On the uniform asymptotic expansion of the Legendre functions. Nail R. Khusnutdinov∗ Department of Physics, Kazan State Pedagogical University, Mezhlauk 1, Kazan, 420021, Russia An uniform expansion of the Legendre functions of large indices are considered by using the WKB approach. We obtain the recurrent formula for the coefficients of uniform expansion and compare them with the uniform expansion of the Bessel function.
arXiv:math-ph/0302054v1 22 Feb 2003
PACS numbers: 02.30.Gp,02.30.Mv,04.62.+v
I.
INTRODUCTION
An uniform expansion of special functions is very useful representation of them which is used in many branches of science. It is well-known, for example, the Debay uniform expansion of the Bessel functions [1]. To obtain the uniform expansion one usually uses the complicate calculations which exploit an contour integral representation of function (see for example [4]). In this paper we use the WKB approach to obtain an uniform expansion for the Legendre functions. Previously, this question was analyzed by Thorne in Ref.[8] by using different approach and in Ref.[2] for particular case of the Legendre equation. We would like to note that this special case of calculations plays an important role in the so called functional methods which are at present the most powerful method (see [7]). The organization of this article is as follows. First of all, in Sec.II we reobtain the Debay formulas for the uniform expansion of Bessel function by using the WKB approach. In Sec.III, we apply the same method to the Legendre functions and their derivative. The Appendix contains the list of the first four coefficients in manifest form. II.
UNIFORM EXPANSION OF THE BESSEL FUNCTIONS
In this section we reobtain the well-known [1] uniform asymptotic expansion for the Bessel functions of second kind In (nλ) and Kn (nλ) for large value of n. These functions obey to the following differential equation 1 1 W ′′ + W ′ = n2 1 + 2 W, (1) λ λ where the prime is the derivative with respect λ. Let us represent the solution of above equation as a series over small value of 1/n: W = CenS−1 +S0
∞ X
n−k ωk
(2)
k=0
with ω0 = 1. Using this expression in Eq. (1) we obtain the chain of equations r 1 ′ (3a) S−1 = ε 1 + 2 , λ 1 ′ 1 ′′ , (3b) S−1 + S−1 S0′ = − ′ 2S−1 λ ′ ε λ(λ2 − 4) λωk′ ε ′ √ − ωk , (3c) ωk+1 = − 2 8 (1 + λ2 )5/2 1 + λ2 √ where ε = ±1 and k = 0, 1, . . . . With new variable t = 1/ 1 + λ2 , the last equation may be rewritten in more simple form · ε ε 2 ω˙ k+1 = t (1 − t2 )ω˙ k + (1 − 5t2 )ωk , (3d) 2 8 ∗ Electronic
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2 where the dot denotes the derivative with respect t. The first integral of the Eqs. (3) has the following form S−1 = ε(η(λ) + C−1 ), 1 S0 = − ln(1 + λ2 ) + C0 , 4 Z ε t ε 2 (1 − 5t′2 )ωk (t′ )dt′ + Ck+1 , ωk+1 = t (1 − t2 )ω˙ k + 2 8 0
(4)
where η(λ) =
p 1 + λ2 + ln
λ √ . 1 + 1 + λ2
(5)
To find the set of constants Ck , k = −1, 0, . . . we take the limit λ → ∞ in our expressions (2), (4) and compare them with well-known asymptotic formulas [1] r π −nλ 1 1 1 nλ e (1 + O( )). (6) In (nλ) ≈ √ e (1 + O( )) , Kn (nλ) ≈ λ 2nλ λ 2πnλ Because of the next term of expansion is O(1/λ) we have to set Ck = 0 for k ≥ 1. Taking this into account we have the following expression for uniform expansion in the limit λ → ∞: 1 W ≈ √ eεnλ CeεnC−1 +C0 . λ
(7)
Therefore, ε = 1 corresponds to the uniform expansion of In (nλ) and ε = −1 to the Kn (nλ).√For coincidence the expression (7) with the asymptotic expansions (6) we have to set C−1 = C0 = 0 and C = 1/ 2πn for ε = 1, and p C = π/2n for ε = −1. Therefore, we arrive at the following well-known formulas for uniform expansion of the Bessel functions r ∞ t nη(λ) X −k e n ωk (t), (8) In (nλ) = 2πn k=0 r ∞ πt −nη(λ) X Kn (nλ) = e (−n)−k ωk (t), 2n k=0
where ωk+1 =
1 1 2 t (1 − t2 )ω˙ k + 2 8
Z
t
0
(1 − 5t′2 )ωk (t′ )dt′ .
