On a generalized secant integral

Radiation Physics and Chemistry 59 (2000) 281±285

www.elsevier.com/locate/radphyschem

Technical note

On a generalized secant integral L.A.-M. Hanna, S.L. Kalla* Department of Mathematics and Computer Science, Faculty of Science, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait Received 18 May 1999; accepted 2 February 2000

Abstract In this paper, we introduce a generalization of the secant integral in the following form …c Ia …c, b, l† ˆ ba e ÿ bsec j …sec j †a …tan j † 2lÿ1 dj 0

with ar0, b > 0, 0 < cR p2 , and l > 0: This new generalization of the secant integral might have some applications in radiation ®eld problems of di€erent source-shield con®gurations. We have obtained two series representations for Ia …c, b, l† in terms of incomplete gamma function. The complete generalized secant integral is expressed in terms of generalized hypergeometric functions. Some recurrence relations are given. 7 2000 Elsevier Science Ltd. All rights reserved.

1. Introduction

integral, de®ned as

For an isotropic line source of uniform strength s with a slab shield, the detector response at a particular point P is given as (Michieli, 1998)

Ia …c, b† ˆ ba

(

Rˆ

3 X  rs I0 …c, mt† ‡ Ai …1 ÿ b† ÿaÿi Ia‡i c, mt…1 4ph iˆ0

ÿ b†



)

…1†

where r is the appropriate ¯ux to detector response conversion factor, and Ia …c, b† is the generalized secant

* Corresponding author. Tel.: +965-532-7901; fax: +965481-7201. E-mail addresses: [email protected] (L.A.-M. Hanna), [email protected] (S.L. Kalla).

…c 0

e ÿb secj …sec j †a dj

…2†

with ar0, b > 0, and 0 < cR p2 : This generalized secant integral has been considered recently by Michieli (1998), and he has obtained some series representations. Hungerford (1962) has considered similar integrals, but only for integral values of a. For a ˆ 0, Eq. (2) is known as Sievert integral. For a ˆ 0 and c ˆ p2 , Eq. (2) reduces to a special case of Bickley function (Abramowitz and Stegun, 1972; Sievert, 1921, 1930; Wood, 1982). In this paper, we introduce another generalization of the secant integral in the following form Ia …c, b, l† ˆ ba

…c 0

e ÿb sec j …sec j †a …tan j † 2lÿ1 dj

with ar0, b > 0, 0 < cR p2 , and l > 0:

0969-806X/00/$ - see front matter 7 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 9 - 8 0 6 X ( 0 0 ) 0 0 2 6 4 - 4

…3†

282

L.A.-M. Hanna, S.L. Kalla / Radiation Physics and Chemistry 59 (2000) 281±285

For l ˆ 12 , Eq. (3) reduces to Eq. (2), that is Ia …c, ˆ Ia …c, b†: This new generalization of the secant integral might have some applications in radiation ®eld problems of di€erent source-shield con®gurations. We have obtained two series representations for Ia …c, b, l† in terms of the incomplete gamma function. The complete generalized secant integral is expressed in terms of generalized hypergeometric functions. Some recurrence relations are given. b, 12 †

On expanding …x 2 ÿ 1†lÿ1 in powers of Eq. (6), by using Eqs. (4) and (7)

1 x

we get from

ÿ
I^a ce , c, b, l ˆ ba

… sec c sec ce

e ÿbx x aÿ1

  2iÿ2l‡2 1 X 1 ki dx, x iˆ0

…8†

p rc > ce > 0, 2 which after integration leads to ÿ
I^a ce , c, b, l

2. Generalized secant integral Using the substitution sec j ˆ x, Eq. (3) can be written as Ia …c, b, l† ˆ ba

… sec c 1

1 X  ÿ
ki b 2iÿ2l‡2 G a ÿ 2i ‡ 2l ÿ 2, b sec ce iˆ0

e ÿbx x aÿ1 …x 2 ÿ 1 †

lÿ1

dx:

1 X ki …cos c† 2iÿ2l‡2 E2iÿ2l‡3 …b sec c†

ÿ G…a ÿ 2i ‡ 2l ÿ 2, b sec c† :

