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Double chebyshev expansions for wave functions
Computer Physics Communications 12 (1976) 125— 134 © North-Holland Publishing Company
DOUBLE CHEBYSHEV EXPANSIONS FOR WAVE FUNCTIONS yB. SHEOREY Physical Research Laboratory, Narrangpura, Ah,nedahad-380 009, India Received 26 May 1976
A double Chebyshev expansion technique is suggested for evaluating positive energy solutions of the radial Coulomb equation. It is shown that, for a particle with a wide range of energies and in either an attractive or a repulsive Coulomb field, the two linearly independent radial solutions can he rapidly evaluated.
I. Introduction
2. The radial Coulomb equation
In a previous paper [11we discussed a method for evaluating Chebyshev expansion coefficients in terms of which the two linearly independent positive energy solutions of the radial Schrbdinger equation for an electron irs a Coulomb field could be expressed. It was shown that, for a fixed value of the particle energy and of the charge, the s-wave regular and irregular solutions could be evaluated to a very high accuracy for the entire range of the independent variable. The solutions br other pattial waves could be obtained from recursion relations. It was pointed out that once these two solutions were evaluated everywhere the continuum Green’s function could also be constructed. These functions are required for solving various problems in atomic physics such as photoionization, charged particle impact excitation and ionization of ions. in such applications the continuum Coulomb functions or the Green’s function are generally required for several values of the particle energy. We propose a double Chebyshev expansion in terms of which the radial s-wave regular and irregular solutions. for a wide range of particle energies and for both attractive and repulsive Coulomb potentials, may be expressed. In this paper we follow the notation of [11and use the results obtained therein.
The radial Schrhdinger equation for a particle in a Coulomb field may be written as [21 2
~
1~
+ 1 -p (1) where us a non negative integer and denotes the angular momentum; t~= z/k, where z is the charge (z <0 for an attractive potential); k, the wave number and k2 is numerically equal to the energy in rydbergs. The
H
Ll~2
1(1 + 1) ~
solution, F(r~IIp),and the irregular solution G(r~lJp), are determined by their asymptotic behaviour ~~@ilIp)
sin 01 ;
G(r~lIp)
p
cos 01,
(2)
p
where 01
=
~ in 2p
--
~lir
+
~
(3)
,
and where 01, the Coulomb phase shift equals arg F(l + 1 + in). As in [11 we consider solutions of eq. (1) in two regions, namely, the inner region, 0
,
(4)
tlB. S/iyoi-c’t’ / Douhic L~he/,~‘s/,yy
1 26
where C(i
7, /)is the normalization. It was shown in ]j that the function h(s~l~ p) could be expanded in terms of shifted Chebyshev polynomials of order n, T,1 . In particular, for /
=
0, we have
A
~A,,(77)T~(p/p0),
=
(5)
nO
where we have now explicitly written the expansion coefficients as functions of 77 and the prime on the summation symbol indicates, here and elsewhere, that the coefficient of the r = 0 term is to be halved. Consider the range of r~ as - 1 <17 < I, then, for any particular value of n, if function A0(77) is continuous and of bounded variation in this interval, it may be expanded as
As before, then. A
n0,,, T(77).
=
‘l’(ril = OIp)
F
~o
~,,,,,T,0(77)T~p/p0)
in= 0
I
2.2. Solutions in the outer region in ill the solutions of eq. (1) in the outer region, Po
+
iF = e’~I)(l~p),
(6)
(12)
A
{A,,(77)
~
N
W,,(77)~T~(p0/p).
A
R,,(77) FFO
a,,,,,T,,,(77)T,,(p/p0).
(7)
X
,i=0 1,1—0
The irregular solution of eq. (1), G(77l~p), may be cx-
[cos
e0(77) + i sin O~(77)1T,~(p0/p) 2
F(~l~p)In 2p
+
Q(771)1
and
=
[A~(77)+ B~(77)]’/
=
tan
(15)
C7~(~ 0)
+D(77l)p~’(77l~p), where Q(pl) is a known function =
(14)
,
where
pressed as
D(pI)
(13)
Tins equation may be written as
M
=
+
0
n
(I)(77l~p)
=
(1])
The expansion coefficients for derivatives of 1 and sji with respect to p are easily obtained using eq. (6) of
~1)(77lI p) =
G(7711p)
.