(9)
In order to find formulas for derivative of the Bessel functions we represent them in the form below ∞
X 1 ′ e nSe−1 +Se0 W = Ce n−k ω ek . n
(10)
k=0
Comparing the derivative of Eq. (2) with respect λ with above formula we obtain Se−1 = S−1 , ′ Se0 = S0 + ln(εS−1 ), e C = εC, ε ω ek = ωk + t(t2 − 1)ωk−1 + εt2 (t2 − 1)ω˙ k−1 . 2
(11)
Therefore, with these expressions we arrive at the well-known formulas for uniform expansion of the derivative of the Bessel functions ∞ 1 1 nη(λ) X −k 1 ′ In (nλ) = √ e n ω k (t), n 2πnt λ k=0
(12)
3 1 ′ K (nλ) = − n n
r
∞
π 1 −nη(λ) X e (−n)−k ω k (t), 2nt λ k=0
where 1 ω k = ωk + t(t2 − 1)ωk−1 + t2 (t2 − 1)ω˙ k−1 . 2 III.
(13)
UNIFORM EXPANSION OF THE LEGENDRE FUNCTIONS
In this section we employ the same approach for the Legendre functions. We consider the following equation n2 2 ′′ ′ 2 2 (1 − x )Ψ − 2xΨ − n γ + + 2ξ Ψ = 0, (14) 1 − x2 which has appeared in context of quantum field theory in curved space-time [5, 6]. Here x ∈ (−1, 1), n, γ and ξ are real numbers, and the prime is the derivative with respect x. The particular case of this equation for ξ = 1/8 has been considered in Ref. [2]. The solutions of this equation are the Legendre functions first and second kind: Pµn [x], Qnµ [x]
(15)
1 1p 1 − 8ξ − 4n2 γ 2 . µ=− + 2 2
(16)
with index
For ξ = 1/8 these functions are called the cone functions [1]. We assume n > 0 and consider the following two independent solutions pnµ [x] = Pµ−n [x],
(17)
n
(−1) (Qnµ [x] + Qn−µ−1 [x]) 2 π = − P n [−x]. 2 sin πµ µ
qµn [x] =
They are real functions for arbitrary µ and obey the following Wronskian condition W (pnµ , qµn ) =
1 . 1 − x2
To obtain the uniform expansion of functions (17) for large number n we represent the solution in the WKB form as below Ψ = CenS−1 (x)+S0 (x)
∞ X
n−k ψk (x)
(18)
k=0
with ψ0 (x) = 1. We would like to note the difference of the uniform expansion in form (18), which is over inverse degree of n, with that considered by Thorne in Ref.[8]. He obtained an expansion over inverse degree of µ + 1/2 = p 1 − 8ξ − 4n2 γ 2 /2. Substituting above expression in Eq. (14) we obtain the chain of equations s γ2 1 ′ S−1 = ε + , (1 − x2 )2 1 − x2 ′′ 2x 1 S−1 , (19) − S0′ = − ′ 2 S−1 1 − x2 2x 2ξ 1 ′ , S0′2 + S0′′ − S − ψ1′ = − ′ 2S−1 1 − x2 0 1 − x2
