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On asymptotic representations of spheroidal functions
ON ASYMPTOTIC REPRESENTATIONS OF SPHEROIDAL FUNCTT.ONS* L. I.
PONOMAREV Dubna
(Received
MANYproblems theory, etc.
of mathematical lead to the
[d)
5 February
physics equations
1966)
(quantum mechanics, electromagnetic for spheroidal functions
$w)g+ [ -$I-VI; +[ x2(1--nZ)+h---
m2
x”(p-l)-A--
where ~~ is a given parameter Liouville problem for equation
5” -
m2
1 -Tp
and 74 is the (1’) in the
The functions X(e) and Y(q)eieq parts of the solution and “angular”
1
1
x=
0,
(1 ‘1
1Y=0.
eigenvalue interval
(I”)
of the Sturm [- 1, 11.
are sometimes also called of the Helmholtz equation
the
-
“radial”
A$ + k2$ = 0 In particular, in spheroidal coordinates. written in prolate ellipsoidal coordinates
equations (1’) as follows:
and (1”)
are
(1”‘) E=---,
ri +
r2
rl=
R
where ‘l*
r2
R is are
the the
--
R
distance distances
R2
ri - r2
X2
’
Zh.
u5;chisZ.
Mat.
k2
-
4
’
$(ri, r2, 9; R) = NX(t.) Y(q)@ms,
between the foci of the coordinate ellipsoid and between the faci and the point of observation.
The solutions of equations known special functions which *
=
mat.
Fiz.
(1) are satisfy 7,
1,
268
the natural generalization of the equations (1) with u2 = 0. 196 - 198,
1967.
On asymptotic
representat
ions
of
spheroidal
funct
269
ions
for the For example, equation (1”) with K2 = 0 becomes the equation although when ~~ # 0 the iunCtiOns Y(q) associated Legendre functions, do not belong to the class of well-studied special functions. The reason for this is the lack of integral relations containing the functions X(t) and Y(q), the impossibility of obtaining explicit expressions for h in terms of ~~ and the number of zeros of Y(q) applicable in the whole table for region of change of K~, and also the absenoe of exhaustive these quantities. (Even the tables in [21 and [31. which are very large, cannot completely solve this problem. ) Under these conditions it Is especially important to have different approxfmation formulae for the functions X(c), Y(q) and the eigenvalues A. Such expressions are given in [d for the limiting cases ~~ -4 0, K2 -*a and also in [41 as K2 -+ 0 and h + co. In [51 there are asymptotic formulae for the functions X(e) and Y(q) when n, the number of zeros, is This approximation corresponds to the WKR method [61 and leads to large. the following result (we shall only consider equation (1”) from now on):
(2)
in
the
region
where
Q2(~)
> 0;
c
Y (Ii) = 21(1 -VI
in
the
region
where
Q2(q)
(3)
lQ(W I
< 0; m2
Q”(q)=
Formulae (2) and (3) are f a, i.e. from the roots
x2+&-
applicable Q2(q)
of
(4)
(1 - q.92 *
a long way from the “turn = 0 lying in [- 1, 11.
points”
Below we obtain a transcendental equation from which it is possible determine the approximate values h = hn,n+m(~2) h] , [31 for n >> 1, i.e. in the region of values where the construction of tables is very corresponding to the WKB difficult. These * puant izat ion conditions” method have the form
to
s
Q(q)dq= n(n + l/z),
(5)
--n
where
n
is
the
number
of zeros
of Y(q).
and
a
is
the
least
positive
L. I.
270
root
of Q(q) A simple
Pononarev
= 0 (a < 1). calculation
gives
2x
eb {kzK(k)-(l where
E(k),
K(k),
- a2)[K(k)-‘(6‘
II(a2,
k)
are
the
k)]}
- l)lI(az,
total
=
elliptic
n(n
+ ‘/2).
integrals
(6)
defined
in
[71, a=
C $(A [ I+
b = k =
I+
+%;(h
IA2 + 4m2x2) I”,
+ VA2 + 4m2x2)
a/b.
‘h 1,
(When using the tables 181 it must tie remembered that the definition of the third elliptic Integral n(q, n, k) used there is not the same as formulae 3.167 given in Id that for fl(q, a 2, k) in [Al. III particular are false: the substitution n -+ - n = a2 must be made.) The normalization of the functions can be calculated just as easily. For (2) we find c2 = wb/K(k), for the function of (1”‘) in normalization on 6(k - k’) 411 xb N3=_.---_.
