B OLLETTINO U NIONE M ATEMATICA I TALIANA Fausto Segala On the asymptotic expansion of the Airy function Bollettino dell’Unione Matematica Italiana, Serie 8, Vol. 2-B (1999), n.3, p. 707–714. Unione Matematica Italiana <http://www.bdim.eu/item?id=BUMI_1999_8_2B_3_707_0> L’utilizzo e la stampa di questo documento digitale è consentito liberamente per motivi di ricerca e studio. Non è consentito l’utilizzo dello stesso per motivi commerciali. Tutte le copie di questo documento devono riportare questo avvertimento. Articolo digitalizzato nel quadro del programma bdim (Biblioteca Digitale Italiana di Matematica) SIMAI & UMI http://www.bdim.eu/ Bollettino dell’Unione Matematica Italiana, Unione Matematica Italiana, 1999. Bollettino U. M. I. (8) 2-B (1999), 707-714 On the Asymptotic Expansion of the Airy Function. FAUSTO SEGALA Sunto. – Si prova una nuova formula di rappresentazione per la famosa funzione di Airy. Ne viene data applicazione per la determinazione di certi bounds significativi per la funzione stessa. 1. – Introduction. The famous Airy function is defined by 1iQ Ai (r) 4 1 se (z 3 /3 2 rz) 2 pi 2iQ dz and solves the Airy equation u 9 4 ru . The Airy function (or the Airy integral) has many and very deep applications to mathematics, physics and engineering. In 1963 Copson [1] gave the following representation of the Airy function for r D 0 : (1.1) Ai (r) 4 1 2 kp r 21 /4 e 22 /3 r 3 /2 q(r) with Q (1.2) q(r) 4 2 se kp 0 2t 2 cos g r 23 /4 t 3 3 h dt . Since q(r) K 1 per r K Q , by (1.1) it follows (1.3) Ai (r) A 1 2 kp 3 /2 r 21 /4 e 22 /3 r , rKQ . In this note we give a new representation of q(r). Precisely we prove that 708 FAUSTO SEGALA q(r) can be written in the form Q (1.4) q(r) 4 2 se kp 2t 2 0 k g h t r 3 /4 dt where k(t) has the graph in the picture below: Fig. 1. As a consequence, q(r) has the graph in the next picture: Fig. 2. From (1.4) it follows in an absolutely elementary way, some fundamental well known properties of the Airy function for positive argument. In fact, by (1.4) we have Ai (r) D 0 on [ 0 , Q[ and (since Ai9 (r) 4 r Ai (r) ) , Ai9 (r) D 0 on ]0 , Q[. Therefore Ai8 (r) is increasing and since Ai8 (r) K 0 when r K Q , we obtain Ai8 (r) E 0 on [ 0 , Q[. In conclusion, Ai (r) is positive and decreasing on [ 0 , Q[. 709 ON THE ASYMPTOTIC EXPANSION OF THE AIRY FUNCTION In the section 2 we will prove the formula (1.4) and in the section 3 we will make use of (1.4) in order to obtain bounds for the Airy function and its derivative on the real positive axis. For small values of the argument, our bounds are better than well known classical asymptotic bounds of the Airy function (see for example Olver [2]). The proof of (1.4) is an application of the Riemann method of steepest descent. 