--0
Thus, from a knowledge of the coefficients a,,,,,, the coefficients A0(77), and hence the function rI~(i7,Ot,o) can rapidly be evaluated for any ~iin the interval. The function r1 can then be written as
(1)(77tp)
.11
=
and the function (I)(77l~p) was expanded as
ti F?!
for nary Jiictioiis
expa/isiolls
-
——-~
(8) of
p and of / and
.
We had shown that the function ‘1J(pl~p) could be cx-
punded as a convergent Chehyshev series, namely, A
B,,(p)T~(p/p0).
0(77)
1 {B 11(77)/A,,(77)~.
(16)
It is evident from eq. (14) that the real and imaginary parts of the expansion coefficients, as functions of p, have a sinusoidal modulation and consequently their Chebyshev expansion would be expected to converge
(2/ + 1) C(p/)
‘I’(pllp) = ~
e
(9)
slowly. There is also one other difficulty. It is clear from eqs. (1), (12) and (13) that for the case of / = 0 and p = 0 the value ofA0 = 2 and A0 = 0 for n >0. The imaginary part, B,, = 0 for n >0. Thus the function e,, for n >0 is indeterminate. In fact, the phase
n0
Agair each of the coefficients B,,() may be expressed as a Chebyshev series in the variable 17. u =
,fl
0
a,,,,, T(p).
(10)
function (—),, has a finite discontinuity of absolute magnitude ~r radians at p = 0. This is evident from the differeutial equation satisfied by b(pl~p)(cf, eqs. (37). (38) and (39) of [11). In fig. I we give a plot of phase function in the interval - 1
=
1. We
FIB. Sheorey
/ Double chebysher
expansions for ,yave jimcrions
127
I
panded as N
f(x) =
3,
~
anTr,~x),
(19)
where N
3.
-
6.1
a
0
=
N1~0 f(cos ~i/~ cos(~nf /~
where the double prime on the stimmation symbol in-
-
dicates that the! = 0 and the
I
N terms are to be
halved. The error, ha0, in a0 is given by
1
Fig. I. Phase function O,,(~)(radians) for n = I and P0 = 8, = 0. Note that scales are different for the ranges --1 © © 0 and 0< ,~< 1.
therefore obtain Chebyshev series expansions for the two functions R11(p) and e0(77) in two ranges of p, namely, —1
R0(p)
(20)
(17)
CnmTm(p)
=
ha0 = a2N
fl
+ a2N+,, + a4N - ,~+ a4N+fl +
(21)
and which is negligible ifNis sufficiently large. In our calculation we have fitted the functions defined by the equations mentioned above to a high accuracy. For most values of it, the expansions in p give the expansion coefficients in eqs. (5), (9) and (14) correct to eight significant digits. For the last few values of n, that is, for n = N, ii = N~ I ,etc. the coefficients are correct to six significant digits. However, the cxpansion coefficients for these values of n are themselves very small and this decrease in accuracy does
- 0
not lead to any significant differences in the final resuit.
and in’
0,,,nTm(P).
=
(18)
0
4. Results
Thus the regular and irregular solutions of eq. (1), for any p in the ranges mentioned above, may be evaluated from a knowledge of the expansion coefficients for the functions R0(p) and The expansion enefficients for the derivative of 1 with respect to p are obtained as before.
e0(77).
3. Numerical method The calculation of Chebyshev expansion coefficients for the functions defined by eqs. (6), (10), (17) and (18) is straightforward [3]. The method is based on the use of the orthogonal property of summation of the Chebyshev polynomials. If the range is
In the set of tables 1 --6 we give Chebyshev cxpansion coefficients for functions A0(p), B0(p), R0(p) and O,,(p) defined by eqs. (6), (10), (17) and (18), respectively. Tables 1 and 2 are for expansion coefficients of functions A,,(p) and B~,(p), respectively, and are valid for any p in the range --1
JIB. Sheorer / Doubt,’ Chebv,vhc,’ e.vpans,ons Jb- ware junctions
128 Table I
o
Expansion coefficients o,,,,, of eq. (6), here labeled C(M) for
n~1 -
-
—1 ~?39Q430
Y —us
10
i~
:l~~
~
11
-1.i?819,9o1~Qv
11
F N
)J
=
=
1
F~ 8
1 /
1,0, 11’
I) 1 2 1 4
711—0)?
