Analytic formulas for the evaluation of the Pearcey integral

Transcription

1 Aalytic formulas for the evaluatio of the Pearcey itegral José L. López 1 ad Pedro J. Pagola 1 arxiv: v1 [math.na] 1 Ja Dpto. de Igeiería Matemática e Iformática, Uiversidad Pública de Navarra jl.lopez@uavarra.es, pedro.pagola@uavarra.es Abstract We ca fid i the literature several coverget ad/or asymptotic expasios of the Pearcey itegral P x, y i differet regios of the complex variables x ad y, but they do ot cover the whole complex x ad y plaes. The purpose of this paper is to complete this aalysis givig ew coverget ad/or asymptotic expasios that, together the kow oes, let the evaluatio of the Pearcey itegral i a large regio of the complex x ad y plaes. The accuracy of the approximatios derived i this paper is illustrated some umerical experimets. Moreover, the expasios derived here are simpler compared other kow expasios, as they are derived from a simple maipulatio of the itegral defiitio of P x, y. 010 AMS Mathematics Subject Classificatio: 33E0; 1A60. Keywords & Phrases: Pearcey itegral. Coverget ad asymptotic expasios. Watso lemma. 1 Itroductio The Pearcey itegral is the secod caoical diffractio itegrals [1, Sec. 36.1]. Apart from their mathematical importace i the uiform asymptotic approximatio of oscillatory itegrals [8], the caoical diffractio itegrals have physical applicatios i the descriptio of surface gravity waves [7], [13], bifurcatio sets, optics, quatum mechaics ad acoustics see [1, Sec. 36.1] ad refereces there i. The Cusp catastrophe or Pearcey itegral [1, p.777, eq ] is the itegral: P x, y := e it +xt +yt dt. 1 This itegral was first evaluated umerically by usig quadrature formulas i [1] i the cotext of the ivestigatio of the electromagetic field ear a cusp. I [1, Chap. 36] 1

2 we ca fid may properties such as symmetries, ilustrarive pictures, bifurcatio sets, scalig relatios, zeros, coverget series expasios, differetial equatios ad leadigorder asymptotic approximatios amog others. But we caot fid may details about aalytic approximatio formulas asymptotic expasios i particular or umerical evaluatio techiques. The itegral 1 exists oly for 0 arg x π ad real y. As it is idicated i [10], after a rotatio of the itegratio path through a agle of π/8 that removes the rapidly oscillatory term e it, the Pearcey itegral may be writte i the form P x, y = e iπ/8 P xe iπ/, ye iπ/8, P x, y := 0 e t xt cosytdt = 1 e t +iyt e xt dt. This itegral is absolutely coverget for all complex values of x ad y ad represets the aalytic cotiuatio of the Pearcey itegral P x, y to all complex values of x ad y [10]. Therefore, it is more coveiet to work the represetatio of the Pearcey itegral. We summarize below the more elemetary aalytic expasios coverget or asymptotic of P x, y that we ca fid i the literature ad give refereces to other more elaborated expasios. A coverget series expasio of P x, y may be foud i [1, p. 787, eqs ad 36.8.]: P x, y = Γ a x, y, 3 a 0 x, y = 1, a 1 x, y = y ad, for =, 3,,..., a x, y = 1 [y a 1x, y + x a x, y]. Aother coverget expasio of P x, y may be obtaied after a expasio of the cosie term i ad iterchage of sum ad itegral [11]: P x := P x, y = 1 +3/ Γ + 1 y! P x, x, y C, U + 1, 1; x 1 Γ + 1 M + 1, 1; x xγ + 3 M + 3, 3; x if Rx 0, if Rx < 0, 5 where Ma, b; z ad Ua, b; z are cofluet hypergeometric fuctios [9, Chap. 13] Apart from coverget expasios, we ca also fid i the literature several asymptotic expasios of P x, y. I [] we ca fid a asymptotic expasio of the Pearcey itegral whe x, y are ear the caustic 8x 3 7y = 0 that remais valid as x. The expasio is give i terms of Airy fuctios ad its derivatives ad the coefficiets are computed recursively. We refer the reader to [] for further details.