4 ′ ψk+1
ε = − 2
(
(1 − x2 )ψk′ p 1 + γ 2 (1 − x2 )
)′
+ ψ1′ ψk , k ≥ 1,
where ε = ±1. The first integral of this chain has the following form " γx S−1 (x) = ε γ arctan p 1 + γ 2 (1 − x2 )
(20)
# p 1 (1 + x)(1 + γ 2 (1 − x) + 1 + γ 2 (1 − x2 )) p + ln + C−1 2 (1 − x)(1 + γ 2 (1 + x) + 1 + γ 2 (1 − x2 ))
1 S0 (x) = − ln(1 + γ 2 (1 − x2 )) 4 1 − x2 ε ψ ′ (x) ψk+1 (x) = Ck+1 (ε) − p 2 1 + γ 2 (1 − x2 ) k " # Z x 2 2 2 − x′ 5x′ γ2 − + ε − 8 (1 + γ 2 (1 − x′ 2 ))3/2 (1 + γ 2 (1 − x′ 2 ))5/2 0 ξ + ψk (x′ )dx′ . 2 (1 + γ (1 − x′ 2 ))1/2 We have already set the constant C0 = 0. This leads to redefinition the constant C, only. The formulas look simpler in terms of new variable x , v=p 2 1 + γ (1 − x2 )
instead of x. This quantity obeys to inequality: |v| ≤ |x| < 1. In terms of this variable we have 1 1−v S−1 (v) = ε − ln + γ arctan γv + C−1 , 2 1+v 1 1 + γ 2 v2 S0 (v) = ln , 4 1 + γ2 ε (1 − v 2 )(1 + γ 2 v 2 ) ˙ ψk+1 (v) = Ck+1 − ψk (v) 2 (1 + γ 2 ) Z v 1 + γ2 1 εγ 2 ′2 ′ ψk (v ′ ). − 1 + (8ξ − 1) 5v + dv + 8(1 + γ 2 ) 0 γ2 γ 2 (1 + γ 2 v ′2 )
(21)
(22a) (22b)
(22c)
In above formulas the dot denotes the derivative with respect new variable v. In order to find constants Ck we have to compare our formulas with exact expressions for the Legendre functions at a fixed point. For this reason we take the limit x → 1 in our formulas −εn/2 1 1−x 2 exp nε C−1 − ln(γ + 1) + γ arctan γ (23) Ψ≈C 2 2 and compare them with well-known expressions [3] for the Legendre functions at point x = 1: n/2 1−x , 2 −n/2 (−1)n n (n − 1)! 1 − x qµn [x] = . (Qµ [x] + Qn−µ−1 [x]) ≈ 2 2 2 pnµ [x] = Pµ−n [x] ≈
1 n!
(24)
Therefore, from Eqs. (23), (24) we observe that ε = −1 corresponds to pnµ [x] with C = 1/n!, and ε = +1 corresponds to qµn [x] with C = (n − 1)!/2, and C−1 =
1 ln(1 + γ 2 ) − γ arctan γ 2
(25)
5 for both signs of ε. Furthermore, the coefficients ψk (v) must obey the following condition ψk (1) = 0.
