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Calculation of the logarithmic derivative of associated Legendre functions
CALCULATION OF THE LOGARITHMIC DERIVATIVE OF ASSOCIATED LEGENDRE FUNCTIONS* A. D. LIZAREV Gomel (Received 19 July 1972) (Revised version 25 January 1973) EXACT expressions are obtained
for the logarithmic
derivatives of the associated Legendre
functions F,“(cos 8) = (d/de) [In Pnm(cos 0) I for the case n=--‘/z+iz products, each factor of which Bk< 1. It is known that an exact solution
of the boundary
and 0=x/2 as infinite
value problems of potential
theory for domains bounded by second-order surfaces, for example, problems of statistics and of the dynamics of spherical envelopes, can be obtained in associated Legendre functions P,” (cos 0). In the calculation of a Legendre function with a complex subscript n= --‘/z+ir difficulties may arise in the range of large values of the parameter r, since in this case the hypergeometric series for the Legendre functions and their derivatives converge extremely
slowly, and the terms of the series take very large values.
However, in the determination of the frequencies of oscillation of spherical envelopes it is sufficient to calculate not the Legendre functions and their derivatives themselves, but the logarithmic
f
derivatives
[~P,~COS
e)
I=
(d/de)P,m (COS e) ~~~~~~~ e) ’
In what follows we will write
f
[h
~~~~~~~~ 8) I = F,”
(~0sa).
We will consider the important case O=n/2, where the calculation of the logarithmic derivative Fern (0) is possible without the use of hypergeometric series. The logarithmic *Zh. vjkhisl. Mat. mat. Fiz., 13, 6, 1588-1591, 1973.
265
A. D. Lizarev
266
derivatives Fnm(0) can be used, for example, to determine by an exact method the frequencies of the natural oscillations of hemispherical envelopes, and also to estimate the accuracy of various approximate methods of the dynamic calculation of spherical envelopes. We determine the logarithmic derivative F,” (0), by using the expressions for the associated Legendre function and its first derivative in terms of the Gamma function [ 1] : r(‘/,+(n+m)/z)
Pnrn(O) = 2”
(n+m)aT cos
2
r(l+(n-m)/2)r(‘/2)
JY(l+(n+m)/2)
dPnm(o) - _‘p+* a0
where
dp,,m (0) /dEl
1
sin r(v,+(n-m)/2)r(vz)
(n+m)n
2
is the value of dPnm (cos Cl)/de for8=3-t/2.
Writing (n+m)/2=a,
(n-m)/2=b,
Fam(0)=
we
-2
obtain
r(l+a)r(l+b) r(‘/z+a)I’(Vz+b)
tg na.
(1)
Using the known relation for the Gamma function r(z)r (1-z)=n/sin Euler’s formula, we can represent expression (1) in the form
F,?“(O)= lr(l+a)r(l+b)* n2
(+--a)
r (f-b)
IXZ
and
chnt.
In further transformations of expression (2) we must distinguish the cases where the order m of the function Pnm(0) is odd or even. We will consider the case of odd m=2q+l, q=o, 1,2,. . . . Applying repeatedly to each of the Gamma functions in expression (2) the recurrence formula r(z+1)=zr(z),we obtain
Fnm(0) = -
JQP
_. A
r2(1+n/2)rz('/z-n/2)chn~ mr II TiI-i
where A
= m
IzJl+~w-l) Ipl+(m-l)(m-2)'
We now use the infinite product [2]
p = n(n+l)= -(++0.25).
267
Short communications
(I--) (I+$)(I-;)
(4+q
r(l+n,2;l,2_n,2) - C3)
*a*=
Substituting in the left side of this equation n= -0.5+iz, we notice that each of the factors in the brackets is a complex number, but the product of two adjacent factors is a real positive number:
(I-+)(I+&)=I+--&-.
(4)
We also represent the hyperbolic cosine as an infinite product [3] : co
IH
chnt=C]p]
4lPl-1 l-t---(2kc1)2
(5)
I[
k=1,3,5,...
