Solutions of elliptic integrals and generalizations ... - Semantic Scholar
Applied Mathematics and Computation 243 (2014) 33–43
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Solutions of elliptic integrals and generalizations by means of Bessel functions Sávio L. Bertoli ⇑, Laércio Ender, Juliano de Almeida, Pollyana M. da Silva Kalvelage Department of Chemical Engineering, Regional University of Blumenau, FURB, São Paulo street, 3250, 89030-000 Blumenau, Brazil
a r t i c l e
i n f o
Keywords: Hyperelliptic integrals Elliptic integrals Epstein–Hubbell integral Bessel functions Lauricella function
a b s t r a c t The elliptic integrals and its generalizations are applied to solve problems in various areas of science. This study aims to demonstrate a new method for the calculation of integrals through Bessel functions. We present solutions for classes of elliptic integrals and generalizations, the latter, refers to the hyperelliptic integrals and the integral called Epstein–Hubbell. The solutions obtained are expressed in terms of power series and/or trigonometric series; under a particular perspective, the final form of a class of hyperelliptic integrals is presented in terms of Lauricella functions. The proposed method allowed to obtain solutions in ways not yet found in the literature. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction The elliptic integrals can be viewed as generalizations of inverse trigonometric functions and can provide solutions to various problems in Electromagnetism [1,2], Fluid Mechanics [3] and Chemical Reactions Kinetics [4]. In particular, in geometry are used to measure the perimeter of an ellipse and the area of an ellipsoid [5,6]. In general, those integrals cannot be expressed in terms of elementary functions (all attempts failed, until the nineteenth century when it was finally proved, in fact, to be impossible to carry out this objective [7]). The solutions obtained in this work are compared with the usual forms of the literature, which are in most cases, different in the representation forms. In the present work, the method used to solve the elliptic integrals is to relate them to the Bessel functions. Under a particular perspective, the complete elliptic integrals are expressed in terms of power series, while the incomplete forms are expressed in terms of power and trigonometric series. The complete elliptic integral of the first kind obtained with the proposed method is the same form found in the literature [8,9]. For the complete elliptic integrals of the second and third kind, the solutions are given in a different form from the usual representations [10,11,8]. The incomplete elliptic integrals of the first, second and third kind are also solved with the Bessel function’s method. In exception of the incomplete elliptic integral of the first kind, the other expressions are not found in the literature. The Epstein–Hubbell integral, called as a generalization of elliptic integrals [9], is of a significant importance and is used in the solving of various problems [12,13]. For this integral, we propose a new solution, through the same method here used in other elliptic integrals.
⇑ Corresponding author. E-mail address: [email protected] (S.L. Bertoli). http://dx.doi.org/10.1016/j.amc.2014.05.084 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
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Hyperelliptic integrals are also addressed in this work, specifically, two types of these integrals are proposed with their respective solutions through Bessel functions. One of these integrals is expressed in its final form as a series of Lauricella functions. 1.1. Definition of elliptic integrals An elliptic integral is any function that can be expressed as,
Z
Rðx;
pffiffiffiffiffiffiffiffiffi PðxÞÞdx;
ð1:1Þ
where PðxÞ is a polynomial of third or fourth degree. With some reductions, each elliptic integral can be represented in a form involving integrals of rational, trigonometric, inverse trigonometric, logarithmic and exponential functions; Bronshtein et al. [14], Jeffrey and Dai [9]. 2. Complete elliptic integral of the first kind In this work, we propose that the complete elliptic integral of the first kind can be written in terms of a modified Bessel function of the first kind zeroth order, as expressed in the following:
Z 1 Z p p Z p2 2 dh pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ F k; I0 ðxk sin hÞex dx dh; ¼ 2 2 2 0 0 0 1 k sin h
ð2:1Þ
Z p p Z 1 2 F k; ex dx I0 ðxk sin hÞdh: ¼ 2 0 0
ð2:2Þ
The relationship between a Bessel function of the first kind zeroth order and its modified Bessel function is represented as found in Bowman [15], v
Iv ðzÞ ¼ i J v ðizÞ;
ð2:3Þ
I0 ðzÞ ¼ J 0 ðizÞ:
ð2:4Þ
The following representation found in Abramowitz and Stegun [10] allows to rewrite the integral form of the Bessel function in Eq. (2.2) in a closed form expression:
Z
p 2
0
¼
J n ð2iz sin hÞdh ¼
Z
p
J 2n ðizÞ;
2
p 2
p
In ð2z sin hÞdh ¼
2
0
I2n ðzÞ:
ð2:5Þ
ð2:6Þ
After this, replacing n ¼ 0 and z ¼ kx=2, the integral forms are represented by:
Z
p 2
J 0 ðikx sin hÞdh ¼
0
¼
Z
kx ; J 20 i 2 2
ð2:7Þ
kx : 2 2
ð2:8Þ
p
p 2
I0 ðkx sin hÞdh ¼
p
0
I2o
Substituting the Eq. (2.8) in Eq. (2.2.), we have,
Z p=2 Z p Z 1 p 1 x 2 kx dx: ¼ F k; ex dx I0 ðxk sin hÞdh ¼ e I0 2 2 2 0 0 0
ð2:9Þ
The Bessel function in Eq. (2.9) can be represented by a series, obtained from the following expression found in Abramowitz and Stegun [10]:
J v ðzÞJ l ðzÞ ¼
1 z v þz X
2
After the substitutions:
I20 ðzÞ ¼
1 X ð2rÞ!z2r r¼0
22r ðr!Þ4
r¼0
r ð1Þr Cðv þ l þ 2r þ 1Þ 14 z2 : Cðv þ r þ 1ÞCðl þ r þ 1ÞCðv þ l þ r þ 1Þr!
ð2:10Þ
v ¼ l ¼ 0, we obtain the form, :
ð2:11Þ
S.L. Bertoli et al. / Applied Mathematics and Computation 243 (2014) 33–43
35
Replacing the series given by Eq. (2.11) in Eq. (2.9), follows the expression, r 2 1 p p Z 1 X ð2rÞ!ðk x2 Þ F k; ex dx: ¼ 2 2 0 22r ðr!Þ4 r¼0
ð2:12Þ
Rearranging it, 2r Z 1 1 p p X ð2rÞ!ðkÞ F k; x2r ex dx: ¼ 4 2r 2 2 r¼0 2 ðr!Þ 0
ð2:13Þ
Finally, solving the improper integral in Eq. (2.13) we find the final form, represented by:
F
r 2 1 pX ð2rÞ!k ;k ¼ : 2 2 r¼0 22r r!2
p
ð2:14Þ
The final expression obtained for the complete elliptic integral of the first kind showed equivalence with the usual form of literature, which can be found in Olver et al. [8] and Gradshteyn and Ryzhik [16]. 3. Complete elliptic integral of the second kind The complete elliptic integral of the second kind, can be expressed through a modified Bessel function of the first kind in the following form,
Z p 2 p p 2 sin hdh 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; E ;k ¼ F ;k k 2 2 2 2 0 1 k sin h
ð3:1Þ
Z 1 Z p=2 p p 2 2 E ;k ¼ F ;k k ex dx sin hI0 ðkx sin hÞdh: 2 2 0 0
ð3:2Þ
With the results of Appendix A, Eq. (A.7) and the Eqs. (2.9) and (2.14), the Eq. (3.2) can be expressed as:
p E ;k ¼ 2
! 2 p p Z 1 k kx 2 2 : ex dxJ1 i 1 F ;k þ k 2 2 2 4 0
ð3:3Þ
Also, from Abramowitz and Stegun [10], we have:
r 2 X 1 ð2r þ 2Þ! 14 z2 1 J 21 ðizÞ ¼ z : 3 4 2 r¼0 ðr þ 2Þðr þ 1Þ r!