…4†

For a ˆ 0 and l ˆ 12 , Eq. (4) becomes the Sievert integral, I0 …c, b, 12 † ˆ I0 …c, b†: Using exponential integrals, „ 1 the En …y† ˆ ynÿ1 y e ÿt t ÿn dt, which satisfy the „ x relation En …y† ˆ ynÿ1 G…1„ ÿ n, y†, where g…a, x† ˆ 0 e ÿt taÿ1 dt, 1 and G…a, x† ˆ x e ÿt taÿ1 dt, a > 0 are the incomplete and the complementary incomplete gamma functions, satisfying g…a, x† ‡ G…a, x† ˆ G…a†: Expanding …x 2 ÿ 1†lÿ1 , Eq. (4) can be written for a ˆ 0 as ÿ
I0 …c, b, l† ˆ I0 p2 , b, l ÿ

ˆ

…5†

iˆ0

lÿ1 †i where ki ˆ …ÿ1†i … † ˆ …1ÿl and I0 … p2 , b, 12 † ˆ I0 … p2 , i! i b† is the Bickley function, …a†n ˆ G…a ‡ n†=G…a† ˆ a…a ‡ 1†


…a ‡ n ÿ 1†: The disadvantage of such a representation is the slow convergence of the expansion for small values of c (Michieli, 1998). For simplicity, we introduce the auxiliary de®nition ÿ ÿ ÿ


I^a c1 , c2 , b, l ˆ Ia c2 , b, l ÿ Ia c1 , b, l ,

For a ˆ 0, the use of exponential integrals leads to ÿ
I^0 ce , c, b, l ˆ

nÿ 1 X
2iÿ2l‡2 ÿ
ki cos ce E2iÿ2l‡3 b sec ce iˆ0

o ÿ …cos c† 2iÿ2l‡2 E2iÿ2l‡3 …b sec c† :

x aÿ1 …x 2 ÿ 1 †

lÿ1

1 X i pa, i …x ÿ 1 † ,

ˆ

where,   i 1 X j a ÿ 1 …1 ÿ l †j … † ÿ 1 iÿj 21ÿl jˆ0 …2j†!!

…12a†

  i …1 ÿ l †iÿj 1 X iÿj a ÿ 1 … †   : ÿ 1 1ÿl j 2 2…i ÿ j† !! jˆ0

…12b†

pa, i ˆ

The generalized secant integral from Eq. (3) can be presented as ÿ ÿ

Ia …c, b, l† ˆ Ia ce , b, l ‡ I^a ce , c, b, l ,

Hence, from Eq. (7) we obtain an expression

where from Eq. (4) as ÿ
Ia ce , b, l ˆ ba

… sec ce 1

ÿ
Ia ce , b, l ˆ ba

… sec ce 1

1 X pa, i …x 1ÿl

e ÿbx …x ÿ 1 †

i

ÿ 1 † dx, e

ÿbx

x

aÿ1 …

x ÿ 1† 2

lÿ1

dx:

…7†

…11†

iˆ0

ˆ

…6†

…10†

Let a ˆ 0, c ˆ p2 and ce ˆ c, then from Eq. (6) we get, I^0 …c, p2 , b, l† ˆ I0 … p2 , b, l† ÿ I0 …c, b, l†: So, Eq. (9) reduces to the Sievert integral, as given in Eq. (5). Consider Taylor's expansion about x ˆ 1;

c2 > c1 > 0:

c > ce > 0,

…9†

which after integration leads to

iˆ0

…13†

L.A.-M. Hanna, S.L. Kalla / Radiation Physics and Chemistry 59 (2000) 281±285 1 X ÿ
 ÿ Ia ce , b, l ˆ e ÿb pa, i baÿlÿi g i ‡ l, b sec ce

jp1, iÿj j <

iˆ0




ÿ1 :

We now consider the following cases for di€erent values of a: (i) For a ˆ 0 (Sievert integral) From Eq. (12a), we have p0, i ˆ

i …1 ÿ l †j … ÿ 1 †i X 21ÿl jˆ0 …2j†!!

