients f s (); g s () can also be expressed in terms of the coecients s () that are introduced in (.9) by deriving the analogues of (3.9).

17 . Expansions from integral representations 5 The system of equations (3.0) remains the same: F 00 + G 0 + G? ()F = 0; G 00 + F 0? ()G = 0; (3.9) and the Wronskian relation becomes F () + F ()G 0 ()? F 0 ()G ()? G () = p h ()?( + ) : (3.30) The right-hand side has the expansion (see (3.8) and (.7)) P (?)s s =( ) s. Observe that ((3.30) is an exact relation, whereas (3.) contains the function g(), of which only an asymptotic expansion is available. 3. Numerical aspects of the Airy-type expansions In [8], (Section ), we solved the system (3.9) (for the case of Bessel functions) by substituting Maclaurin series of F (); G() and (). That is, we wrote F () = n=0 c n () n ; G() = n=0 d n () n ; () = where the coecients n can be considered as known (see (3.9)), and we substituted the expansions in (3.9). This gives for n = 0; ; ; : : : the recursion relations (n + )(n + )c n+ + (n + )d n = n ; n = (n + )(n + )d n+ + (n + )c n+ = n ; n = n=0 nx nx k=0 k=0 n n ; kc n?k ; kd n?k : (3.3) If is large, the recursion relations cannot be solved in forward direction, because of numerical instabilities. For the Bessel function case we have shown that we can solve the system by iteration and backward recursion. The relation in (3.30) can be used for normalization of the coecients in the backward recursion scheme. For details we refer to [8]. The present case is identical to the case of the Bessel functions; only the function () is dierent, and instead of in (3.3) we had the order of the Bessel functions.. Expansions from integral representations The expansions developed by Olver, of which some are given in the previous sections, are all valid if jaj is large. For several cases we gave modied expansions that hold if at least one of the two parameters a; z is large and we have indicated the relations between Olver's expansions and the new expansions. The modied expansions have in fact a double asymptotic property. Initially, we derived these expansions by using integral representations of the parabolic cylinder functions, and later we found the relations with Olver's expansions. In this section we explain how some of the modied expansions can be obtained from the integrals that dene U(a; z) and V (a; z). Again we only consider real values of the parameters.

18 6. Expansions in terms of elementary functions by using integrals.. The case a 0; z 0; a + z 0. We start with the well-known integral representation U(a; z) = e? z?(a + ) Z which we write in the form where U(a; z) = za+ e? z Z?(a + ) 0 0 w a? e? w?zw dw; a >? (.) w? e?z (w) dw; (.) (w) = w + w? ln w; = a z : (.3) The positive saddle point w 0 of the integrand in (.3) is computed from giving d(w) dw w 0 = = w + w? w = 0; (.) h p +? i : (.5) We consider z as the large parameter. When is bounded away from 0 we can use Laplace's method (see [] or []). When a and z are such that! 0 Laplace's method cannot be applied. However, we can use a method given in [5] that allows small values of. To obtain a standard form for this Laplace-type integral, we transform w! t (see [6]) by writing (w) = t? ln t + A; (.6) where A does not depend on t or w, and we prescribe that w = 0 should correspond with t = 0 and w = w 0 with t =, the saddle point in the t?plane. This gives where U(a; z) = za+ e? z?az Z ( + )?(a + ) 0 f(t) = ( + ) r t w r dw dt = ( + ) w t t a? e?zt f(t) dt; (.7) t? w + w? : (.8) By normalizing with the quantity ( + ) we obtain f() =, as can be veried from (.8) and a limiting process (using l'h^opital's rule). The quantity A is given by A = w 0 + w 0? ln w 0? + ln : (.9)