ð3:4Þ
Substituting the Eq. (3.4) in the Eq. (3.3), we obtain:
Z 0
1
2 1 r 2 kx k X ð2rÞ!k ð2r þ 1Þ2 2 ¼ ex dxJ1 i : 2 4 r¼0 22r r!2 ðr þ 1Þðr þ 2Þ
ð3:5Þ
After replacing Eq. (3.5) in Eq. (3.3), it results,
p E ;k ¼ 2
1
! 2 r 2 1 p k4 p X k ð2rÞ!k ð2r þ 1Þ2 F ;k : 2r 2 2 2 8 2 r¼0 2 r! ðr þ 1Þðr þ 2Þ
ð3:6Þ
Substituting the Eq. (2.14) in Eq. (3.6), and grouping the similar terms we find the final representation for the complete elliptic integral of the second kind,
" # r 2 2 4 1 p p X ð2rÞ!k k k ð2r þ 1Þ2 : E ;k ¼ 1 2 2 r¼0 22r r!2 2 8 ðr þ 1Þðr þ 2Þ
ð3:7Þ
The final representation (3.7) is different from the usual form found in the literature, for example in Olver et al. [8], Abramowitz and Stegun [10]:
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The results obtained with Eq. (3.7) are compared with the expression found in Abramowitz and Stegun [10]: k
Eðp2 ; kÞ
Eðp2 ; kÞ
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1.554968546243 1.534833464923 1.505941612360 1.467462209339 1.418083394449 1.355661135572 1.276349943169 1.171697052782
1.554968546243 1.534833464923 1.505941612360 1.467462209339 1.418083394449 1.355661135572 1.276349943169 1.171697052782 Abramowitz and Stegun [10]
4. Complete elliptic integral of the third kind The complete elliptic integral of the third kind can also be solved by means of Bessel function.
p
P
2
Z ; n; k ¼
p
Z
dh
2
2
ð1 n sin hÞ
0
1
ex I0 ðkx sin hÞdx:
ð4:1Þ
0
Also, it is possible to express the Eq. (4.1) in the form,
p
P
2
p 2
dh
2
Z ; n; k ¼
Z
1
2
etð1n sin
hÞ
dt
0
0
p
P
Z ; n; k ¼
1
et dt
0
Z
1
ex I0 ðkx sin hÞdx;
ð4:2Þ
0
Z
1
ex dx
0
Z
p 2
2
etn sin h I0 ðkx sin hÞdh:
ð4:3Þ
0
Using the following series representations [10]:
ez ¼
1 X zr r¼0
r!
ð4:4Þ
;
1 X z2r
I0 ðzÞ ¼
r¼0
22r r!2
ð4:5Þ
:
And substituting in the previous expressions, we have:
p
P
2
Z ; n; k ¼
1
et dt
0
Z 0
1
ex dx
Z 0
p 2
s 1 2r 2 1 X ðtn sin hÞ X ðkx sin hÞ dh : 2r 2 s! 2 r! s¼0 r¼0
ð4:6Þ
Or equivalently,
Z Z 1 Z p 2r 1 X 2 k ns 1 s t 2ðsþrÞ ; n; k ¼ t e dt x2r ex dx sin hdh: 2r 2 2 s! 2 r! 0 0 0 r;s¼0
p
P
ð4:7Þ
The integral of the sine function is represented as:
Z
p 2
2ðsþrÞ
sin
0
hdh ¼
p
½2ðs þ rÞ! : 2 22ðsþrÞ ðs þ rÞ!2
ð4:8Þ
Finally, substituting the Eq. (4.8) in the Eq. (4.7) and solving the integrals, we obtain:
p
P
1 pX ð2rÞ! ½2ðs þ rÞ! s 2r ; n; k ¼ nk : 2 2 r;s¼0 24r r!2 22s ðs þ rÞ!2
ð4:9Þ
The final representation is not found in the literature, the usual form can be found for example in Carlitz [11] or Olver et al. [8].
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The results obtained with Eq. (4.9) are compared with the results obtained from the expression found in Carlitz [11]: k
Pðp2 ; 0:5; kÞ
Pðp2 ; 0:5; kÞ
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
2.24810462594213 2.28335058819339 2.33674613731765 2.41367150420119 2.52390070844927 2.68680199682369 2.94787811582397 3.45910690021046
2.24810462594213 2.28335058819339 2.33674613731765 2.41367150420119 2.52390070844927 2.68680199682369 2.94787811582397 3.45910690021046 Carlitz [11]
5. Incomplete elliptic integral of the first kind Using the Bessel function method proposed in the present work, the incomplete elliptic integral of the first kind is expressed as:
Fðu; kÞ ¼
Z
1
ex dx
Z
0
u
I0 ðxk sin hÞdh:
ð5:1Þ
0
From Abramowitz and Stegun [10] we also can write:
I0 ðzÞ ¼
1 X
z2r
r¼0
2 ðr!Þ2 2r
ð5:2Þ
:
So replacing the Eq. (5.2) in the Eq. (5.1), we have:
Fðu; kÞ ¼
Z 2r 1 X k 2r
r¼0
2 r!2
1
ex x2 dx
Z
u
2r
sin hdh:
ð5:3Þ
0
0
Solving the improper integral, it follows,
Fðu; kÞ ¼
2r Z 1 X ð2rÞ!k 2r
r¼0
2 r!
2
u
2r
sin hdh:
ð5:4Þ
0
The integral of the sine function in Eq. (5.4) is represented by the relationship below from Zhao et al. [17]:
Z
u
2r
sin hdh ¼
0
#
" 2mþ1 r1 2m X 2 ðm!Þ2 sin u : cos u u ð2m þ 1Þ! 22r r!2 m¼0
ð2rÞ!
ð5:5Þ
Finally, the incomplete elliptic integral of the first kind is expressed as:
Fðu; kÞ ¼
1 X r¼0
r 2
ð2rÞ!k
22r r!2
" cos u
2mþ1 r1 2m X 2 m!2 sin u m¼0
ð2m þ 1Þ!
# u :
ð5:6Þ
The incomplete elliptic integral of the first kind was first solved by Qureshi and Chaudhary [18]. This result also shows the applicability of the proposed method to the incomplete elliptic integral. Qureshi and Chaudhary [18] presented a solution for the incomplete elliptic of the first kind in terms of series of hypergeometric functions. The results obtained with Eq. (5.6) are compared with the results obtained from the expression found in Qureshi and Chaudhary [18]:
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Fðp3 ; kÞ
Fðp3 ; kÞ
1.053430587029 1.061489706726 1.073313629047 1.089550670051 1.111233322932 1.140044752769 1.178902299538 1.233446325452
1.053430587029 1.061489706726 1.073313629047 1.089550670051 1.111233322932 1.140044752769 1.178902299538 1.233446325452 Qureshi and Chaudhary [18]
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6. Incomplete elliptic integral of the second kind Applying the proposed method in relating the incomplete elliptic integral of the second kind with the Bessel function, we have,
Z
2
Eðu; kÞ ¼ Fðu; kÞ k
1
ex dx
Z
0
u
2
sin hI0 ðkx sin hÞdh:
ð6:1Þ
0
After replacing the Bessel function in the series form in Eq. (6.1) and solving the improper integral, it follows the expression, 2
Eðu; kÞ ¼ Fðu; kÞ k
" # 2r Z 1 X ð2rÞ!k 2r
2 r!
r¼0
2
u
2rþ2
sin
ð6:2Þ
hdh:
0
The integral of the sine function was given by Eq. (5.5), then we can write,
Z
u
"
2rþ2
sin
0
ð2rÞ! ð2r þ 1Þ hdh ¼ 2r 2 ðrÞ!2 2ðr þ 1Þ
#"
2mþ1 r X 22m ðm!Þ2 sin u
cos u
ð2m þ 1Þ!
m¼0
#
u :
ð6:3Þ
Substituting the expression (6.3) in the Eq. (6.2), we obtain the final expression for the incomplete elliptic integral of the second kind:
Eðu; kÞ ¼
" r 2 1 X ð2rÞ!k 22r r!2
r¼0
!
# 2mþ1 2mþ1 r r1 2m ð2r þ 1Þ 2 X 22m ðm!Þ2 sin u X 2 ðm!Þ2 sin u ð2r þ 1Þ 2 þu 1 : cos u k k 2ðr þ 1Þ m¼0 2ðr þ 1Þ ð2m þ 1Þ! ð2m þ 1Þ! m¼0 ð6:4Þ
This representation for the incomplete elliptic integral of the second kind in terms of power series and trigonometric series is not found in the literature. Qureshi and Chaudhary [18] proposed an expression in terms of a series of hypergeometric functions for the integral analyzed above, as well as for the incomplete elliptic integral of first kind. The results obtained from Eq. (6.4) are compared with the expression found in Qureshi and Chaudhary [18]: k
Eðp3 ; kÞ
Eðp3 ; kÞ
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1.04566021970563 1.03322345140654 1.02213013278769 1.00755555514447 0.989221593549377 0.966723133094528 0.93945480372495 0.906456986263154
1.04566021970563 1.03322345140654 1.02213013278769 1.00755555514447 0.989221593549377 0.966723133094528 0.93945480372495 0.906456986263154 Qureshi and Chaudhary [18]
7. Incomplete elliptic integral of the third kind Applying the present Bessel function method, we have:
Z
Pðu; n; kÞ ¼
u
Z
dh 2
ð1 n sin hÞ
0
1
ex I0 ðkx sin hÞdx:
ð7:1Þ
0
It is also possible to express the last expression in the form:
Z
Pðu; n; kÞ ¼
u
dh
Z
0
or,
Z
Pðu; n; kÞ ¼
1
etð1n sin
The functions etn sin
Pð/; n; kÞ ¼
h
Z
hÞ
dt
Z
0
1
et dt
1
ex I0 ðkx sin hÞdx;
ð7:2Þ
0
Z
0 2
2
1
ex dx
Z
0
u
2
etn sin h I0 ðkx sin hÞdh:
ð7:3Þ
0
and I0 ðkx sin hÞ can be represented by series, then it follows, 1
et dt
0
Z
1
ex dx
0
Z 0
/
s 1 2r 2 1 X ðtn sin hÞ X ðkx sin hÞ dh : s! 22r r!2 s¼0 r¼0
ð7:4Þ
Or equivalently,
Pð/; n; kÞ ¼
2r 1 X k
ns 2r 2 s! r;s¼0 2 r!