1
K(k)
2n
nR
(6) is applicable for any ~2. TTl t!le li7it CZSf)S ‘W?C?l K2--+‘I RIlfi can easily obtain from it expressions for A. For example, in of the elliptic integral E(k), K(k) the case m = 0, using the properties [Gl we obtain Equation
K2
+
cn we
as
a., = (We note the stgn
that when m = 0, of h. Since a2=
‘as, x2+ 00.
‘(zn+1,2-;(2n+i)“+8
-u2+u(2n+1)-
k2 is defined
h-IA1 if------, 2x2
x2+0,
in different
b2=1+-----
n+lJ-l 2x2
ways.
depending
’
we have ,l$ = -
X2 X2+ h
for
h>O,
X2+ h k2 = Y .X2
for
li<
0.
(71
(8) on
On asymptotic
representat
ions
of
spheroidal
functions
271
Although we assumed in deriving (6) that n >> 1,the resulting expressions (7) and (8) give a good approximation for h, right down to n = 1. The exact formulae for A,(&., = A,,,) have the form [II
h,
n(n+i)-;
e
c I--
nZ(n -
1
‘(n+1)2(n+2)2
I)2
- 3) (2n - 1)3(2n+l) 1,
=
-_xz+x(2n+l)-
-1x2+
(2n - 1) (Qn + 3)
-(2n
+-[(2n+1)2+5]
-
+ 1) (2n + 3)3(2n+5)
I
as
x2-+-O,
as
x2+
xl
~(2n+i)1(2n+11)‘+1Ll~
00.
x
It must be stressed, however, that it is not must be inserted into (2) and (3) but the “less since it is the latter which ensure the correct totic solutions as q -4 f 1 and q + 0 [51.
these expressions which accurate” (7) and (8). behaviour of the asymp-
To estimate the accuracy of these formulae it is necessary to make a numerical solution of the boundary problem (1). It has been shown [loI in the similar problem for ~~ < 0 that the error of the method for any n does not exceed 5 per cent over the whole range of change of K~. Therefore in the numerical fractions, say, [21 equation approximations. Acknowledgements.
Gershtein
for
his
solution of equations (6) can be used as the
(l), using continued source of good initial
To conclude, I wish to express my gratitude unfailing stimulating interest and discussions. Translated
by
to 8.5.
R. Feinstein
REFERENCES 1.
and SCHAFKE. F.W. Mathieusche ionen, Springer. Berlin, 1054.
YEIXEER, J. funkt
2.
FLAMMER, K. Tables of spheroidal functions funktsii). VTs AN SSSR, 1062.
3.
STRATTON, J.A., F.J. Spheroidal
CHU, L.J.. wave
Funktionen
(Tablitsy
und
Sph&oid-
sferoidal’nykh
MORSE, P.M., LITTLE, I.D.C. and CORBATO. Chapman and Hall, London, 1056.
functions.
272
L. I.
Pnonarev
4.
KARMAZINA, L.M. The asymptote of spheroidal wave functions. Computat iona 1 nathemat its (Vychislitel’ naya matematika) 72 - 79, Izd-vo AN 8898, Moscow, 1959.
5.
PONOMAREV,L. I., Application of the WKB method in the asymptotic solution of equations. Dokl. Akad. Nauk SSSR, 162, 5, 1023 - 1026, 1965.
6.
ZOMMERFEL’D, A. The stoma 1 spektry).
7.
BYRD, P.F. engineers
8.
structure
Vol.11,
of the atom and the 6ostekhiadat. Moscow,
products
Fizmatgiz,
spectrum
(Stroeniye
1956.
and, FRIEDMAN, M.D. Handbook of elliptic integrals and physicists, Springer, Berlin, 1954.
QRADSHTEIN, N.S. and
In Symp. No. 5,
and RYZHIK, I.M. (Tablitsy integralov Moscow, 1962.
and EMDE, F. Tables Moscow, 1959.
0.
YANKE, Ye. Fizmatgia,
10.
OERSHTEIN. 8.8.. approximation 3, 632 - 643,
of
PONOHAREV,L. I. in the two-centre 1965.
Tables
of
integrals,
summ, ryadov
functions
sums,
for
series
i proixvedenil),
(Tablitsy
funktsii),
and PUZYNINA, T. P. Quasiclassical problem. Zh. e’ksp. teor. Fiz.
