Definite integrals and operational methods

Applied Mathematics and Computation 219 (2012) 3017–3021

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Definite integrals and operational methods D. Babusci a, G. Dattoli b,c, G.H.E. Duchamp c, K. Górska d,e,⇑, K.A. Penson f a

INFN – Laboratori Nazionali di Frascati, v.le E. Fermi, 40, I00044 Frascati (Roma), Italy ENEA – Centro Ricerche Frascati, v.le E. Fermi, 45, I00044 Frascati (Roma), Italy Université Paris XIII, LIPN, Institut Galilée, CNRS UMR 7030, 99 Av. J.-B. Clement, F 93430 Villetaneuse, France d Instituto de Fı´sica, Universidade de São Paulo, P.O. Box 66318, BR 05315-970 São Paulo, SP, Brazil e ´ ski Institute of Nuclear Physics, Polish Academy of Sciences, ul.Eljasza-Radzikowskiego 152, PL 31342 Kraków, Poland H. Niewodniczan f Laboratoire de Physique Théorique de la Matière Condensée (LPTMC), Université Pierre et Marie Curie, CNRS UMR 7600, Tour 13 - 5ième ét., Boîte Courrier 121, 4 place Jussieu, F 75252 Paris Cedex 05, France b c

a r t i c l e

i n f o

Keywords: Integrals Ramanujan’s master theorem Bessel functions Struve functions Hermite polynomials

a b s t r a c t An operational method, already employed to formulate a generalization of the Ramanujan master theorem, is applied to the evaluation of integrals of various types. This technique provides a very flexible and powerful tool yielding new results encompassing different aspects of the special function theory. Crown Copyright Ó 2012 Published by Elsevier Inc. All rights reserved.

The use of operational methods is, sometimes, very much useful to ‘‘guess’’ specific theorems in various fields of analysis. The drawback of such an approach is that any of its consequences should be validated by a more rigorous procedure. This is indeed the case of the Ramanujan master theorem [1,2], according to which if a function f ðxÞ can be expanded around x ¼ 0 in a power series of the form

f ðxÞ ¼

1 X uðnÞ n¼0

n!

ðxÞn ;

ð1Þ

with uð0Þ ¼ f ð0Þ – 0, then

Z

1

xm1 f ðxÞ dx ¼ CðmÞ uðmÞ;

ð2Þ

0

where CðzÞ is the Euler’s Gamma function. The rigorous proof of this identity has been obtained in Refs. [3,4]. The simpler but heuristic proof is given in [5,6]. It is based on setting

f ðxÞ ¼

1 X ^c n n¼0

n!

ðxÞn uð0Þ ¼ e^c x uð0Þ;

ð3Þ

with ^c being an umbral operator such that [7]

^cn uð0Þ ¼ uðnÞ:

ð4Þ

The theorem has been exploited in [5,6] in a generalized version to obtain, in a very simple way, old and new formulas for integrals and successive derivatives of Bessel functions. The noticeable feature emerging from such a procedure is that ⇑ Corresponding author at: H. Niewodniczan´ski Institute of Nuclear Physics, Polish Academy of Sciences, ul.Eljasza-Radzikowskiego 152, PL 31342 Kraków, Poland. E-mail addresses: [email protected] (D. Babusci), [email protected] (G. Dattoli), [email protected] (G.H.E. Duchamp), kasia_gorska@ o2.pl (K. Górska), [email protected] (K.A. Penson). 0096-3003/$ - see front matter Crown Copyright Ó 2012 Published by Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.09.029

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cylindrical Bessel functions can be formally treated as functions involving the operator ^c. For example, the n-th order cylindrical Bessel function can be written as 2

J n ð2xÞ ¼ ð^c xÞn e^c x uð0Þ;

uðnÞ ¼

1

Cð1 þ nÞ

ð5Þ

:

According to the above restyling, and by taking the freedom of treating ^c as an ordinary constant, an interesting plethora of new results concerning integrals of Bessel functions and their derivatives can be obtained. Just to give an example of how the method works, we consider the integral

Bm ða; bÞ ¼

Z

1

2

x J2m ðaxÞeibx dx;

0

ða > 0; b > 0Þ:

ð6Þ

The use of Eq. (5) allows to treat this integral as an elementary integral of exponential type and indeed, under the hypothesis a2 < 4b, we find

Bm ða; bÞ ¼

 mþ1   1 a2m i a2 ; bm i 2 2 b 4b

ð7Þ

where we have introduced the function

pffiffiffiffi 1  x i X Cðm þ k þ 1Þ xk p 12m 2x h  x  x e Im1 bm ðxÞ ¼ þ Imþ1 ; ¼ 2 2 2 2 2 Cð2m þ k þ 1Þ k! k¼0 where Il ðzÞ is the modified Bessel function [10]. In the case

m ¼ 0, Eq. (7) gives

  i a2 exp i B0 ða; bÞ ¼ ; 2b 4b

ð8Þ

in agreement with Eqs. 6.728.3 and 6.728.4 of [8], and Eq. 2.12.18.7 of [9]. We consider now the Struve function [10]