Z 0
1
t s et dt
Z 0
1
x2r ex dx
Z 0
u
2ðsþrÞ
sin
hdh:
ð7:5Þ
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S.L. Bertoli et al. / Applied Mathematics and Computation 243 (2014) 33–43
Solving the two improper integrals, we obtain the expression,
Pð/; n; kÞ ¼
Z 2r 1 X ð2rÞ!k ns 2r
2 r!
r;s¼0
2
u
2ðsþrÞ
sin
ð7:6Þ
hdh:
0
The integral in Eq. (7.6) is given by (see Eq. (5.5)):
Z
"
u
sin
2ðrþsÞ
hdh ¼
0
½2ðr þ sÞ!
#"
rþs1 X
cos u
22ðrþsÞ ðr þ sÞ!2
m¼0
2mþ1
22m ðm!Þ2 sin ð2m þ 1Þ!
u
# u :
ð7:7Þ
After replacing the previous expression in the Eq. (7.6), we obtain the following result:
" #" # 2r 1 rþs1 X X 22m ðm!Þ2 sin2mþ1 u ð2rÞ!k ns ½2ðr þ sÞ! cos u Pðu; n; kÞ ¼ u : 4r 2 ð2m þ 1Þ! 22s ðr þ sÞ!2 r;s¼0 2 r! m¼0
ð7:8Þ
This form for the incomplete elliptic integral of the third kind is not found in the literature. The usual representation is expressed in terms of a series of hypergeometric functions, or in terms of other series, for example by Karp et al. [19], using a partial fraction decomposition. The results obtained from Eq. (7.8) are compared with the expression found in [19] Karp et al. [19]:
k
Pðp3 ; 0; 8; kÞ
Pðp3 ; 0; 8; kÞ
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1.48453402981057 1.49858743111330 1.5192950589409 1.5479055930718 1.5864286332503 1.63818518991736 1.70903214209174 1.81054697605315
1.48453402981057 1.49858743111330 1.5192950589409 1.5479055930718 1.5864286332503 1.63818518991736 1.70903214209174 1.81054697605315 Karp et al. [19]
8. Hyperelliptic integrals The hyperelliptic integral or Abelian integral, in its general form is expressed as in Springer [12]:
Z
dx pffiffiffiffiffiffiffiffiffiffi ; QðxÞ
ð8:1Þ
where Q ðxÞ is a polynomial of degree > 4. In the following, it will be show how the method used anteriorly can also solve integrals of this class. Consider now the integral:
Z
dx pffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 1 xa
ð8:2Þ
This integral is considered a hyperelliptic integral for a > 4. It can be expressed in terms of a Bessel function,
Z
dx pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 1 xa
Z
dx
Z
1
a
I0 ðsx2 Þes ds:
ð8:3Þ
0
Replacing the Bessel function by its power series form in Eq. (8.3), we have:
Z
dx pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 1 xa
Z
1
es ds
Z X a 2r 1 ðsx2 Þ
0
r¼0
r!2
dx:
ð8:4Þ
Or,
Z
1 X dx 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2r 2 a 1x r¼0 2 r!
Z
1
es s2r ds
Z
xar dx:
ð8:5Þ
0
Finally, solving the two integrals in Eq. (8.5), it is obtained the following expression for this particular class of hyperelliptic integral,
Z
1 X dx ð2rÞ! xarþ1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ þ C: 2r 2 ðar þ 1Þ a 1x r¼0 2 r!
ð8:6Þ
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S.L. Bertoli et al. / Applied Mathematics and Computation 243 (2014) 33–43
The integral in Eq. (8.1) is usually solved through the binomial series expansion. The next integral is a more general class of hyperelliptic integral:
Z
1
du pffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 1 a2
0
It can be expressed in the following form:
Z
1
du pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 1 a2
0
Z
Z
1
0
1
ex I0 ðaxÞdxdu;
ð8:7Þ
0
where,
a2 ¼
n Y
ð1 xi uÞ:
ð8:8Þ
i¼1
Following the proposed method:
Z
1 X du 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2r 2 2 r! 2 1a r¼0
1
0
Z
1
ex x2r dx
Z
0
1
a2r du:
ð8:9Þ
0
Solving the improper integral, it results:
Z
1 X du ð2rÞ! pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2r 2 2 1a r¼0 2 r!
1
0
Z
1
a2r du:
ð8:10Þ
0
Substituting the expression (8.8) in the Eq. (8.10), it shows:
Z
1 X du ð2rÞ! pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2r 2 2 2 r! 1a r¼0
1
0
Z
1 0
n Y
ð1 xi uÞr du:
ð8:11Þ
i¼1
Now,
Z
1
n Y
0
ð1 xi uÞr du ¼
i¼1
Cð2Þ ðnÞ F ð1; r; . . . ; r; 2; x1 ; . . . ; xn Þ; Cð1ÞCð1Þ D
ð8:12Þ
ðnÞ
where F D is a Lauricella function. Substituting the expression (8.12) in Eq. (8.11) the following result is obtained:
Z 0
1
1 X du ð2rÞ! ðnÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ F ð1; r; . . . ; r; 2; x1 ; . . . ; xn Þ: 2r 2 D 2 1a r¼0 2 r!
ð8:13Þ
This expression for the hyperelliptic integral in terms of Lauricella functions can also be solved by the binomial expansion, different from the way developed anteriorly. 9. Epstein–Hubbell integral The Epstein–Hubbell integral contains the parameters j and k, represented in the form [16,20]:
Xj ðkÞ ¼
Z p
2
ð1 k cos hÞ
j12
2
dhk < 1;
j ¼ 0; 1; 2; 3 . . .
ð9:1Þ
0
The solution of this integral is usually obtained through the binomial expansion. 2
ð1 k cos hÞ
j
¼
2n 1 X ðjÞn k cosn h j ¼ 1; 2; 3 . . . ; n! n¼0
ð9:2Þ
where ðjÞn is the Pochhammer symbol, which can be found in Gradshteyn and Ryzhik [16]. 9.1. Epstein–Hubbell integral for j ¼ 0 For the case of j ¼ 0, we have from the expression (9.4):
X0 ðkÞ ¼
Z p 0
dh qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 ð1 k cos hÞ
ð9:3Þ
It can be rewritten the integral (9.6) as it was done in terms of the modified Bessel function:
X0 ðkÞ ¼
Z
0
1
ex dx
Z p pffiffiffiffiffiffiffiffiffiffiffi Z 1 X ð2rÞ! 2r p I0 xk cos h dh ¼ k cosr hdh: 2 2r 0 0 r¼0 2 ðr!Þ
ð9:4Þ
S.L. Bertoli et al. / Applied Mathematics and Computation 243 (2014) 33–43
41
The integral of the cosine function in the last equation is represented in the form [21]:
Z p
r
cos hdh ¼
0
8 0; > >
> : p 2r
ð9:5Þ
: r!