3 A exhaustive asymptotic aalysis of this itegral ca be foud i [10]. I particular, a complete asymptotic expasio is give i [10] by usig asymptotic techiques for itegrals applied to the itegral. The asymptotic aalysis of this itegral for large x is divided i two regios: arg x < π ad arg x > π. I the first regio we fid that [10] P x, y 1 x e y /x 1 Γ + 1/ M ; 1/; y. 6!x x The asymptotic expasio i the secod regio is a little bit more cumbersome, we refer to [10] for details. Aother coverget ad asymptotic expasio of P x, y for large x, derived from a differetial equatio satisfied by P x, y, may be foud i [5]. A complete asymptotic expasio for large y is derived i [6], where the asymptotic aalysis is divided i three differet regios: arg y π, π arg y < π ad π < arg y π. We give details here for the last regio ad refer the reader to [6] for further details. Whe π < arg y π we have that: 8 where P x, y π 1 e3 /3e iπ/3y/3 /3xy/3e iπ/3 +x /6+iπ/6 3 5/6 y 1/3 A x := m= +1/ b,m,k x := m b,m,k xc +m k x, xk m k/3 1 m k!m k! m! A x, 7 y/3 ad c + x = x 3 c +1x /3 3 c x, = 0, 1,,... /3 c 0 x = 1 ad c 1 x = x 3. 1/3 I [11] we ca fid the hyperasymptotic evaluatio of the Pearcey itegral for real values of x ad y usig Hadamard expasios. The Pearcey itegral is writte i terms of a ifiite series whose terms are Hadamard series. The terms of these Hadamard series are icomplete gamma fuctios. The aalytic expressio is sophisticated ad we refer the reader to [11] for details. I this paper we exploit the followig simple idea: the itegrad i is a expoetial. The, we maipulate the fuctio i the expoet i order to factorize the expoetial ad replace oe of the factors by its Taylor expasio at the origi. We cosider three differet possibilities. I the followig sectio we derive a complete coverget ad asymptotic expasio of P x, y for small x. I Sectio 3 we derive a complete asymptotic expasio of P x, y for large x. I Sectio we derive a complete asymptotic expasio of P x, y for large x ad y. Sectio 5 cotais some umerical experimets ad a few remarks. Through the paper we use the pricipal argumet arg z π, π] for ay complex umber z. 3

4 A coverget ad asymptotic expasio for small x Because P x, y = P x, y, out loss of geerality, we may restrict ourselves to Ry 0. A coverget expasio of P x, y that is also ad asymptotic expasio whe x 0 ca be obtaied by cosiderig the Taylor expasio of the expoetial at the origi: e xt = 1 x k t k k! + r xt, 8 where r t is the Taylor remaider. Replacig this expasio i the itegrad of the right had side of formula ad iterchagig sum ad itegral we fid: P k y := 1 P x, y = 1 x k P k y + R x, y, 9 k! t k e t +iyt dt, R x, y := 1 r xt e t +iyt dt. After some straightforward maipulatios, we obtai a explicit formula for the coefficiets P k y i 9 k = 0, 1,,...: P k y := k 1 + k Γ 1F 3 ; 1,, 3 y ; y 3 + k 3 + k 8 Γ 1F 3 ; 3, 5, 6 y ;. Itegratig by parts we derive the followig five terms recurrece for the coefficiets P k y: y k + 1k + 1 P k+ y = Pk+1 y P k y+k+1k+7/p k+ y, k = 0, 1,,... 8 P 0 y = 1 1 Γ 0F ; 1, 3 ; y y Γ 0F ; 5, 3 ; y, 56 P 1 y = 1 3 Γ 0F ; 1, 1 ; y y Γ 0F ; 3, 3 ; y, 56 P y = Γ 1F 3 ; 1, 1, 3 ; y y Γ 1F 3 ; 3, 5, 3 ; y 56 P 3 y = Γ 1F 3 ; 1, 1, 3 ; y y Γ 0F ; 3, 5, 3 ; y 56 Therefore, we have obtaied that, for ay complex x ad y, P x, y = 1,. P k y x k + R x, y. 10 k!