2. – The proof. By the change of variable z 4 kr w , we can write 1iQ Ai (r) 4 (2.1) kr se r 3 /2 (w 3 /3 2 w) 2 pi 2iQ dw . We observe that 1 is a critical point of f(w) 4 w 3 /3 2 w and f( 1 ) 4 22 /3 . We prove there exists a curve w 4 w(t) such that f(w(t) ) 4 22 /3 2 t 2 , that is w 323w4223t 2. (2.2) By setting w 4 x 1 iy , we obtain (2.3) x 3 2 3 xy 2 2 3 x 4 22 2 3 t 2 , (2.4) y( 3 x 2 2 y 2 2 3 ) 4 0 and therefore t 24 (2.5) 2 3 (4x 323x21) . For t D 0 , the equation (2.5) has one root x(t) D 1 . We take y(t) 4 k3 x(t)2 2 3 , w(t) 4 x(t) 6 iy(t). Therefore if we integrate along w 4 w(t), from (2.2) we have Ai (r) 4 r 1 /2 p r 21 /4 p Q e 22 /3 r 3 /2 se 2r 3 /2 t 2 y 8 (t) dt 4 0 Q e 22 /3 r 3 /2 where k(t) 4 y 8 (t). se 0 2t 2 y8 g h t r 3 /4 dt 4 r 21 /4 Q 2 2 kp kp e 22 /3 r 3 /2 se 0 2t 2 k g h t r 3 /4 dt 710 FAUSTO SEGALA From (2.3),(2.4) it follows (2.6) t x 84 2 4x 21 y 82 4 (2.7) , 3 xx 8 y 84 9 x 2 x 82 y2 4 y 3 x 2 x 82 x 221 , . By inserting (2.5) in the first equation (2.6), we get 8x 326x22 x 82 4 (2.8) . 3( 4 x 2 2 1 )2 Now, by (2.7) and (2.8), one obtains y 82 4 x2 2 8x 326x22 x 21 2 (4x 21) 2 2x 2(4x 214x11) 4 2 2 ( 4 x 2 1 ) (x 1 1 ) 4 2x 2 ( 2 x 2 1 )2 (x 1 1 ) . Put F(x) 4 k2 x ( 2 x 2 1 ) kx 1 1 4 k2x 3 ( 4 x 2 3 x 1 1 )1 /2 . By a simple calculation we have F 8 (x) 4 2 2x 21x12 1 k2 ( 2 x 2 1 )2 (x 1 1 )3 /2 E0 , xD1 . Finally, by taking into account that y 8 (t) 4 F(x(t) ) , y 9 (t) 4 F 8 (x(t) ) x 8 (t) , we conclude that y 9 is negative and therefore k 4 y 8 is decreasing, k( 0 ) 4 F(x( 0 ) ) 4 F( 1 ) 4 1 . 3. – Bounds of the Airy function. From (1.4) it follows Q (3.1) q(r) 4 2 kp 3 21 /6 r 1 /4 se 0 2t 2 c g h t r 3 /4 with (3.2) c(t) 4 31 /6 k(t) t 1 /3 . t 21 /3 dt ON THE ASYMPTOTIC EXPANSION OF THE AIRY FUNCTION 711 In section 2 we proved that k(t) 4 F(x(t) ) with (3.3) F(x) 4 k2 x . 3 ( 4 x 2 3 x 1 1 )1 /2 By (2.5), we have (3.4) t4 k2 ( 4 x 3 2 3 x 2 1 )1 /2 k3 and therefore, after some easy calculations, by using (3.2), (3.3) and (3.4) we get c(t) 4 G(x(t) ) (3.5) with (3.6) G(x) 4 x(x 3 2 3 /4 x 2 1 /4 )1 /6 . (x 3 2 3 /4 x 1 1 /4 )1 /2 If we put L 4 max G , we can write [ 1 , Q[ Q r 21 /4 q(r) G L 2 3 21 /6 kp se 2t 2 21 /3 t dt 4 L 0 321 /6 G kp gh 1 3 4 2 kp L Ai ( 0 ) and since L 4 1.067 , R , we conclude with the estimate 1 2 kp r 21 /4 q(r) G ( 1.067 , R) Ai ( 0 ) , rD0 . On the other hand, from (3.1) (or