Hm ðxÞ ¼

 2kþmþ1 ð1Þk 2x   ; C k þ 32 C k þ m þ 32 k¼0

1 X



ð9Þ

^ m as that can be rewritten with the umbral operator h

Hm ðxÞ ¼

 2

^m x exp h uð0Þ; 2

 x mþ1 2

ð10Þ

with

Cðn þ 1Þ   : C n þ 32 C n þ m þ 32

uðnÞ ¼ ðh^m Þn uð0Þ ¼ 

ð11Þ

By applying our method, and using the Euler’s reflection formula for Gamma functions, we find:

Z

1

Hm ðbxÞ dx ¼

0

    1 C 1 þ 2m C  2m 1 1m 1þm ¼   ; b C 2 C 2 b tan p2m

ð2 < Rem < 0; b > 0Þ;

ð12Þ

that is formula 6.811.1 of [8]. Further application of Eq. (10) furnishes

Z

1

xðmþ1Þ Hm ðxÞ dx ¼

1

pffiffiffiffi

p ^12

2m

hm uð0Þ ¼

p 2m Cð1 þ mÞ

ð13Þ

(see formula 6.813.2 of [8]). It is evident that the method can easily be applied also to more complicated integrals involving the Struve functions. As a further evidence of the usefulness of the method, we apply it to the search of close form of the following generating function

Gðx; tj mÞ ¼

1 n X t n¼0

n!

J m n ð2xÞ:

ð14Þ

With the help of Eq. (5) it can be written as

Gðx; tj mÞ ¼ exp½ð^cxÞm t  ^c x2  uð0Þ:

ð15Þ

If we introduce the higher, (m P 2), order Hermite polynomials [11]

HðmÞ n ðu; v Þ ¼ n!

½n=m X k¼0

unmk v k ; ðn  mkÞ! k!

ð16Þ

D. Babusci et al. / Applied Mathematics and Computation 219 (2012) 3017–3021

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their generating function is [11–13]: 1 X zn n¼0

n!

HnðmÞ ðu; v Þ ¼ expðuz þ v zm Þ:

n n=2 (Note that Hð2Þ Hn n ðx; yÞ ¼ ðiÞ y (15) in the operator form

Gðx; tj mÞ ¼

1 X ^c n

n!

n¼0





ix pffiffi 2 y

ð17Þ

, where Hn ðzÞ are conventional Hermite polynomials [10]). This allows us to recast Eq.

HnðmÞ ðx2 ; x m tÞuð0Þ:

ð18Þ

Furthermore, as a consequence of the definitions (5) and (16), it is possible to show that ðmÞ  2

Gðx; tj mÞ ¼ H C 0

x ; ðxÞm t ;

ð19Þ

where ðmÞ H C n ðx; yÞ

¼

1 X

ð1Þk ðmÞ H ðx; yÞ; k! ðn þ kÞ! k k¼0

ð20Þ

are the so called Hermite-based Tricomi functions [14]. The correctness of the above identity has been thoroughly checked numerically. We stress that its derivation with conventional means is much more involved. In the same spirit, we will use operational methods to evaluate various families of integrals. In particular, we consider the integrals of the type

IðxÞ ¼

Z

1

f ðx gðtÞÞ dt:

ð21Þ

½gðtÞx @x f ðxÞ ¼ f ½x gðtÞ

ð22Þ

1

The identity [13]

can be used to write

IðxÞ ¼

Z

1

½gðtÞx @x f ðxÞ dt;

ð23Þ

1

and, by assuming that the integral (a ¼ constant)

FðaÞ ¼

Z

1

½gðtÞa dt

1

exists, one can express IðxÞ as

IðxÞ ¼ b F ðx @ x Þf ðxÞ:

ð24Þ

If the function f ðxÞ admits a power series expansion

f ðxÞ ¼

1 X

aðkÞxmkþp ;

ð25Þ

k¼0

as a consequence of the identity

b F ðx @ x Þxn ¼ FðnÞxn ;

ð26Þ

we get

IðxÞ ¼

1 X

aðkÞFðmk þ pÞxmkþp :

ð27Þ

k¼0 2

To clarify the content of this result, we choose the case gðtÞ ¼ et , f ðxÞ ¼ J n ðxÞ. By using Eq. (26) one has

Z

1

1   pffiffiffiffi X 2 J n xet dt ¼ p

1

k¼0

 x2kþn ð1Þk 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; k! ðk þ nÞ! 2k þ n

n > 0:

ð28Þ

The above series has been proved to be convergent and the correctness of the proposed procedure has been checked numerically. 2 As a further example we consider the evaluation of the integral (21) for gðtÞ ¼ ð1 þ t 2 Þ1 ; f ðxÞ ¼ x2 ex . Since

Z

1

1

ð1 þ t 2 Þa dt ¼

  pffiffiffiffi C a  12 p CðaÞ

ð29Þ

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we get

Z

1

h i x2 exp  ð1þt 2 Þ2

1

ð1 þ t 2 Þ2

dt ¼

    1 pffiffiffiffi X ðx2 Þk C 2k þ 32 p 3 5 3 ; ; 1; ; x2 ; p ¼ 2F2 4 4 2 k! 2 2k þ 2 k¼0

ð30Þ

where p F q ða1 ; . . . ; ap ; b1 ; . . . ; bq ; yÞ is the generalized hypergeometric function, see [9,10]. Let us now consider the following Laplace-type transform integral

LðxÞ ¼

Z

1

et gðx tÞ dt;

ð31Þ

0

also known as the Borel transform. The use of Eq. (22) leads to the operatorial identity

LðxÞ ¼ Cðx @ x þ 1Þ gðxÞ

ð32Þ

from which, under the assumption that LðxÞ can be expanded as in Eq. (1), as a consequence of Eq. (26), one obtains

gðxÞ ¼

1 X uðkÞ k¼0

ðk!Þ2

ðxÞk :

ð33Þ

See of [15, Chapter 2] for the use of the operator Cðx @ x þ 1Þ. The use of Borel transform is of noticeable importance in the derivation of Quantum Chromodynamics (QCD) sum rules [16]. For a general introduction see [17]. We will just touch on this topic and leave further developments for a forthcoming paper. Just to quote a few examples we note that the polynomials given in Eq. (16) are the Borel transform of their hybrid counterpart [18]

e ðmÞ ðx; yÞ ¼ H n

½n=m X

xnmk yk

k¼0

k! ½ðn  mkÞ!2

;

m P 2;

ð34Þ

e ðmÞ if we choose gðx tÞ ¼ H n ðx t; yÞ and treat y as a parameter. They are called hybrid because they have properties in between those of Hermite and Laguerre polynomials. It is worth stressing that the Borel transform with respect to the y variable P½n=m xnmk yk ðmÞ e ðmÞ [namely choosing gðytÞ ¼ H n ðx; ytÞ], is en ðx; yÞ ¼ k¼0 ½ðnmkÞ!2 corresponding to a family of truncated polynomials. See [20,21] for discussions of related polynomial families. Furthermore, for gðxÞ ¼ ð1 þ xm Þ1 ; jxj < 1, we find that the associated ðmÞ transform is the pseudo-trigonometric function [19] c0 ðxÞ, where ðmÞ

ck ðxÞ ¼

1 X ð1Þr r¼0

xmrþk ; ðmr þ kÞ!

0 6 k < m:

ð35Þ

The following relationship can be viewed as an extension of the Borel transform

Ia;b ðxÞ ¼

Z

1

ua1 ð1  uÞb1 f ðu xÞ du ðRea; Reb > 0Þ:

ð36Þ

0

By using Eq. (22), this integral can be written in terms of the Beta function as follows

Ia;b ðxÞ ¼ Bða þ x @ x ; bÞf ðxÞ; i.e., in terms of the Gamma function

Ia;b ðxÞ ¼

CðbÞ Cða þ x @ x Þ f ðxÞ: Cð a þ b þ x @ x Þ

ð37Þ

If the function f ðxÞ satisfies Eq. (1), one has 1 X WðnÞ

ðxÞn ;

ð38Þ

WðnÞ ¼ Bða þ n; bÞ uðnÞ:

ð39Þ

Ia;b ðxÞ ¼

n¼0

n!

where

In conclusion, in this paper we have provided a few elements proving the usefulness of formal procedures to get results encompassing various aspects of operators and of special functions. Other examples could be discussed to corroborate the validity of the method. We believe, however, that the main plot of the whole technique can be quite well understood via the examples discussed here. In closing, we remark that the results we have obtained are at the level of sound conjectures. The relevant rigorous proof of them is out of the scope of the present article. There are similar techniques that kept being revived, see for example [22] and references therein.

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Acknowledgements One of us (G.D.) recognizes the warm hospitality and the financial support of the University of Paris XIII, whose stimulating atmosphere provided the necessary conditions for the elaboration of the ideas leading to this paper. K. G. thanks Fundação de Amparo á Pesquisa do Estado de São Paulo (FAPESP, Brazil) under Program No. 2010/15698-5. The authors acknowledge support from Agence Nationale de la Recherche (Paris, France) under Program PHYSCOMB No. ANR-08BLAN-243-2. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]

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