5 This expasio is coverget for ay value of x ad y ad is a asymptotic expasio of P x, y as x 0. Moreover, usig the Lagrage formula for the remaider i 8 we have r xt = r xt < e ξ Rx! xt 0 < ξ < t t! x if Rx 0, t e Rx t x if Rx < 0,! I the first case Rx 0, we immediately coclude that R x, y x! P i Iy. I secod case Rx < 0, we have R x, y = 1 r xt e t +iyt dt 1 x! t e Rx t e t Iyt dt. Writig the domiat term e t of itegrad i the form e t e t we ca rewrite the above itegral as 1 x! gt e t Iyt t dt gt := e t + Rx t. The fuctio gt attais it absolute maximum at t = Rx ad the gt g Rx = e R x. Therefore, we coclude that, for Rx < 0, R x, y x! R e x P i Iy. 3 A asymptotic expasio for large x Usig the Cauchy s residue theorem we fid that we ca replace the itegratio path, i by ay straight Γ := {u = se iσ : < s < }, σ < π/8: Defie the parameter α := u x 1/ e iθ/ we fid P x, y = 1 e iσ e iσ e u xu +iyu du. y x ad θ := arg x. After the chage of variable t = u x = P x, y = 1 e iσ+θ/ e t +iαt t /x e dt. 11 x e iσ+θ/ 5

6 From here, a asymptotic expasio of the itegral 11 for large x ca be obtaied cosiderig the Taylor expasio of the expoetial at the origi: e t /x = 1 1 k t k k!x k + r t x, 1 where r t is the Taylor remaider. From the Lagrage s formula for the remaider we have: t r = r x s e i σ = t!x e ξ ei σ 0 < ξ < s. The r t x < t. 13! x Replacig expasio 1 i the above itegral ad iterchagig sum ad itegral we fid y π e 1 x P x, y = 1 k x k! x Q kα + R k x, y, Q k α := e y x e iσ+θ/ π e iσ+θ/ t k e t +iαt dt, R x, y := e y x e iσ+θ/ t r e t +iαt dt. π e x iσ+θ/ 1 These itegrals exist whe arg x + σ < π/. For ay value of x such that arg x < 3π/, we ca chose σ σ < π/8 such that these itegrals are fiite. After some maipulatios ad usig the itegral represetatio of Hermite polyomials [eq , [3]] we fid H x = i e x π Q k α = k H k α e t t e ixt dt, k = 0, 1,,... Moreover, itroducig 13 i 1 we fid R x, y cos arg y arg x σ cosσ+arg x y x e! x cos +1/ σ + arg x H y siσ + arg y i. x cosσ + arg x It is clear that R x, y = Ox as x uiformly i y bouded y/ x ad arg x < 3π/. Therefore, we have obtaied that P x, y = π e y x x 1 1 k y k!x H k k + R x, y x, 15 6

7 is a asymptotic expasio of P x, y whe x uiformly i y bouded y/ x ad arg x < 3π/. This expasio is just the expasio 3. derived i [10]. The derivatio of 3. is much more cumbersome as it is obtaied from a cotour itegral represetatio of the Pearcey itegral i terms of a parabolic cylider fuctio. A error boud for the remaider is ot provided there. O the other had, a asymptotic expasio for arg x < π/ is give i [10]. A asymptotic expasio for large x ad y Defie the parameter γ := y. We ca write the Pearcey itegral i the form x P x, y = 1 e xft t dt, 16 phase fuctio ft := iγt t. The uique saddle poit of this phase fuctio is the poit t 0 = iγ. From [] we kow that the asymptotic aalysis of this itegral does ot require the applicatio of the stadard saddle poit method, but just a chage of variable. Usig the Cauchy s residue theorem we fid that we ca replace the itegratio path, i 16 by ay straight Γ := {iγ + t; t = re iγ : < r < }, σ < π/8: x P x, y = e γ e xt e iγ+t dt. 17 We write the above itegral i the form P x, y = e α x α e iσ x 6α t e e iσ htdt, 18 ht := e iγ3 t iγt 3 t. From here, a asymptotic expasio of the itegral 17 for large x ca be derived from Watso s lemma. From the Taylor expasio of the expoetial at the origi we have that ht = A γt, A γ := / k/3 j=0 iγ 3 k 3j iγ j 1 k. k!j! k 3j! Replacig this expasio i the itegral 18 ad iterchagig sum ad itegral we fid that /x y /16x P x, y e y A γp γ, x, P γ, x := t e x 6γ t dt, t = re iγ, < r <. 7