ð2r Þ!2
; if r even
From Eq. (9.8) it is know that only the r even terms will contribute for X0 ðkÞ, then:
X0 ðkÞ ¼ p
1 X ð2rÞ! r¼0 r–odd
r 22r ðr!Þ2 2
r! 2r r 2 k : ! 2
ð9:6Þ
Now, making the following change r ! 2r:
X0 ðkÞ ¼ p
1 X
ð4rÞ!
r¼0
26r ð2rÞ!ðr!Þ2
4r
k :
ð9:7Þ
The final form given by the Eq. (9.10) can be found Srivastava [22]. 9.2. Epstein–Hubbell integral for j ¼ 1; 2; 3 . . . For the Epstein–Hubbell integral with argument j > 0 it follows the expression,
Xj ðkÞ ¼
Z p
2
ð1 k cos hÞ
j12
2
dhk < 1;
j ¼ 1; 2; 3 . . . :
ð9:8Þ
0
It can be represented in the following way,
Xj ðkÞ ¼
j Z p 2 ð1 k cos hÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dh: 2 0 ð1 k cos hÞ
ð9:9Þ
Relating it to the modified Bessel function of the first kind, it is obtained:
Xj ðkÞ ¼
Z
1
ex dx
Z p
pffiffiffiffiffiffiffiffiffiffiffi j 2 I0 ðxk cos hÞð1 k cos hÞ dh:
ð9:10Þ
0
0
Now, substituting the binomial expansion found in expression (9.13):
Xj ðkÞ ¼
2n 1 2r Z p 1 X ðjÞn k X ð2rÞ!k cosnþr hdh: n! r¼0 22r r!2 0 n¼0
ð9:11Þ
The integral of the cosine function is from Eq. (9.8):
Z p
cosrþn hdh ¼
0
8 0; > >
pðrþnÞ! > : 2rþn ; if n þ r ¼ even !2 ðrþn 2 Þ
ð9:12Þ
:
Now, following the same procedure as it was done for j ¼ 0, it results:
Xj ðkÞ ¼ p
1 X ðjÞn ð2rÞ! ðr þ nÞ! 2ðrþnÞ : 2 k n 3r 2 rþn n!2 2 r! ! n;r¼0 2
ð9:13Þ
nþr¼ev en
There are two conditions for ðn þ r) to be an even number: (a) when ðn; rÞ are even numbers; (b) when ðn; rÞ are odd numbers. For case (a) is made the following substitution in Eq. (9.16): r ! 2r and n ! 2n, which results:
Xj ðkÞ ¼ p
1 X
ðjÞ2n
n;r¼0 ð2nÞ!2
ð4rÞ! 2n
ð2r þ 2nÞ!
26r ð2rÞ!2 22n ðr þ nÞ!2
k
4ðrþnÞ
:
ð9:14Þ
For case (b) is made the following substitution in Eq. (9.16): r ! 2r þ 1 and n ! 2n þ 1, then we have:
Xj ðkÞ ¼ p
1 X ðjÞ2nþ1 ð4r þ 2Þ! ð2r þ 2n þ 2Þ! 4ðrþnþ1Þ k : 6r 2 2nþ4 ð2n þ 1Þ! 2 ð2r þ 1Þ! 2 ðr þ n þ 1Þ!2 n;r¼0
ð9:15Þ
Finally adding the expressions (9.14) and (9.15), it is obtained the solution for the Epstein–Hubbel integral:
" #" " ## 4ðrþnÞ 4 1 X ð4rÞ!ð2r þ 2nÞ!k ðjÞ2n 1 ðjÞ2nþ1 ð4r þ 1Þð2r þ 2n þ 1Þk : Xj ðkÞ ¼ p þ 2 2 6rþ2n ð2nÞ! 4 ð2n þ 1Þ! ð2r þ 1Þðr þ n þ 1Þ n;r¼0 ð2rÞ! ðr þ nÞ! 2
ð9:16Þ
42
S.L. Bertoli et al. / Applied Mathematics and Computation 243 (2014) 33–43
The expression (9.16) is the final form for the Epstein–Hubbell integral with argument j > 0. This form of representation is not found in the literature. The usual forms may be found, for example, in Al-Zamel and Kalla [23], Cengiz [21], Gradshteyn and Ryzhik [16], Kalla and Montanus [24], Kalla et al. [25], Kalla and Tuan [26], Srivastava [22] and Galué [5]. The comparison with the expression in Al-Zamel and Kalla [23], it follows: j
Xj ð0; 5Þ
Xj ð0; 5Þ
1 2 3
3.337745013 3.616446839 1.8105469760
3.337745013 3.616446839 1.8105469760 Al-Zamel and Kalla [23]
10. Conclusions This paper proposed a new method for solving elliptic integrals and some of its generalizations, based on the modified Bessel function of first kind and order zero. By the present method it was possible to obtain various solutions representations similar to those found in the literature and also some integral representations obtained in unusual ways. The proposed method suggests a broad applicability that may be more general than was exploited. Acknowledgments The authors are thankful to the Department of Chemical Engineering at the University of Blumenau. The authors also thanks to Marta H. Caetano and Luiz H. da Silva for the English revision of the manuscript. Appendix A. Alternative representation for the integral:
R p=2 0
2
sin hI 0 ðkx sin hÞdh
Starting from Abramowitz and Stegun [10], it is know that,
Z p
J 0 ð2z sin tÞ cos 2t dt ¼ pJ 21 ðzÞ;
0
ðA:1Þ
also, the following trigonometric identity is valid, 2
cos 2t ¼ 1 2 sin t;
ðA:2Þ
with the two last relations, it can be written,
Z p
J 0 ð2z sin tÞsen2 tdt ¼
0
Z p
1 2
0
J 0 ð2z sin tÞdt
p 2
J 21 ðzÞ:
ðA:3Þ
The functions sen2 t and J 0 ð2zsen tÞ are symmetrical to each other in the interval ½0; p=2 and ½p=2; p, then using this property, it can be established,
Z
p 2
0
J 0 ð2z sin tÞsen2 tdt ¼
1 2
Z
p 2
0
J 0 ð2z sin tÞdt
p 4
J 21 ðzÞ:
ðA:4Þ
Now making the following substitution z ¼ iz, we have from Eq. (A.4),
Z
p 2
2
J 0 ð2iz sin tÞ sin tdt ¼
0
¼
Z
1 2
p 2
2
I0 ð2z sin tÞ sin tdt ¼
0
Z
p 2
J 0 ð2iz sin tÞdt
0
1 2
Z
p 4
p 2
J 21 ðizÞ;
p
ðA:5Þ
J 21 ðizÞ:
ðA:6Þ
kx : J 21 i 2 4
ðA:7Þ
I0 ð2z sin tÞdt
0
4
Finally, substituting z ¼ kx=2, it results,
Z
p 2
I0 ðkx sin tÞsen2 tdt ¼
0
1 2
Z 0
p 2
I0 ðkx sin tÞdt
p
References [1] S. Datta, Electric and magnetic fields from a circular coil using elliptic integrals, Phys. Educ. (2007) 203–212. [2] F.L. Derek, Elliptic Functions and Applications, Springer, 1989. [3] J.T. Sommerfeld, M.P. Stallybrassa, Elliptic integral solutions for fluid discharge rates from punctured horizontal cylindrical vessels, J. Loss Prev. Process Ind. 6 (1993) 11–13.