  i …1 ÿ l †…2 ÿ l †


…j ÿ l† 1 j … ÿ 1 †i X ˆ 1ÿl , 2 j! 2 jˆ0 which when compared with the expansion of 1 1 lÿ1 …1 ÿ x†lÿ1 , with x ˆ 12 we get jp0; i j < 21ÿl …2† ˆ 1, and hence, jp0, i jR1: (ii) For a ˆ 1 From Eq. (12a) we have … ÿ 1 †i …1 ÿ l †i , 21ÿl …2i †!!

jpn, i jR

<

ˆ

for irn:

…20†

 nÿ1  X nÿ1 jp1, iÿj j j jˆ0

 …nÿ1 †ÿj nÿ1  1 X 1 nÿ1 …1 † 2iÿn‡1 21ÿl jˆ0 j 2 1

1

1

2iÿn‡1

21ÿl

 1‡

1 2

nÿ1 ˆ

j

3nÿ1 : 2i‡1ÿl

can

… ÿ 1 † …i ÿ l † p1, iÿ1 ˆ i 2   …1 ÿ l † … ÿ 1 †i …i ÿ l † …i ÿ 1 † ÿ l


p1, 0 : ˆ i iÿ1 i 1 2 So, …17†

(iii) For a ˆ 2,3, . . . ,n From Eqs. (16) and (12b) we get a recurrence relation to express p1, iÿj in terms of p1, iÿk where k satis®es jRkRi, 0RjRi, that is 0RjRkRi

p1, iÿj ˆ

…19†

(vi) For a is a noninteger and 0 < a < 1 Following Michieli (1998), we can show that jpa; i j < jp0; i jR1: (v) For a is a noninteger and a > 1 Following Michieli (1998), and using Eq. (17), it

… ÿ 1 † …i ‡ 1 † ÿ l p1, i and p1, i 2 i‡1

jp1, i‡1 j < jp1, i jR2 ÿiÿ1‡l :

1 : 2iÿj‡1ÿl

 nÿ1  X nÿ1 ˆ p1, iÿj j jˆ0

…16†

… ÿ 1 †i‡1 …1 ÿ l †i‡1 ˆ 21ÿl …2…i ‡ 1 ††!! ˆ

2 ÿi‡n‡lÿ2 2nÿ1ÿj

So, from Eq. (19),

From Eq. (16), p1, i‡1

jp1, iÿn‡1 jR

From Eqs. (12b) and (16) we have pa; i ˆ Pi a ÿ 1 †p1; iÿj , and hence jˆ0 … j  i  X nÿ1 pn, i ˆ p1, iÿj for i < n, j jˆ0

pn, i …15†

p1, i ˆ

ˆ

…14†

1 2nÿ1ÿj

283

be easily shown that jpa; i jRra; i ˆ Pi a ÿ 1 1 iÿj †j… 2 † : It is also evident that ra, i jˆ0 j… j 21ÿl satis®es the relation 1

ra, i ˆ

  1 1 aÿ1 ra, iÿ1 ‡ 1ÿl j j: i 2 2

The following series representations of Ia …c, b, l† were used as input to the mathematica computer package to generate a number of tables for the generalized secant integral as in Eq. (3) for di€erent values of the parameters. For lack of space the tables are not reproduced here.


ÿ ÿ
ÿ
… ÿ 1 †kÿj …i ÿ j† ÿ l …i ÿ j ÿ 1† ÿ l …i ÿ j ÿ 2† ÿ l ……i ÿ k ‡ 1 † ÿ l † p1, iÿk ,


…i ÿ k ‡ 1 † 2kÿj …i ÿ j ÿ 1† …i ÿ j ÿ 2† …i ÿ j†

1 and so, jp1; iÿj j < 2kÿj jp1; iÿk j: For k ˆ n ÿ 1, by using Eq. (17) we have

For 0 < cRce < p3 , from Eq. (14) we have

…18†

284

L.A.-M. Hanna, S.L. Kalla / Radiation Physics and Chemistry 59 (2000) 281±285

Ia …c, b, l† ˆ e ÿb

  4n ‡ 3 p, b, l , Ia …c, b, l† ˆ Ia c ‡ 2

1 X   pa, i baÿlÿi g i ‡ l, b…sec c ÿ 1† : iˆ0

…21† For ce < c <

p 2,

3. Complete secant integral

iˆ0


‡ 2l ÿ 2, b sec ce ÿ G…a ÿ 2i ‡ 2l ÿ 2, b sec c† :