8 This itegral exists whe argx 6t = re iγ : < r < + σ < π/. For ay value of x ad y such that argx 3y /x < 3π/, we ca chose σ σ < π/8 such that this itegral is fiite: Γ + 1/ P γ, x = x 6γ. +1/ Therefore, we have obtaied that, whe x, /x y /16x P x, y e y x x 3 3y y x A Γ + 1/, 19 x x 3 3y uiformly i y bouded y/x ad argx 3y /x < 3π/. 5 Fial remarks ad umerical experimets Figure 1 summarizes the regios for x, y covered by the above metioed algorithms except [11] that is more sophisticated ad valid oly for real x, y ad by the three algorithms derived below i this paper. y x Small Moderate Large Small 3 Sec. [] 6 Sec. 3 Moderate Sec. 1 [] 6 Sec. 3 Large 7 7 Sec. Figure 1:I blue, the regios of x, y for which there are aalytic formulas for the computatio of P x, y available i the literature ad detailed i the itroductio sectio. The regios for which the ew formulas derived here are useful are dashed. The curve i the large x ad y box idicates the caustic. The expasio give i Sectio is ew ad valid for ay complex values of x ad y. The expasio give i Sectio 3 is just the oe give i eq. 3. of [10], although derived i a more straightforward way that permits a accurate error boud for the remaider. It is valid for arg x < 3π/ ad ay complex y. The expasio give i Sectio is also ew ad permits a wider regio for x, y tha the expasio give i []. The expasio 8

9 give i [] is restricted to the viciity of the caustic 8x 3 7y = 0, whereas the oe derived i Sectio above is valid for argx 3y /x < 3π/. Fially, we illustrate the accuracy of the above expasios. I the followig tables we show the relative error obtaied these algorithms for several values of x, y ad differet orders of the approximatio. As we do ot have at our disposal the exact value of the Pearcey itegral, we have take the umerical itegratio obtaied the program Mathematica workig precisio= 00 as the exact value of P x, y. x, y , i e iπ/, /5, i /10 i/8, i/0, i Table 1: Relative error for several values of x, y ad the umber of terms of the approximatio give by 10. x, y , i e iπ/, i, 1 i , i, i i, i Table : Relative error for several values of x, y ad the umber of terms of the approximatio give by 15. x, y , 3i e 5iπ/8, i, 0e iπ/ , 0i i, Table 3: Relative error for several values of x, y ad the umber of terms of the approximatio give by 19. 9

10 6 Ackowledgmets This research was supported by the Spaish Miistry of Ecoomía y Competitividad, project MTM The Uiversidad Pública de Navarra is ackowledged by its fiacial support. Refereces [1] M. V. Berry ad C. J. Howls, Itegrals coalescig saddles, i: NIST Hadbook of Mathematical Fuctios, Cambridge Uiversity Press, Cambridge, 010, pp Chapter 36. [] D. Kamiski, Asymptotic expasio of the Pearcey itegral ear the caustic, J. SIAM J. Math. Aal., , [3] T. H. Koorwider, R. Koekoek, R. F. Swarttouw, R. Wog, Orthogoal Polyomials, i: NIST Hadbook of Mathematical Fuctios, Cambridge Uiv. Press, 010, pp Chapter 18. [] J. L. López ad N. Temme, Uiform approximatios of Beroulli ad Euler polyomials i terms of hyperbolic fuctios, Stud. Appl. Math., , [5] J. L. López ad P. Pagola, Coverget ad asymptotic expasios of the Pearcey itegral. J. Math. Aal. Appl., 30, o. 1, [6] J. L. López ad P. Pagola, The Pearcey itegral i the highly oscillatory regio. Submitted. [7] Lord Kelvi, Deep water ship-waves, Phil. Mag., , [8] A. B. Olde Daalhuis, O the asymptotics for late coefficiets i uiform asymptotic expasios of itegrals coalescig saddles Meth. Appl. Aal., 7 o. 000, [9] A. B. Olde Daalhuis, Cofluet Hypergeometric Fuctios, i: NIST Hadbook of Mathematical Fuctios, Cambridge Uiversity Press, Cambridge, 010, pp Chapter 13. [10] R. B. Paris, The asymptotic behaviour of Pearcey s itegral for complex variables, Proc. Roy. Soc. Lodo Ser. A., 3 o , [11] R. B. Paris ad D. Kamiski, Hyperasymptotic evaluatio of the Pearcey itegral via Hadamard expasios, J. Comput. Appl. Math., , [1] T. Pearcey, The structure of a electromagetic field i the eighbourhood of a cusp of a caustic, Phi. Mag., , [13] F. Ursell, Itegrals a large parameter: several early coicidet saddle poits, Proc. Camb. Phil. Soc., 7 197, [1] R. Wog, Asymptotic approximatios of itegrals, Academic Press, New York,