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[4] P.W. Hart, J.T. Sommerfeld, Elliptic integral solutions for the kinetics of autocatalytic termolecular reactions with reactive intermediate species, Chem. Eng. J. 66 (1997) 231–234. [5] L. Galué, Asymptotic expansions for generalized elliptic-type integrals, Radiat. Phys. Chem. 58 (2000) 307–313. [6] I.M. Mladenov, New geometrical applications of the elliptic integrals: the Mylar balloon, J. Nonlinear Math. Phys. (2004) 55–65. [7] R. Courant, Differential and Integral Calculus, vol. 1, Wiley-Interscience, 1988. [8] F.W.J. Olver, D.W. Lozier, R.F. Boisvert, C.W. Clark (Eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, 2010. [9] A. Jeffrey, H. Dai, Handbook of Mathematical Formulas and Integrals, Academic Press, New York, 2003. [10] M. Abramowitz, I. Stegun, Handbook of Mathematical Functions, Dover, New York, 1972. [11] L. Carlitz, On the complete elliptic integral of the third kind, Bollettino dell’Unione Matematica Italiana, serie 3 (15) (1960) 373–378. [12] G. Springer, Introduction to Riemann Surfaces, American Mathematical Society, 2002. [13] S.L. Kalla, S. Conde, J.H. Hubbell, Some results on generalized elliptic-type integrals, Appl. Anal. 22 (1986) 273–287. [14] I.N. Bronshtein, K.A. Semendyayev, G. Musiol, H. Muehlig, Handbook of Mathematics, Springer, Berlin, 2004. [15] F. Bowman, Introduction to Bessel Functions, Dover Publications, New York, 1958. [16] I.S. Gradshteyn, I.M. Ryzhik, Table of Series, Integrals and Products, Academic Press, New York, 2007. [17] J. Zhao, J. Jia, X. He, H. Wang, Post-buckling and snap-through behavior of inclined slender beams, J. Appl. Mech. (2008) 1–7. [18] M.I. Qureshi, M.P. Chaudhary, Ramanujan’s entries and solutions of incomplete elliptic integrals in terms of hypergeometric functions, Int. J. Open Prob. Comput. Math. 1 (2009) 85–97. [19] D. Karp, A. Savenkova, S.M. Sitnik, Series expansions for the third incomplete elliptic integral via partial fraction decompositions, J. Comput. Appl. Math. 207 (2007) 331–337. [20] L.F. Epstein, J.H. Hubbell, Evaluation of generalized elliptic-type integral, J. Res. N. B. S. 67 (1963) 1–17. [21] A. Cengiz, A study on the Epstein–Hubbell generalized elliptic-type integral using residue theory, Appl. Math. Comput. (1997) 19–26. [22] R. Srivastava, A note on the Epstein–Hubbell generalized elliptic-type integral, Appl. Math. Comput. (1995) 255–262. [23] A. Al-Zamel, S. Kalla, Epstein–Hubbell elliptic type integral and its generalizations, Appl. Math. Comput. (1996) 9–32. [24] S.L. Kalla, J.M.C. Montanus, A series approximation for disk galaxies by means of the Epstein–Hubbell integral, Math. Comput. Model. 40 (2004) 611– 626. [25] S.L. Kalla, C. Leubner, J.H. Hubbell, Further results on generalized elliptic-type integrals, Appl. Anal. 25 (1987) 269–274. [26] S.L. Kalla, V.K. Tuan, Asymptotic formulas for generalized elliptic-type integrals, Comput. Math. Appl. 32 (1966) 49–55.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Solutions of elliptic integrals and generalizations by means of Bessel functions Sávio L. Bertoli ⇑, Laércio Ender, Juliano de Almeida, Pollyana M. da Silva Kalvelage Department of Chemical Engineering, Regional University of Blumenau, FURB, São Paulo street, 3250, 89030-000 Blumenau, Brazil
a r t i c l e
i n f o
Keywords: Hyperelliptic integrals Elliptic integrals Epstein–Hubbell integral Bessel functions Lauricella function
a b s t r a c t The elliptic integrals and its generalizations are applied to solve problems in various areas of science. This study aims to demonstrate a new method for the calculation of integrals through Bessel functions. We present solutions for classes of elliptic integrals and generalizations, the latter, refers to the hyperelliptic integrals and the integral called Epstein–Hubbell. The solutions obtained are expressed in terms of power series and/or trigonometric series; under a particular perspective, the final form of a class of hyperelliptic integrals is presented in terms of Lauricella functions. The proposed method allowed to obtain solutions in ways not yet found in the literature. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction The elliptic integrals can be viewed as generalizations of inverse trigonometric functions and can provide solutions to various problems in Electromagnetism [1,2], Fluid Mechanics [3] and Chemical Reactions Kinetics [4]. In particular, in geometry are used to measure the perimeter of an ellipse and the area of an ellipsoid [5,6]. In general, those integrals cannot be expressed in terms of elementary functions (all attempts failed, until the nineteenth century when it was finally proved, in fact, to be impossible to carry out this objective [7]). The solutions obtained in this work are compared with the usual forms of the literature, which are in most cases, different in the representation forms. In the present work, the method used to solve the elliptic integrals is to relate them to the Bessel functions. Under a particular perspective, the complete elliptic integrals are expressed in terms of power series, while the incomplete forms are expressed in terms of power and trigonometric series. The complete elliptic integral of the first kind obtained with the proposed method is the same form found in the literature [8,9]. For the complete elliptic integrals of the second and third kind, the solutions are given in a different form from the usual representations [10,11,8]. The incomplete elliptic integrals of the first, second and third kind are also solved with the Bessel function’s method. In exception of the incomplete elliptic integral of the first kind, the other expressions are not found in the literature. The Epstein–Hubbell integral, called as a generalization of elliptic integrals [9], is of a significant importance and is used in the solving of various problems [12,13]. For this integral, we propose a new solution, through the same method here used in other elliptic integrals.
⇑ Corresponding author. E-mail address: [email protected] (S.L. Bertoli). http://dx.doi.org/10.1016/j.amc.2014.05.084 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
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S.L. Bertoli et al. / Applied Mathematics and Computation 243 (2014) 33–43
Hyperelliptic integrals are also addressed in this work, specifically, two types of these integrals are proposed with their respective solutions through Bessel functions. One of these integrals is expressed in its final form as a series of Lauricella functions. 1.1. Definition of elliptic integrals An elliptic integral is any function that can be expressed as,
Z
Rðx;
pffiffiffiffiffiffiffiffiffi PðxÞÞdx;
ð1:1Þ
where PðxÞ is a polynomial of third or fourth degree. With some reductions, each elliptic integral can be represented in a form involving integrals of rational, trigonometric, inverse trigonometric, logarithmic and exponential functions; Bronshtein et al. [14], Jeffrey and Dai [9]. 2. Complete elliptic integral of the first kind In this work, we propose that the complete elliptic integral of the first kind can be written in terms of a modified Bessel function of the first kind zeroth order, as expressed in the following:
Z 1 Z p p Z p2 2 dh pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ F k; I0 ðxk sin hÞex dx dh; ¼ 2 2 2 0 0 0 1 k sin h
ð2:1Þ
Z p p Z 1 2 F k; ex dx I0 ðxk sin hÞdh: ¼ 2 0 0
ð2:2Þ
The relationship between a Bessel function of the first kind zeroth order and its modified Bessel function is represented as found in Bowman [15], v
Iv ðzÞ ¼ i J v ðizÞ;
ð2:3Þ
I0 ðzÞ ¼ J 0 ðizÞ:
ð2:4Þ
The following representation found in Abramowitz and Stegun [10] allows to rewrite the integral form of the Bessel function in Eq. (2.2) in a closed form expression:
Z
p 2
0
¼
J n ð2iz sin hÞdh ¼
Z
p
J 2n ðizÞ;
2
p 2
p
In ð2z sin hÞdh ¼
2
0
I2n ðzÞ:
ð2:5Þ
ð2:6Þ
After this, replacing n ¼ 0 and z ¼ kx=2, the integral forms are represented by:
Z
p 2
J 0 ðikx sin hÞdh ¼
0
¼
Z
kx ; J 20 i 2 2
ð2:7Þ
kx : 2 2
ð2:8Þ
p
p 2
I0 ðkx sin hÞdh ¼
p
0
I2o
Substituting the Eq. (2.8) in Eq. (2.2.), we have,
Z p=2 Z p Z 1 p 1 x 2 kx dx: ¼ F k; ex dx I0 ðxk sin hÞdh ¼ e I0 2 2 2 0 0 0
ð2:9Þ
The Bessel function in Eq. (2.9) can be represented by a series, obtained from the following expression found in Abramowitz and Stegun [10]:
J v ðzÞJ l ðzÞ ¼
1 z v þz X
2
After the substitutions:
I20 ðzÞ ¼
1 X ð2rÞ!z2r r¼0
22r ðr!Þ4
r¼0
r ð1Þr Cðv þ l þ 2r þ 1Þ 14 z2 : Cðv þ r þ 1ÞCðl þ r þ 1ÞCðv þ l þ r þ 1Þr!
ð2:10Þ
v ¼ l ¼ 0, we obtain the form, :
ð2:11Þ
S.L. Bertoli et al. / Applied Mathematics and Computation 243 (2014) 33–43
35
Replacing the series given by Eq. (2.11) in Eq. (2.9), follows the expression, r 2 1 p p Z 1 X ð2rÞ!ðk x2 Þ F k; ex dx: ¼ 2 2 0 22r ðr!Þ4 r¼0
ð2:12Þ
Rearranging it, 2r Z 1 1 p p X ð2rÞ!ðkÞ F k; x2r ex dx: ¼ 4 2r 2 2 r¼0 2 ðr!Þ 0
ð2:13Þ
Finally, solving the improper integral in Eq. (2.13) we find the final form, represented by:
F
r 2 1 pX ð2rÞ!k ;k ¼ : 2 2 r¼0 22r r!2
p
ð2:14Þ
The final expression obtained for the complete elliptic integral of the first kind showed equivalence with the usual form of literature, which can be found in Olver et al. [8] and Gradshteyn and Ryzhik [16]. 3. Complete elliptic integral of the second kind The complete elliptic integral of the second kind, can be expressed through a modified Bessel function of the first kind in the following form,
Z p 2 p p 2 sin hdh 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; E ;k ¼ F ;k k 2 2 2 2 0 1 k sin h
ð3:1Þ
Z 1 Z p=2 p p 2 2 E ;k ¼ F ;k k ex dx sin hI0 ðkx sin hÞdh: 2 2 0 0
ð3:2Þ
With the results of Appendix A, Eq. (A.7) and the Eqs. (2.9) and (2.14), the Eq. (3.2) can be expressed as:
p E ;k ¼ 2
! 2 p p Z 1 k kx 2 2 : ex dxJ1 i 1 F ;k þ k 2 2 2 4 0
ð3:3Þ
Also, from Abramowitz and Stegun [10], we have:
r 2 X 1 ð2r þ 2Þ! 14 z2 1 J 21 ðizÞ ¼ z : 3 4 2 r¼0 ðr þ 2Þðr þ 1Þ r!