…22†

As studied by Michieli (1998), it turns out that ce ˆ p4 is a good choice for both series in Eqs. (21) and (22) to converge fast enough for all practical cases. For c ˆ p2 , Eq. (22) leads to (for, sec p2 ˆ 1 and G…x, 1† ˆ 0) 

   X 1 ÿ p p ki b 2iÿ2l‡2 G a ÿ 2i ‡ 2l , b, l ˆ Ia , b, l ‡ 2 4 iˆ0 p
ÿ 2, b 2

where ce ˆ p4 : Using Eq. (21), and replacing c by ce ˆ p4 , we get  a



p , b, l ˆ Ie ÿb 2 ‡

1 X

h
i ÿp pa, i baÿlÿi g i ‡ l, b 2 ÿ 1

iˆ0

1 ÿ X p
ki b 2iÿ2l‡2 G a ÿ 2i ‡ 2l ÿ 2, b 2 …23† iˆ0

By using the functional relations, g…r ‡ 1, y† ˆ rg…r, y† ÿ yr e ÿy and G…r ‡ 1, y† ˆ rG…r, y† ‡ yr e ÿy , we can easily deduce that " # iÿ1 X 1 ÿy l‡k g…i ‡ l, y† ˆ …l †i g…l, y† ÿ e …24† y …l †k‡1 kˆ0

and G…r ÿ i, y† ˆ

G…r ÿ i † G…r, y † G…r† ÿ e ÿy

iÿ1 X G…r ÿ i † kˆ0

G…r ÿ k †

yrÿkÿ1

1 Ia‡2 …c, b, l ÿ 1† ÿ Ia …c, b, l ÿ 1†: b2

Ia …c, b, l† ˆ Ia …c ‡ 2pn, b, l†,

In Eq. (4), let c ˆ p2 , then Ia … p2 , b, l† can be expressed in terms of the hypergeometric function as follows (Prudnikov et al., 1993, p. 326, Eq. (3)) (      p a a 1 a a 1 Ia , b, l ˆ b B l, 1 ÿ l ÿ ; , 1 F2 2 2 2 2 2 2    b2 1 1ÿa ‡ l; ÿ l 1 F2 ÿ bB l, 2 2 4   1‡a 3 1‡a b2 ; , ‡ l; 2 2 2 4  2ÿaÿ2l … ‡b G a ‡ 2l ÿ 2 †1 F2 1 ÿ l; 2 ÿ l ) a lÿa b2 ÿ , ÿ l; , 2 2 4 Re…l † > 0, Re…b † > 0: For l ˆ 1, we have (       p a a 1 2 ‡ a b2 a 1 Ia , b, 1 ˆ b B 1, ÿ ; , ; 1 F2 2 2 2 2 2 2 4   1 1‡a ÿ bB 1, ÿ 2 2 )   1 ‡ a 3 3 ‡ a b2 ÿa ; , ; ‡ b G…a† , 1 F2 2 2 2 4 Re…l † > 0, Re…b † > 0:

…28†

Eq. (28) can be written as: 82   < X 1  p b Ia , b, 1 ˆ ba 4 : kˆ0 …2k ‡ 1 †…2k ‡ 1 ‡ a † 2 9 3 = 2 †k … b 5 1 ÿ ‡ b ÿa G…a† ; 2k ‡ a …2k †! 