19 . Expansions from integral representations 7 A rst uniform expansion can be obtained by writing f(t) = n=0 a n ()(t? ) n : (.0) Details on the computation of a n () will be given in the appendix. By substituting (.0) into (.7) we obtain U(a; z) e? z?az z a+ ( + ) n=0 a n ()P n (a)z?n ; (.) where Z P n (a) = za+n+?(a + ) t a? e?zt (t? ) n dt; n = 0; ; ; : : : : (.) 0 The P n (a) are polynomials in a. They follow the recursion relation P n+ (a) = (n + )P n(a) + anp n? (a); n = 0; ; ; : : : ; with initial values P 0 (a) = ; P (a) = : We can obtain a second expansion U(a; z) e? z?az z a+ ( + ) k=0 f k () ; (.3) zk with the property that in the series the parameters and z are separated, by introducing a sequence of functions ff k g with f 0 (t) = f(t) and by dening f k+ (t) = p t d pt f k (t)? f k () ; k = 0; ; ; : : : : (.) dt t? The coecients f k () can be expressed in terms of the coecients a n () dened in (.0). To verify this, we write f k (t) = n=0 a (k) n ()(t? ) n ; (.5) and by substituting this in (.) it follows that a (k+) n () = (n + )a (k) () + n+ n + a (k) n+ (); k 0; n 0: (.6) Hence, the coecients f k () of (.3) are given by f k () = a (k) 0 (); k 0: (.7)

20 8 We have f 0 () = ; f () = [a () + a ()] ; f () = f 3 () = 8 a () + a 3 () + 3a () ; 03 a 6 () + 0 a 5 () + 6a () + 5a 3 () : (.8) Explicitly, f 0 () = ; f () =? (0? 0? ); f () = 5 (660? ); f 3 () =? 3 70 (77006? ? ? ); (.9) where = z (? ) + p ; = a + z = z p a + z (z + p a + z ) : (.0) We observe that f k () is a polynomial of degree k in multiplied with k. If a and z are positive then [0; ]. Furthermore, the sequence f k =z k g is an asymptotic scale when one or both parameters a and z are large. The expansion in (.3) is valid for z! and holds uniformly for a 0. It has a double asymptotic property in the sense that it is also valid as a!, uniformly with respect to z 0. As follows from the coecients given in (.9) and relations to be given later, we can indeed let z! 0 in the expansion. The expansion in (.3) can be obtained by using an integration by parts procedure. We give a few steps in this method. Consider the integral F a (z) = Z?(a + ) t a? e?zt f(t) dt; (.) 0 We have (with = a=z ) F a (z) = f() R?(a + ) t a? e?zt R dt + 0?(a + ) 0 = z?a? f()? = z?a? f() + z?(a + ) R 0 t [f(t)? f()] t? t a? e?zt [f(t)? f()] dt de?z (t? ln t) z?(a + ) R 0 t a? e?zt f (t) dt; where f is given in (.) with f 0 = f. Repeating this procedure we obtain (.3). More details on this method and proofs of the asymptotic nature of the expansions (.) and (.3) can be found in our earlier papers. We concentrate on expansion (.3) because (.) cannot be compared with Olver's expansions.

21 . Expansions from integral representations 9 To compare (.3) with Olver's expansion (.6), we write a = ; z = p t: (.) Then the parameters and dened in (.0) become + = e + ; = = t p + t t p + t (t + p + t ) ; (.3) where e is given in (.3). After a few manipulations we write (.3) in the form (cf.(.9)) where U ; t p e h() e? e = (t + ) e = ef (z); e F (z) k=0 e (?) k k () ; (.) k h p t + t + ln t p i + + t ; (.5) and Explicitly, h() = e??? (.6) e k () = (?)k (t ) k f k(): (.7) e 0 () = ; e () =? (0? 0? ); (? ) e () = (660 88? ); (? )3 e 3 () = 580 (77006? ? ? ); (.8) where is given in (.3). Comparing (.) with (.9) we obtain e k () = k (e); k 0, because = + e... The case a 0; z 0; a? z 0. To derive the rst expansion in (.3) we use the contour integral U(a;?z) = p e z?(a + ) H a(z); H a (z) =?(a + i ) Z C e zs+ s s?a? ds; (.9)