ð3:4Þ
Substituting the Eq. (3.4) in the Eq. (3.3), we obtain:
Z 0
1
2 1 r 2 kx k X ð2rÞ!k ð2r þ 1Þ2 2 ¼ ex dxJ1 i : 2 4 r¼0 22r r!2 ðr þ 1Þðr þ 2Þ
ð3:5Þ
After replacing Eq. (3.5) in Eq. (3.3), it results,
p E ;k ¼ 2
1
! 2 r 2 1 p k4 p X k ð2rÞ!k ð2r þ 1Þ2 F ;k : 2r 2 2 2 8 2 r¼0 2 r! ðr þ 1Þðr þ 2Þ
ð3:6Þ
Substituting the Eq. (2.14) in Eq. (3.6), and grouping the similar terms we find the final representation for the complete elliptic integral of the second kind,
" # r 2 2 4 1 p p X ð2rÞ!k k k ð2r þ 1Þ2 : E ;k ¼ 1 2 2 r¼0 22r r!2 2 8 ðr þ 1Þðr þ 2Þ
ð3:7Þ
The final representation (3.7) is different from the usual form found in the literature, for example in Olver et al. [8], Abramowitz and Stegun [10]:
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S.L. Bertoli et al. / Applied Mathematics and Computation 243 (2014) 33–43
The results obtained with Eq. (3.7) are compared with the expression found in Abramowitz and Stegun [10]: k
Eðp2 ; kÞ
Eðp2 ; kÞ
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1.554968546243 1.534833464923 1.505941612360 1.467462209339 1.418083394449 1.355661135572 1.276349943169 1.171697052782
1.554968546243 1.534833464923 1.505941612360 1.467462209339 1.418083394449 1.355661135572 1.276349943169 1.171697052782 Abramowitz and Stegun [10]
4. Complete elliptic integral of the third kind The complete elliptic integral of the third kind can also be solved by means of Bessel function.
p
P
2
Z ; n; k ¼
p
Z
dh
2
2
ð1 n sin hÞ
0
1
ex I0 ðkx sin hÞdx:
ð4:1Þ
0
Also, it is possible to express the Eq. (4.1) in the form,
p
P
2
p 2
dh
2
Z ; n; k ¼
Z
1
2
etð1n sin
hÞ
dt
0
0
p
P
Z ; n; k ¼
1
et dt
0
Z
1
ex I0 ðkx sin hÞdx;
ð4:2Þ
0
Z
1
ex dx
0
Z
p 2
2
etn sin h I0 ðkx sin hÞdh:
ð4:3Þ
0
Using the following series representations [10]:
ez ¼
1 X zr r¼0
r!
ð4:4Þ
;
1 X z2r
I0 ðzÞ ¼
r¼0
22r r!2
ð4:5Þ
:
And substituting in the previous expressions, we have:
p
P
2
Z ; n; k ¼
1
et dt
0
Z 0
1
ex dx
Z 0
p 2
s 1 2r 2 1 X ðtn sin hÞ X ðkx sin hÞ dh : 2r 2 s! 2 r! s¼0 r¼0
ð4:6Þ
Or equivalently,
Z Z 1 Z p 2r 1 X 2 k ns 1 s t 2ðsþrÞ ; n; k ¼ t e dt x2r ex dx sin hdh: 2r 2 2 s! 2 r! 0 0 0 r;s¼0
p
P
ð4:7Þ
The integral of the sine function is represented as:
Z
p 2
2ðsþrÞ
sin
0
hdh ¼
p
½2ðs þ rÞ! : 2 22ðsþrÞ ðs þ rÞ!2
ð4:8Þ
Finally, substituting the Eq. (4.8) in the Eq. (4.7) and solving the integrals, we obtain:
p
P
1 pX ð2rÞ! ½2ðs þ rÞ! s 2r ; n; k ¼ nk : 2 2 r;s¼0 24r r!2 22s ðs þ rÞ!2
ð4:9Þ
The final representation is not found in the literature, the usual form can be found for example in Carlitz [11] or Olver et al. [8].
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S.L. Bertoli et al. / Applied Mathematics and Computation 243 (2014) 33–43
The results obtained with Eq. (4.9) are compared with the results obtained from the expression found in Carlitz [11]: k
Pðp2 ; 0:5; kÞ
Pðp2 ; 0:5; kÞ
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
2.24810462594213 2.28335058819339 2.33674613731765 2.41367150420119 2.52390070844927 2.68680199682369 2.94787811582397 3.45910690021046
2.24810462594213 2.28335058819339 2.33674613731765 2.41367150420119 2.52390070844927 2.68680199682369 2.94787811582397 3.45910690021046 Carlitz [11]
5. Incomplete elliptic integral of the first kind Using the Bessel function method proposed in the present work, the incomplete elliptic integral of the first kind is expressed as:
Fðu; kÞ ¼
Z
1
ex dx
Z
0
u
I0 ðxk sin hÞdh:
ð5:1Þ
0
From Abramowitz and Stegun [10] we also can write:
I0 ðzÞ ¼
1 X
z2r
r¼0
2 ðr!Þ2 2r
ð5:2Þ
:
So replacing the Eq. (5.2) in the Eq. (5.1), we have:
Fðu; kÞ ¼
Z 2r 1 X k 2r
r¼0
2 r!2
1
ex x2 dx
Z
u
2r
sin hdh:
ð5:3Þ
0
0
Solving the improper integral, it follows,
Fðu; kÞ ¼
2r Z 1 X ð2rÞ!k 2r
r¼0
2 r!
2
u
2r
sin hdh:
ð5:4Þ
0
The integral of the sine function in Eq. (5.4) is represented by the relationship below from Zhao et al. [17]:
Z
u
2r
sin hdh ¼
0
#
" 2mþ1 r1 2m X 2 ðm!Þ2 sin u : cos u u ð2m þ 1Þ! 22r r!2 m¼0
ð2rÞ!
ð5:5Þ
Finally, the incomplete elliptic integral of the first kind is expressed as:
Fðu; kÞ ¼
1 X r¼0
r 2
ð2rÞ!k
22r r!2
" cos u
2mþ1 r1 2m X 2 m!2 sin u m¼0
ð2m þ 1Þ!
# u :
ð5:6Þ
The incomplete elliptic integral of the first kind was first solved by Qureshi and Chaudhary [18]. This result also shows the applicability of the proposed method to the incomplete elliptic integral. Qureshi and Chaudhary [18] presented a solution for the incomplete elliptic of the first kind in terms of series of hypergeometric functions. The results obtained with Eq. (5.6) are compared with the results obtained from the expression found in Qureshi and Chaudhary [18]:
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Fðp3 ; kÞ
Fðp3 ; kÞ
1.053430587029 1.061489706726 1.073313629047 1.089550670051 1.111233322932 1.140044752769 1.178902299538 1.233446325452
1.053430587029 1.061489706726 1.073313629047 1.089550670051 1.111233322932 1.140044752769 1.178902299538 1.233446325452 Qureshi and Chaudhary [18]
38
S.L. Bertoli et al. / Applied Mathematics and Computation 243 (2014) 33–43
6. Incomplete elliptic integral of the second kind Applying the proposed method in relating the incomplete elliptic integral of the second kind with the Bessel function, we have,
Z
2
Eðu; kÞ ¼ Fðu; kÞ k
1
ex dx
Z
0
u
2
sin hI0 ðkx sin hÞdh:
ð6:1Þ
0
After replacing the Bessel function in the series form in Eq. (6.1) and solving the improper integral, it follows the expression, 2
Eðu; kÞ ¼ Fðu; kÞ k
" # 2r Z 1 X ð2rÞ!k 2r
2 r!
r¼0
2
u
2rþ2
sin
ð6:2Þ
hdh:
0
The integral of the sine function was given by Eq. (5.5), then we can write,
Z
u
"
2rþ2
sin
0
ð2rÞ! ð2r þ 1Þ hdh ¼ 2r 2 ðrÞ!2 2ðr þ 1Þ
#"
2mþ1 r X 22m ðm!Þ2 sin u
cos u
ð2m þ 1Þ!
m¼0
#
u :
ð6:3Þ
Substituting the expression (6.3) in the Eq. (6.2), we obtain the final expression for the incomplete elliptic integral of the second kind:
Eðu; kÞ ¼
" r 2 1 X ð2rÞ!k 22r r!2
r¼0
!