…25†

where i ˆ 1, 2, 3, . . . : The following recurrence relations can be easily veri®ed: Ia …c, b, l† ˆ

n is an integer:

we get from Eqs. (9) and (6)

1 ÿ  ÿ
X Ia …c, b, l† ˆ Ia ce , b, l ‡ ki b 2iÿ2l‡2 G a ÿ 2i

Ia

…26†

n is an integer:

ˆ G…a† ÿ

1 1 X X … ÿ 1 † 2k‡1 b 2k‡1‡a … ÿ 1 † 2k b 2k‡a ÿ …2k †!…2k ‡ a † …2k ‡ 1 †!…2k ‡ 1 ‡ a † kˆ0 kˆ0

ˆ G…a† ÿ g…a, b† ˆ G…a, b†:

L.A.-M. Hanna, S.L. Kalla / Radiation Physics and Chemistry 59 (2000) 281±285

In Eq. (4) let c ˆ p2 and a ˆ n ‡ 2, then we get (Prudnikov et al., 1993, p. 326, Eq. (4)),  In‡2



1 lÿ 2

†n n‡2

…ÿ1 b p 2 p , b, l ˆ 2 p Re…l † > 0, Re…b † > 0:

G…l †

n

@ @ bn



1 b 2 ÿl K

 … † b , 1 l‡ 2

I2

which should be corrected from Eq. (20) as pn, i ˆ

 i  X nÿ1 p1, iÿj : j jˆ0

Also, Eq. (12) in Section 3.1.5., that is

where Kl …x† is the modi®ed Bessel function of the second kind (Abramowitz and Stegun, 1972; Prudnikov et al., 1993). For n ˆ 0, we get 

285

1 3  p 2lÿ 2 b 2 ÿl p G…l †K 1 …b †, , b, l ˆ l‡ 2 2 p

Re…l † > 0, Re…b † > 0:

Ia …c, l† ˆ e ÿb

  1 X 1 1 pa, i b1ÿiÿ 2 g i ‡ , b…sec c ÿ 1† , 2 iˆ0

should be read as Ia …c, l† ˆ e ÿb

  1 X 1 1 pa, i baÿiÿ 2 g i ‡ , b…sec c ÿ 1† : 2 iˆ0

Further in Prudnikov et al., 1993, p. 326, Eq. (3), there seems to be a misprint in the last term: …2p† 2ÿaÿ2b should be replaced by …p† 2ÿaÿ2b :

Acknowledgements The authors acknowledge the support of Kuwait University (Project: SM. 181). Appendix A

References

In Michieli's (1998) paper, there are some misprints which we would like to point out here:Eq. (9), that is i 1 X … ÿ 1† Pa, i ˆ p 2 jˆ0

j



 a ÿ 1 …2j ÿ 1†!! , 1 ÿ j …2j†!!2 j

should be corrected to i 1 X … ÿ 1† pa, i ˆ p 2 jˆ0

j



aÿ1 iÿj



…2j ÿ 1†!! : …2j†!!2 j

In Eq. (12a) put l ˆ 12 , and notice that !! … 12 †j ˆ …2jÿ1† 2 j :The equation in Section 3.1.3. in Michieli (1998) is Pn, i ˆ

 nÿ1  X nÿ1 uˆ0

j

p1, iÿj ,

Abramowitz, M., Stegun, I.A., 1972. Handbook of Mathematical Functions. Dover, New York. Hungerford, H.E., 1962. Tables of secant integrals of the ®rst and second kinds. Nuclear Science Engineering (14), 312. Michieli, I., 1998. Point kernel calculation of dose ®elds from line sources using expanded polynomial form of buildup factor data: generalized secant integral-series representation. Radiat. Phys. Chem. 51 (2), 121±128. Prudnikov, A.P., Brychkov, Yu.A., Marichev, O.I., 1993. Integrals and Series, vol. 1. Gordon and Breach, London, UK. Sievert, R.M., 1921. Die IntensitaÈtsverteilung der primaÈren gStrahlung in der NaÈhe medizinischer radiumpraÈparate. Acta Radiologica 1, 89±128. Sievert, R.M., 1930. Die g-StrahlungsintensitaÈt an der Ober¯aÈche und in der naÈchsten Umgebung von Radiumnadeln. Acta Radiologica 11, 249±267. Wood, J., 1982. Computational Methods in Reactor Shielding. Pergamon Press, New York.