We introduce the notation
Taking into account the relations (3) - (S), we finally obtain for odd m 9+i
~,m(o)=$~Am( jj_ Bk)-‘. k=i,3,5,...
m==l
After similar transformations we can find the logarithmic derivative Fnm (0)inthe case of even m also
F,m (0) = -
“‘,“I f-jA_fi
Bh.
h=i,3,5,...
m=2
Finally, if m=O, we have F” (0) = -
4bl 3-l
m II
Bk
h=f,3,5,...
We will mention some features of the calculation of the logarithmic derivatives ~,,m(0) ,defmed by the products (6) - (8).
(6)
A. D. Lizurev
268
1. The products n A, and n A, are independent of the number k, and the co m-1 m=2 Bk is independent of the order m, which considerably simplifies the product II k=1,3,5,...
tabulation of the functions F,,” (0) for various values of m. For the same degrees n and different values of m the functions F,” (0) can be calculated by the recurrence formula (0) = Am+zF,,"'(O). F,m+2
2. For any lpl and any number inequalities 4lPl-1
-<(2lc+i)2
kal
each factor Bhcl,
IPI
4lPl--1
IPI
(2k+3)2
k(k+l)
-<-----. and
k(k+l)
which follows from the
As k+ m the value of the factor asymptotically approaches unity.
3. When Ip I is increased the number of factors Bk in the infinite product
II
Bk,
h=l,S,S,...
required for the calculation of the logarithmic derivative Fnm(0) with a specified degree of accuracy, increases, so that for very large values of IpI it is more convenient to use asymptotic expressions for Fnm (0). It can be shown that for large Ip I the following expression holds: 0
l-I
at
Bk w--_ 42
(9)
h=i,S,5,..,
This result agrees with the well-known asymptotic expression for the Legendre function for large r: P-y,+i% (cos 0) =
F,(O)
ere ,‘(2nz
sin 0) *
(10)
Indeed, if 8=3-c/2, it follows from (10) that F_‘h+ir (0) -r.Substituting this value of into the left side of (8) and solving the resulting equation for the product
oy Bk, we
II k=1,3,5....
obtain expression (9).
Using Eqs. (6) and (7), expression (9) and taking into account the remainder terms, we find the following asymptotic expressions for the logarithmic derivatives:
Short communications
269
for odd m
for even m m/2
l-I
A,
Fnn(0)=T
--&.
l?L=2
We give the asymptotic expressions for the logarithmic derivatives F,“”(0) for m=O, 1, 2, 3:
(11) 1 1 F1 ‘A+ir(O)‘Y z + + -, $1” 4t 8~ 1 = z +- $/a ’ 4t2+1
F:t,&O)
FS
128 %+ir
(0)
(12)
m Z +
$
+--+-. 9
z
4?+9
(13) 1
(14)
I?/2
The values of F, (0)) calculated by Eq. (8) on the Minsk-22 computer, were verified for values ~~25 by tables [4,5] . The values of Fsm (0)) found by the asymptotic relations (11) - (14) were compared with the values found by the exact expressions (6) - (8). It was then found that the asymptotic relations give 5 correct figures of the mantissa for 2>50, 6 correct figures for ~>I30 and 7 correct figures for 2>340. Translated by
J. Berry
REFERENCES 1.
HOBSON, E. W., l?reory of spherical and ellipsoidalharmonics (Teoriya sfericheskikh i ellipsoidal’nykh funktsii), Izd-vo in. lit., Moscow, 1952.
2.
BATEMAN, H., and ERDELYI, A., Higher transcendentalfkctions. Hypergeometric functions. Legendre finctions. (Visshie transtsendentnye funktsii. Gipergeometricheskaya funktsii, Funktsii Lezhandra), “Nauka”, Moscow, 1965.
3.
GRADSHTEIN, I. S., and RYZHIK, I. M., Tables of integrals, sums, series and products (Tablitsy integralov, summ, ryadov i proizvedenii), Fizmatgiz, Moscow, 1963.
270
A. D. Lizarev
4.
ZHURINA, M. I., and KARMAZINA, L. N., Tables ofthe Legendrefunctions P_I,,+~,(s). (Tablitsy funktsii Lezhandra P- l/n+ir(z ), Izd-vo Aka. Nauk SSSR, Moscow, 1960.
5.