22 0 where C is a vertical line in the half plane <s > 0. This integral can be transformed into a standard form that involves the same mapping as in the previous subsection. We rst write (by transforming via s = zw) H a (z) = z?a?(a + ) Z e z (w+ w )?a? w dw = z?a?(a + ) Z e z (w) i C i C dw p w ; (.30) where (w) is dened in (.3). By using the transformation given in (.6) it follows that H a (z) = z?a?(a + Z )eaz e zt t?a? f(t) dt: (.3) i C The integration by parts method used for (.) gives the expansion (see [5]) za e Az H a (z) (a + z ) k=0 (?) k f k() ; (.3) zk where the f k () are the same as in (.3); see also (.8). This gives the rst expansion of (.3). Remark.. The rst result in (.3) can also be obtained by using (.) with z < 0. The integral for U(a;?z) can be written as in (.), now with (w) = w? w? ln ; = a=z. In this case the relevant saddle point at w 0 = ( + p + )= is always inside the interval [; ) and the standard method of Laplace can be used. The same expansion will be obtained with the same structure and coecients as in (.3), because of the unicity of Poincare-type asymptotic expansions. See also Section.. where Laplace's method will be used for an integral that denes V (a; z)...3 The case a 0; z > p?a;?a + z 0. Olver's starting point (.) can also be obtained from an integral. Observe that (.) is not valid for a?. We take as integral (see [], p. 687, 9.5.) U(?a; z) =?( + a) Z e? z e zs? s s?a? ds; (.33) i where is a contour that encircles the negative s?axis in positive direction. Using a transformation we can write this in the form (cf. (.)) where U(?a; z) =?( + a) Z z?a e? z e (w) w? dw; (.3) i (w) = w? w? ln w; = a z : (.35) The relevant saddle point is now given by w 0 =? p? ; 0 < < : (.36)

23 . Expansions from integral representations When! 0 the standard saddle point method is not applicable, and we can again use the methods of our earlier papers ([5] and [6]) and transform (w) = t? ln t + A; (.37) where the points at? in the w? and t? plane should correspond, and w = w 0 with t =. We obtain U(?a; z) =?( + a) Z (? ) i z?a e? z +z A e zt t?a? f(t) dt; (.38) where is a contour that encircles the negative t?axis in positive direction and r r f(t) = (? ) t dw w dt = (? ) w t? t w? w? : (.39) Expanding f(t) as in (.0), and computing f k () as in the procedure that yields the relations in (.8), we nd that the same values f k () as in (.9), up to a factor(?) k and a dierent value of and. By using the integration by parts method for contour integrals [5], that earlier produced (.3), we obtain the result U(?a; z) za e Az? z (z? a) k=0 where the rst f k () are given in (.9) with z (? = ) + p ; = z? a This expansion can be written in the form (.9). (?) k f k() ; (.0) zk = z p z? a+(z + p z? a) : (.).. The case a 0; z <? p?a;?a? z 0. We use the relation (see (.7)) U(?a;?z) = sin a U(?a; z) +?( V (?a; z); (.)? a) and use the result of U(?a; z) given in (.0) or the form (.9). An expansion for V (?a; z) in (.) can be obtained from the integral (see [9]) V (a; z) = e? z Z[ e? s +zs s a? ds; (.3) where and are two horizontal lines, in the upper half plane =s > 0 and in the lower half plane =s < 0; the integration is from <s =? to <s = +. (Observe that when we integrate on in the other direction (from <s = + to <s = ) the contour [ can be deformed into of (.33), and the integral denes U(a; z), up to a factor.) We can apply Laplace's method to obtain the expansion given in (.) (see Remark.).