# 2mþ1 2mþ1 r r1 2m ð2r þ 1Þ 2 X 22m ðm!Þ2 sin u X 2 ðm!Þ2 sin u ð2r þ 1Þ 2 þu 1 : cos u k k 2ðr þ 1Þ m¼0 2ðr þ 1Þ ð2m þ 1Þ! ð2m þ 1Þ! m¼0 ð6:4Þ
This representation for the incomplete elliptic integral of the second kind in terms of power series and trigonometric series is not found in the literature. Qureshi and Chaudhary [18] proposed an expression in terms of a series of hypergeometric functions for the integral analyzed above, as well as for the incomplete elliptic integral of first kind. The results obtained from Eq. (6.4) are compared with the expression found in Qureshi and Chaudhary [18]: k
Eðp3 ; kÞ
Eðp3 ; kÞ
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1.04566021970563 1.03322345140654 1.02213013278769 1.00755555514447 0.989221593549377 0.966723133094528 0.93945480372495 0.906456986263154
1.04566021970563 1.03322345140654 1.02213013278769 1.00755555514447 0.989221593549377 0.966723133094528 0.93945480372495 0.906456986263154 Qureshi and Chaudhary [18]
7. Incomplete elliptic integral of the third kind Applying the present Bessel function method, we have:
Z
Pðu; n; kÞ ¼
u
Z
dh 2
ð1 n sin hÞ
0
1
ex I0 ðkx sin hÞdx:
ð7:1Þ
0
It is also possible to express the last expression in the form:
Z
Pðu; n; kÞ ¼
u
dh
Z
0
or,
Z
Pðu; n; kÞ ¼
1
etð1n sin
The functions etn sin
Pð/; n; kÞ ¼
h
Z
hÞ
dt
Z
0
1
et dt
1
ex I0 ðkx sin hÞdx;
ð7:2Þ
0
Z
0 2
2
1
ex dx
Z
0
u
2
etn sin h I0 ðkx sin hÞdh:
ð7:3Þ
0
and I0 ðkx sin hÞ can be represented by series, then it follows, 1
et dt
0
Z
1
ex dx
0
Z 0
/
s 1 2r 2 1 X ðtn sin hÞ X ðkx sin hÞ dh : s! 22r r!2 s¼0 r¼0
ð7:4Þ
Or equivalently,
Pð/; n; kÞ ¼
2r 1 X k
ns 2r 2 s! r;s¼0 2 r!
Z 0
1
t s et dt
Z 0
1
x2r ex dx
Z 0
u
2ðsþrÞ
sin
hdh:
ð7:5Þ
39
S.L. Bertoli et al. / Applied Mathematics and Computation 243 (2014) 33–43
Solving the two improper integrals, we obtain the expression,
Pð/; n; kÞ ¼
Z 2r 1 X ð2rÞ!k ns 2r
2 r!
r;s¼0
2
u
2ðsþrÞ
sin
ð7:6Þ
hdh:
0
The integral in Eq. (7.6) is given by (see Eq. (5.5)):
Z
"
u
sin
2ðrþsÞ
hdh ¼
0
½2ðr þ sÞ!
#"
rþs1 X
cos u
22ðrþsÞ ðr þ sÞ!2
m¼0
2mþ1
22m ðm!Þ2 sin ð2m þ 1Þ!
u
# u :
ð7:7Þ
After replacing the previous expression in the Eq. (7.6), we obtain the following result:
" #" # 2r 1 rþs1 X X 22m ðm!Þ2 sin2mþ1 u ð2rÞ!k ns ½2ðr þ sÞ! cos u Pðu; n; kÞ ¼ u : 4r 2 ð2m þ 1Þ! 22s ðr þ sÞ!2 r;s¼0 2 r! m¼0
ð7:8Þ
This form for the incomplete elliptic integral of the third kind is not found in the literature. The usual representation is expressed in terms of a series of hypergeometric functions, or in terms of other series, for example by Karp et al. [19], using a partial fraction decomposition. The results obtained from Eq. (7.8) are compared with the expression found in [19] Karp et al. [19]:
k
Pðp3 ; 0; 8; kÞ
Pðp3 ; 0; 8; kÞ
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1.48453402981057 1.49858743111330 1.5192950589409 1.5479055930718 1.5864286332503 1.63818518991736 1.70903214209174 1.81054697605315
1.48453402981057 1.49858743111330 1.5192950589409 1.5479055930718 1.5864286332503 1.63818518991736 1.70903214209174 1.81054697605315 Karp et al. [19]
8. Hyperelliptic integrals The hyperelliptic integral or Abelian integral, in its general form is expressed as in Springer [12]:
Z
dx pffiffiffiffiffiffiffiffiffiffi ; QðxÞ
ð8:1Þ
where Q ðxÞ is a polynomial of degree > 4. In the following, it will be show how the method used anteriorly can also solve integrals of this class. Consider now the integral:
Z
dx pffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 1 xa
ð8:2Þ
This integral is considered a hyperelliptic integral for a > 4. It can be expressed in terms of a Bessel function,
Z
dx pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 1 xa
Z
dx
Z
1
a
I0 ðsx2 Þes ds:
ð8:3Þ
0
Replacing the Bessel function by its power series form in Eq. (8.3), we have:
Z
dx pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 1 xa
Z
1
es ds
Z X a 2r 1 ðsx2 Þ
0
r¼0
r!2
dx:
ð8:4Þ
Or,
Z
1 X dx 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2r 2 a 1x r¼0 2 r!
Z
1
es s2r ds
Z
xar dx:
ð8:5Þ
0
Finally, solving the two integrals in Eq. (8.5), it is obtained the following expression for this particular class of hyperelliptic integral,
Z
1 X dx ð2rÞ! xarþ1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ þ C: 2r 2 ðar þ 1Þ a 1x r¼0 2 r!
ð8:6Þ
40
S.L. Bertoli et al. / Applied Mathematics and Computation 243 (2014) 33–43
The integral in Eq. (8.1) is usually solved through the binomial series expansion. The next integral is a more general class of hyperelliptic integral:
Z
1
du pffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 1 a2
0
It can be expressed in the following form:
Z
1
du pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 1 a2
0
Z
Z
1
0
1
ex I0 ðaxÞdxdu;
ð8:7Þ
0
where,
a2 ¼
n Y
ð1 xi uÞ:
ð8:8Þ
i¼1
Following the proposed method:
Z
1 X du 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2r 2 2 r! 2 1a r¼0
1
0
Z
1
ex x2r dx
Z
0
1
a2r du:
ð8:9Þ
0
Solving the improper integral, it results:
Z
1 X du ð2rÞ! pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2r 2 2 1a r¼0 2 r!
1
0
Z
1
a2r du:
ð8:10Þ
0
Substituting the expression (8.8) in the Eq. (8.10), it shows:
Z
1 X du ð2rÞ! pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2r 2 2 2 r! 1a r¼0
1
0
Z
1 0
n Y
ð1 xi uÞr du:
ð8:11Þ
i¼1
Now,
Z
1
n Y
0
ð1 xi uÞr du ¼
i¼1
Cð2Þ ðnÞ F ð1; r; . . . ; r; 2; x1 ; . . . ; xn Þ; Cð1ÞCð1Þ D
ð8:12Þ
ðnÞ
where F D is a Lauricella function. Substituting the expression (8.12) in Eq. (8.11) the following result is obtained:
Z 0
1
1 X du ð2rÞ! ðnÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ F ð1; r; . . . ; r; 2; x1 ; . . . ; xn Þ: 2r 2 D 2 1a r¼0 2 r!