ZHURINA, M. I., and KARMAZINA, L. N., Tables of the Legendre functions P1_-‘/2+ir(Z). (Tablitsy funktsii Lezhandra P’_1/,+ir(x ), VTS Akad. Nauk SSSR, MOSCOW, 1963.
for the logarithmic
derivatives of the associated Legendre
functions F,“(cos 8) = (d/de) [In Pnm(cos 0) I for the case n=--‘/z+iz products, each factor of which Bk< 1. It is known that an exact solution
of the boundary
and 0=x/2 as infinite
value problems of potential
theory for domains bounded by second-order surfaces, for example, problems of statistics and of the dynamics of spherical envelopes, can be obtained in associated Legendre functions P,” (cos 0). In the calculation of a Legendre function with a complex subscript n= --‘/z+ir difficulties may arise in the range of large values of the parameter r, since in this case the hypergeometric series for the Legendre functions and their derivatives converge extremely
slowly, and the terms of the series take very large values.
However, in the determination of the frequencies of oscillation of spherical envelopes it is sufficient to calculate not the Legendre functions and their derivatives themselves, but the logarithmic
f
derivatives
[~P,~COS
e)
I=
(d/de)P,m (COS e) ~~~~~~~ e) ’
In what follows we will write
f
[h
~~~~~~~~ 8) I = F,”
(~0sa).
We will consider the important case O=n/2, where the calculation of the logarithmic derivative Fern (0) is possible without the use of hypergeometric series. The logarithmic *Zh. vjkhisl. Mat. mat. Fiz., 13, 6, 1588-1591, 1973.
265
A. D. Lizarev
266
derivatives Fnm(0) can be used, for example, to determine by an exact method the frequencies of the natural oscillations of hemispherical envelopes, and also to estimate the accuracy of various approximate methods of the dynamic calculation of spherical envelopes. We determine the logarithmic derivative F,” (0), by using the expressions for the associated Legendre function and its first derivative in terms of the Gamma function [ 1] : r(‘/,+(n+m)/z)
Pnrn(O) = 2”
(n+m)aT cos
2
r(l+(n-m)/2)r(‘/2)
JY(l+(n+m)/2)
dPnm(o) - _‘p+* a0
where
dp,,m (0) /dEl
1
sin r(v,+(n-m)/2)r(vz)
(n+m)n
2
is the value of dPnm (cos Cl)/de for8=3-t/2.
Writing (n+m)/2=a,
(n-m)/2=b,
Fam(0)=
we
-2
obtain
r(l+a)r(l+b) r(‘/z+a)I’(Vz+b)
tg na.
(1)
Using the known relation for the Gamma function r(z)r (1-z)=n/sin Euler’s formula, we can represent expression (1) in the form
F,?“(O)= lr(l+a)r(l+b)* n2
(+--a)
r (f-b)
IXZ
and
chnt.
In further transformations of expression (2) we must distinguish the cases where the order m of the function Pnm(0) is odd or even. We will consider the case of odd m=2q+l, q=o, 1,2,. . . . Applying repeatedly to each of the Gamma functions in expression (2) the recurrence formula r(z+1)=zr(z),we obtain
Fnm(0) = -
JQP
_. A
r2(1+n/2)rz('/z-n/2)chn~ mr II TiI-i
where A
= m
IzJl+~w-l) Ipl+(m-l)(m-2)'
We now use the infinite product [2]
p = n(n+l)= -(++0.25).
267
Short communications
(I--) (I+$)(I-;)
(4+q
r(l+n,2;l,2_n,2) - C3)
*a*=
Substituting in the left side of this equation n= -0.5+iz, we notice that each of the factors in the brackets is a complex number, but the product of two adjacent factors is a real positive number:
(I-+)(I+&)=I+--&-.
(4)
We also represent the hyperbolic cosine as an infinite product [3] : co
IH
chnt=C]p]
4lPl-1 l-t---(2kc1)2
(5)
I[
k=1,3,5,...
We introduce the notation
Taking into account the relations (3) - (S), we finally obtain for odd m 9+i
~,m(o)=$~Am( jj_ Bk)-‘. k=i,3,5,...
m==l
After similar transformations we can find the logarithmic derivative Fnm (0)inthe case of even m also
F,m (0) = -
“‘,“I f-jA_fi
Bh.
h=i,3,5,...
m=2
Finally, if m=O, we have F” (0) = -
4bl 3-l
m II
Bk
h=f,3,5,...