24 . The singular points of the mapping (.6) The mapping dened in (.6) is singular at the saddle point w? =? p + + : (.) If = 0 then w? =? and the corresponding t?value is?. For large values of we have the estimate: t(w? )?0:785? 0:356 p : (.5) This estimate is obtained as follows. The value t? = t(w? ) is implicitly dened by equation (.6) with w = w?. This gives p t?? ln t?? + ln =? + i + ln ( + p + ) = i? p + + O(? ) ; (.6) as!. The numerical solution of the equation s? ln s? = i is given by s = 0:785 e i. This gives the leading term in (.6). The other term follows by a further simple step..3 Expansions in terms of Airy functions All results for the modied Airy-type expansions given in Section 3.3 can be obtained by using certain loop integrals. The integrals in (.33) and (.3) can be used for obtaining (3.3) and (3.), respectively. The method is based on replacing (w) in (.3) by a cubic polynomial, in order to take into account the inuence of both saddle points of (w). This method is rst described in [6]; see also [] and []. 5. Numerical verifications We verify several asymptotic expansions by computing the error in the Wronskian relation for the series in the asymptotic expansions. Consider Olver's expansions of Section.3 for the oscillatory region? < t < with negative a. We verify the relation in (.8). Denote the left-hand side of the rst line in (.8) by W (; t). Then we dene as the error in the expansions (; t) := W (; t)? ? Taking 3 terms in the series of (.3), (.) and (.7), we obtain for several values of and t the results given in Table 5.. We clearly see the loss of accuracy when t is close to. Exactly the same results are obtained for negative values of t in this interval. (5.)

25 5. Numerical verications t.00.3e-09.78e-3.3e-7.3e-.78e-5.0.6e-09.63e-3.e-7.6e-.63e-5.0.8e-0.0e-3.33e-8.8e-.0e e-08.39e-.65e-7.6e-0.39e-.0.88e-08.e-.36e-6.89e-0.e e-07.3e-0.e-5.5e-9.3e-.60.0e-06.99e-0.7e-.0e-8.99e e-05.3e-08.e-3.5e-7.3e e-03.50e-07.8e-.0e-5.50e e-00.e-0.e-09.0e-.5e-6 Table 5.. Relative accuracy (; t) dened in (5.) for the asymptotic series of Section.3. Next we consider the modied expansions of Section.. Denote the left-hand side of (.0) by W (; t). Then we dene as the error in the expansions (; t) := W (; t)? (5.) When we use the series in (.9), (.), (.8) and (.9) with 5 terms, we obtain the results given in Table 5.. We observe that the accuracy improves as or t increase. This shows the double asymptotic poperty of the modied expansions of Section t..5e-0.8e-05.7e-0.8e-3.3e-7..39e-0.79e-08.3e-.3e-6.78e e-06.9e-09.3e-.78e-8.9e-..56e-07.3e-0.3e-5.55e-9.3e-.5.7e-08.7e-.9e-6.70e-0.7e-3.0.0e-0.5e-.3e-9.0e-.5e-6.5.e-.5e-6.87e-.e-.5e e-6.8e-0.8e-5.e-8.8e e-0.8e-.8e-9.0e-3.8e e-5.73e-9.e-33.30e-37.73e- Table 5.. Relative accuracy (; t) dened in (5.) for the asymptotic series of Section.. Finally we consider the expansions of Sections. and.5. Let the left-hand side of (.35) be denoted by W (; t). Then we dene as the error in the expansions (; t) := W (; t)? (5.3)