ð8:13Þ
This expression for the hyperelliptic integral in terms of Lauricella functions can also be solved by the binomial expansion, different from the way developed anteriorly. 9. Epstein–Hubbell integral The Epstein–Hubbell integral contains the parameters j and k, represented in the form [16,20]:
Xj ðkÞ ¼
Z p
2
ð1 k cos hÞ
j12
2
dhk < 1;
j ¼ 0; 1; 2; 3 . . .
ð9:1Þ
0
The solution of this integral is usually obtained through the binomial expansion. 2
ð1 k cos hÞ
j
¼
2n 1 X ðjÞn k cosn h j ¼ 1; 2; 3 . . . ; n! n¼0
ð9:2Þ
where ðjÞn is the Pochhammer symbol, which can be found in Gradshteyn and Ryzhik [16]. 9.1. Epstein–Hubbell integral for j ¼ 0 For the case of j ¼ 0, we have from the expression (9.4):
X0 ðkÞ ¼
Z p 0
dh qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 ð1 k cos hÞ
ð9:3Þ
It can be rewritten the integral (9.6) as it was done in terms of the modified Bessel function:
X0 ðkÞ ¼
Z
0
1
ex dx
Z p pffiffiffiffiffiffiffiffiffiffiffi Z 1 X ð2rÞ! 2r p I0 xk cos h dh ¼ k cosr hdh: 2 2r 0 0 r¼0 2 ðr!Þ
ð9:4Þ
S.L. Bertoli et al. / Applied Mathematics and Computation 243 (2014) 33–43
41
The integral of the cosine function in the last equation is represented in the form [21]:
Z p
r
cos hdh ¼
0
8 0; > >
> : p 2r
ð9:5Þ
: r!
ð2r Þ!2
; if r even
From Eq. (9.8) it is know that only the r even terms will contribute for X0 ðkÞ, then:
X0 ðkÞ ¼ p
1 X ð2rÞ! r¼0 r–odd
r 22r ðr!Þ2 2
r! 2r r 2 k : ! 2
ð9:6Þ
Now, making the following change r ! 2r:
X0 ðkÞ ¼ p
1 X
ð4rÞ!
r¼0
26r ð2rÞ!ðr!Þ2
4r
k :
ð9:7Þ
The final form given by the Eq. (9.10) can be found Srivastava [22]. 9.2. Epstein–Hubbell integral for j ¼ 1; 2; 3 . . . For the Epstein–Hubbell integral with argument j > 0 it follows the expression,
Xj ðkÞ ¼
Z p
2
ð1 k cos hÞ
j12
2
dhk < 1;
j ¼ 1; 2; 3 . . . :
ð9:8Þ
0
It can be represented in the following way,
Xj ðkÞ ¼
j Z p 2 ð1 k cos hÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dh: 2 0 ð1 k cos hÞ
ð9:9Þ
Relating it to the modified Bessel function of the first kind, it is obtained:
Xj ðkÞ ¼
Z
1
ex dx
Z p
pffiffiffiffiffiffiffiffiffiffiffi j 2 I0 ðxk cos hÞð1 k cos hÞ dh:
ð9:10Þ
0
0
Now, substituting the binomial expansion found in expression (9.13):
Xj ðkÞ ¼
2n 1 2r Z p 1 X ðjÞn k X ð2rÞ!k cosnþr hdh: n! r¼0 22r r!2 0 n¼0
ð9:11Þ
The integral of the cosine function is from Eq. (9.8):
Z p
cosrþn hdh ¼
0
8 0; > >
pðrþnÞ! > : 2rþn ; if n þ r ¼ even !2 ðrþn 2 Þ
ð9:12Þ
:
Now, following the same procedure as it was done for j ¼ 0, it results:
Xj ðkÞ ¼ p
1 X ðjÞn ð2rÞ! ðr þ nÞ! 2ðrþnÞ : 2 k n 3r 2 rþn n!2 2 r! ! n;r¼0 2
ð9:13Þ
nþr¼ev en
There are two conditions for ðn þ r) to be an even number: (a) when ðn; rÞ are even numbers; (b) when ðn; rÞ are odd numbers. For case (a) is made the following substitution in Eq. (9.16): r ! 2r and n ! 2n, which results:
Xj ðkÞ ¼ p
1 X
ðjÞ2n
n;r¼0 ð2nÞ!2
ð4rÞ! 2n
ð2r þ 2nÞ!
26r ð2rÞ!2 22n ðr þ nÞ!2
k
4ðrþnÞ
:
ð9:14Þ
For case (b) is made the following substitution in Eq. (9.16): r ! 2r þ 1 and n ! 2n þ 1, then we have:
Xj ðkÞ ¼ p
1 X ðjÞ2nþ1 ð4r þ 2Þ! ð2r þ 2n þ 2Þ! 4ðrþnþ1Þ k : 6r 2 2nþ4 ð2n þ 1Þ! 2 ð2r þ 1Þ! 2 ðr þ n þ 1Þ!2 n;r¼0
ð9:15Þ
Finally adding the expressions (9.14) and (9.15), it is obtained the solution for the Epstein–Hubbel integral:
" #" " ## 4ðrþnÞ 4 1 X ð4rÞ!ð2r þ 2nÞ!k ðjÞ2n 1 ðjÞ2nþ1 ð4r þ 1Þð2r þ 2n þ 1Þk : Xj ðkÞ ¼ p þ 2 2 6rþ2n ð2nÞ! 4 ð2n þ 1Þ! ð2r þ 1Þðr þ n þ 1Þ n;r¼0 ð2rÞ! ðr þ nÞ! 2
ð9:16Þ
42
S.L. Bertoli et al. / Applied Mathematics and Computation 243 (2014) 33–43
The expression (9.16) is the final form for the Epstein–Hubbell integral with argument j > 0. This form of representation is not found in the literature. The usual forms may be found, for example, in Al-Zamel and Kalla [23], Cengiz [21], Gradshteyn and Ryzhik [16], Kalla and Montanus [24], Kalla et al. [25], Kalla and Tuan [26], Srivastava [22] and Galué [5]. The comparison with the expression in Al-Zamel and Kalla [23], it follows: j
Xj ð0; 5Þ
Xj ð0; 5Þ
1 2 3
3.337745013 3.616446839 1.8105469760
3.337745013 3.616446839 1.8105469760 Al-Zamel and Kalla [23]
10. Conclusions This paper proposed a new method for solving elliptic integrals and some of its generalizations, based on the modified Bessel function of first kind and order zero. By the present method it was possible to obtain various solutions representations similar to those found in the literature and also some integral representations obtained in unusual ways. The proposed method suggests a broad applicability that may be more general than was exploited. Acknowledgments The authors are thankful to the Department of Chemical Engineering at the University of Blumenau. The authors also thanks to Marta H. Caetano and Luiz H. da Silva for the English revision of the manuscript. Appendix A. Alternative representation for the integral:
R p=2 0
2
sin hI 0 ðkx sin hÞdh
Starting from Abramowitz and Stegun [10], it is know that,
Z p
J 0 ð2z sin tÞ cos 2t dt ¼ pJ 21 ðzÞ;
0
ðA:1Þ
also, the following trigonometric identity is valid, 2
cos 2t ¼ 1 2 sin t;
ðA:2Þ
with the two last relations, it can be written,
Z p
J 0 ð2z sin tÞsen2 tdt ¼
0
Z p
1 2
0
J 0 ð2z sin tÞdt
p 2
J 21 ðzÞ:
ðA:3Þ
The functions sen2 t and J 0 ð2zsen tÞ are symmetrical to each other in the interval ½0; p=2 and ½p=2; p, then using this property, it can be established,
Z
p 2
0
J 0 ð2z sin tÞsen2 tdt ¼
1 2
Z
p 2
0
J 0 ð2z sin tÞdt
p 4
J 21 ðzÞ:
ðA:4Þ
Now making the following substitution z ¼ iz, we have from Eq. (A.4),
Z
p 2
2
J 0 ð2iz sin tÞ sin tdt ¼
0
¼
Z
1 2
p 2
2
I0 ð2z sin tÞ sin tdt ¼
0
Z
p 2
J 0 ð2iz sin tÞdt
0
1 2
Z
p 4
p 2
J 21 ðizÞ;
p
ðA:5Þ
J 21 ðizÞ:
ðA:6Þ
kx : J 21 i 2 4
ðA:7Þ
I0 ð2z sin tÞdt
0
4
Finally, substituting z ¼ kx=2, it results,
Z
p 2
I0 ðkx sin tÞsen2 tdt ¼
0
1 2
Z 0
p 2
I0 ðkx sin tÞdt
p
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