from (1.1)) it follows lim rK0 1 2 kp r 21 /4 q(r) 4 Ai ( 0 ) . Hence we have (3.7) Ai ( 0 ) G sup [ 0 , Q[ 1 r y 2 kp 21 /4 z q(r) G ( 1.067 , R) Ai ( 0 ) . Now we examine Ai8 (r). From the calculations developed in section 2, we have (3.8) Ai8 (r) 4 2 r 1 /4 2 kp e 22 /3 r 3 /2 p(r) 712 FAUSTO SEGALA with Q (3.9) 2 se kp p(r) 4 2t 2 h 0 g h t dt r 3 /4 and d (xy)(t) 4 H(x(t) ) , (3.10) h(t) 4 (3.11) H(x) 4 k2 dt 2x 221 . kx 1 1( 2 x 2 1 ) By setting (3.12) R(x) 4 H(x) k2 x , k2 m(t) 4 3 kx(t) 2 1 t 1 /3 1 /6 , from (3.9) we obtain Q (3.13) g g hh o g h s g g hh yo g h z g g hh p(r) 4 Q 2 k2 se kp 2t 2 Q kp 2Q3 se kp 2t 2 r r kp se 2t 2 Q 2 k2 se kp 0 dt1 3/4 2t 2 x 3 /4 r 3/4 dt 4 t 2 e 2t R x r 0 t x t r 3 /4 r 3 /4 x 3/4 t 21 dt4 r 3/4 21 ( 31 /6 /k2)(t/r 3 /4 )1 /3 dt1 2Q31/6 kp Q r se 21/4 0 By (3.12) and (3.4) one has (3.14) r 3 /4 dt 1 g g hh t t Q 2 k2 g h o g g hh R x R x r kp r 3 /4 0 t R x 0 t Q 21 /4 2t 2 g g hh R x 0 1 /6 se t R x 0 2 k2 2 k2 m(t) 4 Q(x(t) ) 2t 2 t 1 /3 dt 4 g g hh g h R x t r 3/4 m t r 3/4 t 1/3dt . ON THE ASYMPTOTIC EXPANSION OF THE AIRY FUNCTION 713 with Q(x) 4 kx 2 1 . 3 (x 2 3 /4 x 2 1 /4 )1 /6 We have the following two estimates about R and Q : sup R(x) 4 lim R(x) 4 1 , (3.15) xKQ [ 1 , Q[ sup Q(x) 4 lim Q(x) 4 1 . (3.16) xKQ [ 1 , Q[ We give the proof of (3.16). Q(x 2 ) 4 (x 2 1 ) /(x 6 2 3 /4 x 2 2 1 /4 )1 /6 E 1 on [ 1 , Q[ is equivalent to f (x) D 0 on [ 0 , Q[, where f (x) 4 24 x 5 2 60 x 4 1 80 x 3 2 63 x 2 1 24 x 2 5 . On the other hand, f( 1 ) 4 0 , f 8 ( 1 ) 4 18 , f 9 ( 1 ) 4 114 , fR(x) 4 480( 3 x 2 2 3 x 1 1 ) D 0 , (x R . From (3.13), (3.15), (3.16) we obtain p(r) G k2 1 2 Q 31 /6 kp Q r 21 /4 se 2t 2 1 /3 t dt 4 0 k2 1 2 kp321 /3 G( 1 /3 ) r 21 /4 4 k2 1 2 kpNAi8 ( 0 ) Nr 21 /4 , that is (3.17) r 1 /4 p(r) k2 p G 1 k2 p r 1 /4 1 NAi8 ( 0 ) N . Finally (3.17) and (3.8) imply the estimate (3.18) 3 /2 NAi8 (r) N G NAi8 ( 0 ) N( 1 1 ar 1 /4 ) e 22 /3 r , rF0 , with a 4 1 /NAi8 ( 0 ) Nk2 p . The estimate (3.18) is, in some sense, the best possible in virtue of (3.15) and (3.16). We may compare (3.18) with the known formula (3.19) NAi8 (r) N G 1 2 kp g r 1 /4 1 which is the best possible for r K Q . 7 48 r 3 /2 h e 22 /3 r 3 /2 714 FAUSTO SEGALA REFERENCES [1] E. T. COPSON, On the asymptotic expansion of Airy’s integral, Proc. Glasgow Math. Ass., 6 (1963), 113-115. [2] F. W. J. OLVER, Asymptotics and Special Functions, Academic Press, New York and London (1974). Università degli Studi di Ferrara, Dipartimento di Matematica, Via Macchiavelli 35, Ferrara Pervenuta in Redazione l’8 maggio 1998