We will mention some features of the calculation of the logarithmic derivatives ~,,m(0) ,defmed by the products (6) - (8).
(6)
A. D. Lizurev
268
1. The products n A, and n A, are independent of the number k, and the co m-1 m=2 Bk is independent of the order m, which considerably simplifies the product II k=1,3,5,...
tabulation of the functions F,,” (0) for various values of m. For the same degrees n and different values of m the functions F,” (0) can be calculated by the recurrence formula (0) = Am+zF,,"'(O). F,m+2
2. For any lpl and any number inequalities 4lPl-1
-<(2lc+i)2
kal
each factor Bhcl,
IPI
4lPl--1
IPI
(2k+3)2
k(k+l)
-<-----. and
k(k+l)
which follows from the
As k+ m the value of the factor asymptotically approaches unity.
3. When Ip I is increased the number of factors Bk in the infinite product
II
Bk,
h=l,S,S,...
required for the calculation of the logarithmic derivative Fnm(0) with a specified degree of accuracy, increases, so that for very large values of IpI it is more convenient to use asymptotic expressions for Fnm (0). It can be shown that for large Ip I the following expression holds: 0
l-I
at
Bk w--_ 42
(9)
h=i,S,5,..,
This result agrees with the well-known asymptotic expression for the Legendre function for large r: P-y,+i% (cos 0) =
F,(O)
ere ,‘(2nz
sin 0) *
(10)
Indeed, if 8=3-c/2, it follows from (10) that F_‘h+ir (0) -r.Substituting this value of into the left side of (8) and solving the resulting equation for the product
oy Bk, we
II k=1,3,5....
obtain expression (9).
Using Eqs. (6) and (7), expression (9) and taking into account the remainder terms, we find the following asymptotic expressions for the logarithmic derivatives:
Short communications
269
for odd m
for even m m/2
l-I
A,
Fnn(0)=T
--&.
l?L=2
We give the asymptotic expressions for the logarithmic derivatives F,“”(0) for m=O, 1, 2, 3:
(11) 1 1 F1 ‘A+ir(O)‘Y z + + -, $1” 4t 8~ 1 = z +- $/a ’ 4t2+1
F:t,&O)
FS
128 %+ir
(0)
(12)
m Z +
$
+--+-. 9
z
4?+9
(13) 1
(14)
I?/2
The values of F, (0)) calculated by Eq. (8) on the Minsk-22 computer, were verified for values ~~25 by tables [4,5] . The values of Fsm (0)) found by the asymptotic relations (11) - (14) were compared with the values found by the exact expressions (6) - (8). It was then found that the asymptotic relations give 5 correct figures of the mantissa for 2>50, 6 correct figures for ~>I30 and 7 correct figures for 2>340. Translated by
J. Berry
REFERENCES 1.
HOBSON, E. W., l?reory of spherical and ellipsoidalharmonics (Teoriya sfericheskikh i ellipsoidal’nykh funktsii), Izd-vo in. lit., Moscow, 1952.
2.
BATEMAN, H., and ERDELYI, A., Higher transcendentalfkctions. Hypergeometric functions. Legendre finctions. (Visshie transtsendentnye funktsii. Gipergeometricheskaya funktsii, Funktsii Lezhandra), “Nauka”, Moscow, 1965.
3.
GRADSHTEIN, I. S., and RYZHIK, I. M., Tables of integrals, sums, series and products (Tablitsy integralov, summ, ryadov i proizvedenii), Fizmatgiz, Moscow, 1963.
270
A. D. Lizarev
4.
ZHURINA, M. I., and KARMAZINA, L. N., Tables ofthe Legendrefunctions P_I,,+~,(s). (Tablitsy funktsii Lezhandra P- l/n+ir(z ), Izd-vo Aka. Nauk SSSR, Moscow, 1960.
5.
ZHURINA, M. I., and KARMAZINA, L. N., Tables of the Legendre functions P1_-‘/2+ir(Z). (Tablitsy funktsii Lezhandra P’_1/,+ir(x ), VTS Akad. Nauk SSSR, MOSCOW, 1963.