(26)
Taking into account above formulas we arrive at the following expression for uniform expansion of the Legendre’s functions 1/4 ∞ X 1 1 + γ 2 v2 nS−1 (v) pnµ [x] = e n−k ψk (v) (27a) n! 1 + γ 2 k=0 1/4 ∞ X (n − 1)! 1 + γ 2 v 2 −nS−1 (v) e (−n)−k ψk (v), (27b) qµn [x] = 2 1 + γ2 k=0
where 1−v 1 ln − γ [arctan γv − arctan γ] , 2 (1 + v)(1 + γ 2 ) (1 − v 2 )(1 + γ 2 v 2 ) ˙ ψk (v) ψk+1 (v) = 2(1 + γ 2 ) Z v 1 1 + γ2 γ2 ′2 ′ ψk (v ′ ). 5v + − 1 + (8ξ − 1) dv − 8(1 + γ 2 ) 1 γ2 γ 2 (1 + γ 2 v ′2 ) S−1 (v) =
(27c) (27d)
Taking into account the same procedure as we used above for the derivative of the Bessel functions we obtain the following formulas for uniform expansion of the derivative of functions pnµ and qµn 3/4 ∞ 1 d n 1 + γ 2 nS−1 (v) X −k 1 1 + γ 2 v2 e pµ [x] = − n ψ k (v) n dx n! 1 + γ 2 1 − v2 k=0 3/4 ∞ 1 + γ 2 −nS−1 (v) X (n − 1)! 1 + γ 2 v 2 1 d n e qµ [x] = (−n)−k ψ k (v) n dx 2 1 + γ2 1 − v2
(28a) (28b)
k=0
ψ k (v) = ψk (v) −
(1 − v 2 )(1 + γ 2 v 2 ) ˙ γ 2 v(1 − v 2 ) ψk−1 (v). ψ (v) − k−1 2(1 + γ 2 ) 1 + γ2
(28c)
The first four coefficients ψk and ψ k are listed in Appendix. From the recurrent formula (27d) it is possible to find the value of the coefficients ψ(v) for γ → ∞. Indeed, comparing Eq. (27d) in the limit v → 0 and Eq. (9) in the limit t → 1 we obtain the following relation ψk (0) = (−1)k+1 ωk (1).
(29)
Now we represent formulas obtained in slightly different form which is close to expansion the Bessel functions. We set x = cos ǫ and γ = λ/ sin ǫ and use the asymptotic expansion for gamma function from Ref.[4] ∞
X 1 1 1 B2k ln n! = (n + ) ln n − n + ln 2π + , 2k−1 2 2 2k(2k − 1) n
1 1 ln(n − 1)! = (n − ) ln n − n + ln 2π + 2 2
k=1 ∞ X k=1
1 B2k , 2k(2k − 1) n2k−1
where Bk are the Bernoulli numbers. With these notations one has r
n sin ǫ = , λn k=0 r −n ∞ sin ǫ πt −n˜η X n , e (−n)−k ψk+ (v) qµ [cos ǫ] = 2n λn k=0 r n ∞ 1 n˜η X −k + 1 sin ǫ 1 dpnµ [x] , = − e n ψ k (v) n dx |x=cos ǫ 2πnt λn sin2 ǫ k=1 pnµ [cos ǫ]
∞
t n˜η X −k + e n ψk (v) 2πn
(30)
6
Dpm
Dp0
0.0005
0.01
m=3 0.005
1 1
2
4
3
5
l
2
3
4
5
l
m=2
-0.0005
-0.005 -0.01
-0.001
m=0
-0.015
-0.0015
-0.02
-0.002
m=1 n n n FIG. 1: The plot of the relative errors ∆pm = (pn µ − (pµ )m )/pµ versus of λ for ǫ = 0.1, ξ = 0 and n = 4. Here (pµ )m is the uniform expansion of the Legendre function pn up to degree m. µ
1 dqµn [x] = n dx |x=cos ǫ
r
∞
π −n˜η X + e (−n)−k ψ k (v) 2nt k=1
sin ǫ λn
n
1 , sin2 ǫ
where λ sin ǫ λ tan ǫ − arctan + 1, η˜ = ln √ − arctan λ λt 1 + λ2 + cos ǫ sin ǫ 1 , v = t cos ǫ, t = √ 1 + λ2 r 1 1 4n2 λ2 µ = − + 1 − 8ξ − , 2 2 sin2 ǫ
(31a) (31b) (31c)
and the coefficients ψk+ are found from relation ∞ X
k=0
n−k ψk+ (v) = exp −
∞ X
k=1
B2k 2k(2k − 1)n2k−1
!