48,
PONOMAREV Dubna
(Received
MANYproblems theory, etc.
of mathematical lead to the
[d)
5 February
physics equations
1966)
(quantum mechanics, electromagnetic for spheroidal functions
$w)g+ [ -$I-VI; +[ x2(1--nZ)+h---
m2
x”(p-l)-A--
where ~~ is a given parameter Liouville problem for equation
5” -
m2
1 -Tp
and 74 is the (1’) in the
The functions X(e) and Y(q)eieq parts of the solution and “angular”
1
1
x=
0,
(1 ‘1
1Y=0.
eigenvalue interval
(I”)
of the Sturm [- 1, 11.
are sometimes also called of the Helmholtz equation
the
-
“radial”
A$ + k2$ = 0 In particular, in spheroidal coordinates. written in prolate ellipsoidal coordinates
equations (1’) as follows:
and (1”)
are
(1”‘) E=---,
ri +
r2
rl=
R
where ‘l*
r2
R is are
the the
--
R
distance distances
R2
ri - r2
X2
’
Zh.
u5;chisZ.
Mat.
k2
-
4
’
$(ri, r2, 9; R) = NX(t.) Y(q)@ms,
between the foci of the coordinate ellipsoid and between the faci and the point of observation.
The solutions of equations known special functions which *
=
mat.
Fiz.
(1) are satisfy 7,
1,
268
the natural generalization of the equations (1) with u2 = 0. 196 - 198,
1967.
On asymptotic
representat
ions
of
spheroidal
funct
269
ions
for the For example, equation (1”) with K2 = 0 becomes the equation although when ~~ # 0 the iunCtiOns Y(q) associated Legendre functions, do not belong to the class of well-studied special functions. The reason for this is the lack of integral relations containing the functions X(t) and Y(q), the impossibility of obtaining explicit expressions for h in terms of ~~ and the number of zeros of Y(q) applicable in the whole table for region of change of K~, and also the absenoe of exhaustive these quantities. (Even the tables in [21 and [31. which are very large, cannot completely solve this problem. ) Under these conditions it Is especially important to have different approxfmation formulae for the functions X(c), Y(q) and the eigenvalues A. Such expressions are given in [d for the limiting cases ~~ -4 0, K2 -*a and also in [41 as K2 -+ 0 and h + co. In [51 there are asymptotic formulae for the functions X(e) and Y(q) when n, the number of zeros, is This approximation corresponds to the WKR method [61 and leads to large. the following result (we shall only consider equation (1”) from now on):
(2)
in
the
region
where
Q2(~)
> 0;
c
Y (Ii) = 21(1 -VI
in
the
region
where
Q2(q)
(3)
lQ(W I
< 0; m2
Q”(q)=
Formulae (2) and (3) are f a, i.e. from the roots
x2+&-
applicable Q2(q)
of
(4)
(1 - q.92 *
a long way from the “turn = 0 lying in [- 1, 11.
points”
Below we obtain a transcendental equation from which it is possible determine the approximate values h = hn,n+m(~2) h] , [31 for n >> 1, i.e. in the region of values where the construction of tables is very corresponding to the WKB difficult. These * puant izat ion conditions” method have the form
to
s
Q(q)dq= n(n + l/z),
(5)
--n
where
n
is
the
number
of zeros
of Y(q).
and
a
is
the
least
positive
L. I.
270
root
of Q(q) A simple
Pononarev
= 0 (a < 1). calculation
gives
2x
eb {kzK(k)-(l where
E(k),
K(k),
- a2)[K(k)-‘(6‘
II(a2,
k)
are
the
k)]}
- l)lI(az,
total
=
elliptic
n(n
+ ‘/2).
integrals
(6)
defined
in
[71, a=
C $(A [ I+
b = k =
I+
+%;(h
IA2 + 4m2x2) I”,
+ VA2 + 4m2x2)
a/b.
‘h 1,
(When using the tables 181 it must tie remembered that the definition of the third elliptic Integral n(q, n, k) used there is not the same as formulae 3.167 given in Id that for fl(q, a 2, k) in [Al. III particular are false: the substitution n -+ - n = a2 must be made.) The normalization of the functions can be calculated just as easily. For (2) we find c2 = wb/K(k), for the function of (1”‘) in normalization on 6(k - k’) 411 xb N3=_.---_.