26 When we use the series in (.9), (.33) and (.3) with 5 terms, we obtain the results of Table 5.3. We again observe that the accuracy improves as or t increase. This shows the double asymptotic poperty of the modied expansions of Sections. and t.00.3e-09.78e-3.3e-7.3e-.78e-5.5.e-09.8e-3.7e-8.e-.8e e-.e-.9e-9.6e-3.e e-.e-.e-9.58e-3.e-6.0.7e-.65e-5.e-9.7e-3.65e-7.5.9e-3.70e-7.e-.9e-5.70e-9.0.0e-3.8e-7.8e-.0e-5.8e-9.5.3e-.e-7.8e-.3e-6.e e-7.e-0.8e-5.5e-9.e e-0.38e-.6e-9.6e-3.38e-36 Table 5.3. Relative accuracy (; t) dened in (5.3) for the asymptotic series of Sections. and Concluding remarks As mentioned in Section. of the Introduction, several sources for numerical algorithms for evaluating parabolic cylinder functions are available in the literature, but not so many algorithms make use of asymptotic expansions. The paper [0] is a rich source for asymptotic expansions, for all combinations of real and complex parameters, where always jaj has to be large. There are no published algorithms that make use of Olver's expansions, although very ecient algorithms can be designed by using the variety of these expansions; [3] is the only reference we found in which Olver's expansions are used for numerical computations. We started our eorts in making algorithms for the case of real parameters. We selected appropriate expansions from Olver's paper and for some cases we modied Olver's expansions in order to get expansions having a double asymptotic property. A serious point is making ecient use of the powerful Airy-type expansions that are valid near the turning points of the dierential equation (and in much larger intervals and domains of the complex plane). In particular, constructing reliable software for all possible combinations of the complex parameters a and z is a challengeing problem. A point of research interest is also the construction of error bounds for Olver's expansions and the modied expansions. Olver's paper is written before he developed the construction of bounds for the remainders, which he based on methods for dierential equations, and which are available now in his book []. 7. Appendix: Computing the coefficients f k () of (.3) We give the details on the computation of the coecients f k () that are used in (.3). The rst step is to obtain coecients d k in the expansion w = d 0 + d (t? ) + d (t? ) + : : : ; (7.) where d 0 = w 0. From (.6) we obtain dw dt = w t t? w + w? : (7.)

27 7. Appendix: Computing the coecients f k () of (.3) 5 Substituting (7.) we obtain d = w 0 ( + w 0 ) ; (7.3) where the saddle point w 0 is dened in (.5). From the conditions on the mapping (.6) it follows that d > 0. Higher order coecients d k can be obtained from the rst ones by recursion. When we have determined the coecients in (7.) we can use (.8) to obtain the coecients a n () of (.0). For computing in this way a set of coecients f k (), say f 0 (); : : : ; f 5 (), we need more than 35 coecients d k in (7.). Just taking the square root in (7.3) gives for higher coecients d k very complicated expressions, and even by using computer algebra programs, as Maple, we need suitable methods in computing the coecients. The computation of the coecients d k ; a n () and f k () is done with a new parameter [0; ) which is dened by = tan : (7.) We also write = cos ; (7.5) which is introduced earlier in (.0) and (.3). Then w 0 =? (? ) ; =? (? ) ; d =? : (7.6) In particular the expressions for w 0 and d are quite convenient, because we can proceed without square roots in the computations. Higher coecients d k can be obtained by using (7.). The rst relation f 0 () = a (0) 0 () = easily follows from (.3), (.8), (7.) and (7.6): r f 0 () = ( + ) d = : w 0 Then using (.8) we obtain a 0 () = ; a () =? cos ( + c) 6(c + )c ; a () = cos (0c + 0c c + c + 3) (c + ) c ; where c = p = cos. Using the scheme leading to (.7) one obtains the coecients f k(). The rst few coecients are given in (.9). We observe that f k () is a polynomial of degree k in multiplied with k. If a and z are positive then [0; ]. It follows that the sequence f k =z k g is an asymptotic scale when one or both parameters a and z are large, and, hence, that ff k ()=z k g of (.3) is an asymptotic scale when one or both parameters a and z are large. Because of the relation in (.7) and e k () = k (e), higher coecients f k () can also be obtained from the recursion relation (.), which is obtained by using the dierential equation of the parabolic cylinder functions.

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29 References 7 9. W.J. Thompson (997), Atlas for Computing Mathematical Functions: An illustrated guide for practitioners, with programs in Fortran 90 and Mathematica. There is also an edition with programs in C. John Wiley and Sons, New York. 0. H.F. Weber (869), Uber die Integration der partiellen u=@x u=@y + k u = 0, Math. Annal.,, {36.. S. Wolfram (996), The Mathematica book, 3rd Ed. Wolfram Media, Cambridge University Press, Champaign (IL).. R. Wong (989), Asymptotic approximations of integrals, Academic Press, New York. 3. S. Zhang and J. Jin (996), Computation of Special Functions. John Wiley and Sons, New York. Diskette with software included.