∞ X
n−k ψk (v)
(31d)
k=0
by comparing the same degree of n in the left and right hand sides. The expressions (30) have the form similar to that for the Bessel functions expansion given by Eq. (8). Furthermore, it is easy to see that in the limit ǫ → 0 (argument of the Legendre functions tends to unit and lower index tends to infinity) the uniform expansion obtained is transformed to the uniform expansion of the Bessel functions below lim µn pnµ [cos ǫ] = in In (nλ),
ǫ→0
(32)
lim µ−n qµn [cos ǫ] = i−n Kn (nλ)
ǫ→0
as it should be according with well-known formulas [3] h xi lim z n Pz−n cos = Jn (x), z→∞ zi h π x = − Yn (x), lim z n Q−n cos z z→∞ z 2
(33)
where x = inλ and z = inλ/ǫ. In this limit the function η˜ given by Eq. (31a) coincides with function η (5) in the uniform expansion of Bessel functions: p λ lim ηe = ln √ + 1 + λ2 . 2 ǫ→0 1+λ +1
(34)
The numerical calculation of the relative errors ∆pm = (pnµ − (pnµ )m )/pnµ and ∆qm = (qµn − (qµn )m )/qµn are plotted in Fig. 1 and Fig. 2 for different m = 0, 1, 2, 3 as function λ, where (pnµ )m and (qµn )m are the uniform expansions of the Legendre functions pnµ and qµn up to degree n−m . The difference is smaller the greater λ.
7
Dq0
Dqm m=3
0.0005
0.02 0.015
1
m=0
0.01
-0.0005
2
3
4
5
l
m=2
0.005 -0.001
1
2
3
4
5
l -0.0015
-0.005
m=1 -0.002
FIG. 2: The plot of the relative errors ∆qm = (qµn − (qµn )m )/qµn versus of λ for ǫ = 0.1, ξ = 0 and n = 4. Here (qµn )m is the uniform expansion of the Legendre function qµn up to degree m.
In conclusion we would like to summarize the results. In this paper we obtain thepuniform expansion for the Legendre functions pnµ [x] and qµn [x] given by Eq. (17) for large indices n and µ = − 21 + 12 1 − 8ξ − 4n2 γ 2 as a series over inverse degree on n. These expansions of the functions are given by Eq. (27) and by Eq. (28) for their derivatives with respect of argument x. The coefficients of expansion may be found from recurrent chain of equations (27d) and (28c). The first four coefficients are listed in Appendix. Acknowledgments
The author would like to thank Dr. M. Bordag for stimulation of this work and for reading this manuscript. The work was supported by part the Russian Foundation for Basic Research grant N 02-02-17177. APPENDIX: MANIFEST FORM OF FIRST FOUR COEFFICIENTS.