1
K(k)
2n
nR
(6) is applicable for any ~2. TTl t!le li7it CZSf)S ‘W?C?l K2--+‘I RIlfi can easily obtain from it expressions for A. For example, in of the elliptic integral E(k), K(k) the case m = 0, using the properties [Gl we obtain Equation
K2
+
cn we
as
a., = (We note the stgn
that when m = 0, of h. Since a2=
‘as, x2+ 00.
‘(zn+1,2-;(2n+i)“+8
-u2+u(2n+1)-
k2 is defined
h-IA1 if------, 2x2
x2+0,
in different
b2=1+-----
n+lJ-l 2x2
ways.
depending
’
we have ,l$ = -
X2 X2+ h
for
h>O,
X2+ h k2 = Y .X2
for
li<
0.
(71
(8) on
On asymptotic
representat
ions
of
spheroidal
functions
271
Although we assumed in deriving (6) that n >> 1,the resulting expressions (7) and (8) give a good approximation for h, right down to n = 1. The exact formulae for A,(&., = A,,,) have the form [II
h,
n(n+i)-;
e
c I--
nZ(n -
1
‘(n+1)2(n+2)2
I)2
- 3) (2n - 1)3(2n+l) 1,
=
-_xz+x(2n+l)-
-1x2+
(2n - 1) (Qn + 3)
-(2n
+-[(2n+1)2+5]
-
+ 1) (2n + 3)3(2n+5)
I
as
x2-+-O,
as
x2+
xl
~(2n+i)1(2n+11)‘+1Ll~
00.
x
It must be stressed, however, that it is not must be inserted into (2) and (3) but the “less since it is the latter which ensure the correct totic solutions as q -4 f 1 and q + 0 [51.
these expressions which accurate” (7) and (8). behaviour of the asymp-
To estimate the accuracy of these formulae it is necessary to make a numerical solution of the boundary problem (1). It has been shown [loI in the similar problem for ~~ < 0 that the error of the method for any n does not exceed 5 per cent over the whole range of change of K~. Therefore in the numerical fractions, say, [21 equation approximations. Acknowledgements.
Gershtein
for
his
solution of equations (6) can be used as the
(l), using continued source of good initial
To conclude, I wish to express my gratitude unfailing stimulating interest and discussions. Translated
by
to 8.5.
R. Feinstein
REFERENCES 1.
and SCHAFKE. F.W. Mathieusche ionen, Springer. Berlin, 1054.
YEIXEER, J. funkt
2.
FLAMMER, K. Tables of spheroidal functions funktsii). VTs AN SSSR, 1062.
3.
STRATTON, J.A., F.J. Spheroidal
CHU, L.J.. wave
Funktionen
(Tablitsy
und
Sph&oid-
sferoidal’nykh
MORSE, P.M., LITTLE, I.D.C. and CORBATO. Chapman and Hall, London, 1056.
functions.
272
L. I.
Pnonarev
4.
KARMAZINA, L.M. The asymptote of spheroidal wave functions. Computat iona 1 nathemat its (Vychislitel’ naya matematika) 72 - 79, Izd-vo AN 8898, Moscow, 1959.
5.
PONOMAREV,L. I., Application of the WKB method in the asymptotic solution of equations. Dokl. Akad. Nauk SSSR, 162, 5, 1023 - 1026, 1965.
6.
ZOMMERFEL’D, A. The stoma 1 spektry).
7.
BYRD, P.F. engineers
8.
structure
Vol.11,
of the atom and the 6ostekhiadat. Moscow,
products
Fizmatgiz,
spectrum
(Stroeniye
1956.
and, FRIEDMAN, M.D. Handbook of elliptic integrals and physicists, Springer, Berlin, 1954.
QRADSHTEIN, N.S. and
In Symp. No. 5,
and RYZHIK, I.M. (Tablitsy integralov Moscow, 1962.
and EMDE, F. Tables Moscow, 1959.
0.
YANKE, Ye. Fizmatgia,
10.
OERSHTEIN. 8.8.. approximation 3, 632 - 643,
of
PONOHAREV,L. I. in the two-centre 1965.
Tables
of
integrals,
summ, ryadov
functions
sums,
for
series
i proixvedenil),
(Tablitsy
funktsii),
and PUZYNINA, T. P. Quasiclassical problem. Zh. e’ksp. teor. Fiz.
48,