Below are the expressions for first four coefficients ψk and ψ k in which we introduced for simplicity the following notations: x 1 δ = arctan[γ] − arctan[γv], ζ = ξ − , v = p . 2 8 1 + γ (1 − x2 ) ψ0 = 1, 2 δζ 2γ + 3 v(γ 2 − 1) 5v 3 γ 2 1 ψ1 = , + 2 + − γ γ +1 24 8 24 2 2 ζ(−1 + v 2 ) δζ 1 2γ + 3 v(γ 2 − 1) 5v 3 γ 2 1 δζ + + + − ψ2 = 2 2 γ γ γ +1 24 8 24 2(γ 2 + 1) 4 1 4γ + 84γ 2 − 63 v(γ 2 − 1)(2γ 2 + 3) v 2 (9γ 4 − 58γ 2 + 9) + + + (γ 2 + 1)2 1152 192 128 4 2 2 6 4 3 2 2 5v γ (2γ + 3) 77v γ (γ − 1) 385v γ , − + − 576 192 1152 3 2 2 1 δζ δζ 1 δζ 1 2γ + 3 v(γ 2 − 1) 5v 3 γ 2 1 ψ3 = + + + − 2 2 6 γ 2 γ γ +1 24 8 24 γ (γ + 1)2 4 4γ + 84γ 2 − 63 v(γ 2 − 1)(2γ 2 + 3) v 2 (9γ 4 − 58γ 2 + 9) 5v 3 γ 2 (2γ 2 + 3) + + − × 1152 192 128 576 4 2 2 2 2 2 6 4 77v γ (γ − 1) 385v γ ζ 2γ + 1 v ζ (1 − v) − + 2 − + 2 2 + + 192 1152 γ +1 2γ 2 2 2γ (γ + 1)
(A.1)
(A.2)
8 + + + − +
2γ 2 + 7 v(3γ 2 − 11) v 2 (2γ 2 + 3) v 3 (44γ 2 − 29) 35v 5 γ 2 ζ − − + + − (γ 2 + 1)2 48 16 48 48 48 6 4 2 6 4 2 1 1112γ + 1116γ − 918γ + 5265 v(4γ + 728γ − 4323γ + 711) + − (γ 2 + 1)3 414720 9216 2 2 4 2 3 6 4 v (2γ + 3)(9γ − 58γ + 9) v (2005γ − 37671γ + 37566γ 2 − 2025) + 3072 27648 77v 4 (γ 2 − 1)(2γ 2 + 3) 13v 5 γ 2 (1053γ 4 − 3706γ 2 + 1053) 385v 6 γ 4 (2γ 2 + 3) − + 4608 15360 27648 17017v 7γ 4 (γ 2 − 1) 85085v 9γ 6 − 9216 82944 ψ0+ = ψ0 = 1, 1 ψ1+ = ψ1 − , 12 1 + ψ2 = ψ2 − ψ1 + 12 1 + ψ3 = ψ3 − ψ2 + 12
(A.3)
1 , 288 1 139 ψ1 + . 288 51840
ψ 0 = 1, 2 δζ 2γ + 3 v(3γ 2 + 1) 7v 3 γ 2 1 ψ1 = + 2 − + , γ γ +1 24 8 24 2 2 1 δζ ζ(1 − v 2 ) δζ 1 2γ + 3 v(3γ 2 + 1) 7v 3 γ 2 ψ2 = + + − + 2 2 γ γ γ +1 24 8 24 2(γ 2 + 1) 1 (2γ 2 − 27)(2γ 2 − 3) v(3γ 2 + 1)(2γ 2 + 3) v 2 (15γ 4 − 62γ 2 + 7) + − − 2 2 (γ + 1) 1152 192 128 3 2 2 4 2 2 6 4 7v γ (2γ + 3) v γ (99γ − 79) 455v γ + , + − 576 192 1152 3 2 2 1 δζ δζ 1 δζ 1 2γ + 3 v(3γ 2 + 1) 7v 3 γ 2 1 ψ3 = + + − + 2 2 6 γ 2 γ γ +1 24 8 24 γ (γ + 1)2 2 2 2 2 2 4 2 (2γ − 3)(2γ − 27) v(3γ + 1)(2γ + 3) v (15γ − 62γ + 7) 7v 3 γ 2 (2γ 2 + 3) × − − + 1152 192 128 576 v 4 γ 2 (99γ 2 − 79) 455v 6 γ 4 ζ ζ 2 (1 − v) 1 v2 + − 2 + 2 2 + − 192 1152 γ + 1 2γ 2 2 2γ (γ + 1) 2 2 2 2 3 2 2γ − 1 v(3γ − 7) v (2γ + 3) v (44γ − 25) 35v 5 γ 2 ζ + − − + + (γ 2 + 1)2 48 16 48 48 48 6 4 2 6 4 1112γ + 5436γ + 1242γ − 1215 v(12γ + 904γ − 4281γ 2 + 585) 1 − − + 2 3 (γ + 1) 414720 9216 2 2 4 2 3 6 4 2 v (2γ + 3)(15γ − 62γ + 7) v (2807γ − 42897γ + 37458γ − 1863) − − 3072 27648 v 4 (99γ 2 − 79)(2γ 2 + 3) 11v 5 γ 2 (1521γ 4 − 4762γ 2 + 1241) 455v 6 γ 4 (2γ 2 + 3) + − + 4608 15360 27648 385v 7 γ 4 (51γ 2 − 47) 95095v 9γ 6 − + 9216 82944
(A.4)
9
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