-1 . 100010111 <1 —o.5,031uo90—l,t -~ . O4700 ((IN—i —,,.5o’uo.-uo 4.017/9o46o—l,3 5.107)120070—,/3
5
r7
1,215037I140—,l S.5)I1109’P26l,—
3
1.119110231—04
0
04 91076O’1)Su~l5 0.(iS499olI,—1j1,
“
—
10
o.15163107t,-l,(
7
11
5.41,15)11530—00
I’ 10
17
3.057(35090—UP
11~ 12
13 13 1.
10.1?57050’i—))O 3.3,37,6,1?I—I)9
5=
2
l-OPIINFNI
1’
NZ
I
F?P0Nf?.T~ —1
6
(10)
II
0
—R.95500601P’U? —50)0,51 lOll—Ill —4.55501134,I—ul 2.ll’73570,’l’
O
—4,3760 111-42 —5,17925407—1)!
4 5
?.44761’l’,’N[I II) 5.30432)1071) 1)11 1,1~1,1S9.1u00 ‘Ill —4.92326510))_lh1 - 1. ~ 0.411I51)l~O1
3)11)
I)
1
1 7 S
—~
0(11)
Ii 1 2
4.7~43l1)41 —u? 8.4,9),0”1,—114
‘1
F0600F,,To
F’
/
1.1F.1)1532’,)—hl’l 9.?SS55(,Si-sl
1.1739o185’
—5.54563971—09
‘~10
FXPOOFF.l=
1
—4
—0.000005040—u( 9.4755072l1)I-03
U 1
—4.892s9895”—u1
7
-0. 1l)14937701)’.
2 5
1)) 11
_4.793p/130l~fl1 2.?35011)1’,l’Ol 7.0S50044I.U—3? 3.l)6700174102 —6.010625891o—OC
12
2.?459’727109
l.316560,740—U’) ~• 11434444)) 4—1)1
TALFIF 1—ti
4
N
FXP)ll,FN1
Ii
F’
C)??)
0 1
9.532911100-02 6.73171)5740—02 1.731094160—01 5.828596570—02 2.3361030’11’—t3 —2.547493030—03 —6.7628091110—04 —8.070604790—05 —3.953168640—06 2.787049550—07 7.409427620-OFF 8.0095384611—09 5.979872840—10
3 4 0 7 0 ‘5 10 11 12
6
F0PIINFNT
6=
H
—1
0 1 2 3 4 5 6 7 8 9 10 11 12
07
Cl??) o
1 2 3 4 5 7
5
—1.923101825—01 —3,990737981—02 —2.1)1)690541 u—01 —7. 40431 951))— 02 1 .158u66600—0? 6. 3168u(1?51’—0? 710075124’,l1—04 —9.038015040—09
—R.1(23556o-02
4 S 7
7
9 8 1))
F0PU6EN1~
F)
—R.638509311—94 —~.105O1)9u74l0~O5
‘5
1,) 11
—F.
((RI -1.696426100
00
—1.61)pooo#up Oh —1.194876)51) 1.3~.7oooolo 01) lu 0.101l((ll510—u1 —4.619034fl0-u,’
—0.ôol3309?4—uo -1.6906u7315-l’3 4,071931100—05 2.34553’.410—0’. 1.902706110—hJp
11 12
—N.104974u111—ur
1?
—0,
0=11
Cl’))
6
12 1 1
1~
1.5101o00 —04 —5.3609010P11-’07 6.1)) —8.?18153761—uO
2.56711761-04 4.81 444/6411—1)5 5’100?141611—lJ0 5,31,~,~535ir_~7 3,,10397?49’o—uO
1)) 11
1. F,
1.OO92))31’~u5 330/p—O1)
4179)33-u,
1.1 1194055)1-1,4 /,21’0040190—(15 /.55393505l,-l)6 0.1170317071)—)l(
S2
1.3)913551 —51 5.01.04l117(i0? —2.SOoplpS’. ‘—01
N
7
‘1
1) 1
1.9278op3’ ~—o1 1.2723613’.) OIl 5l456u11—01 S.211
5
P
F’ 00
1,
0
1—111
1 1 P1) N 1. 1
I.
1 .1,100030,i ))~ 0.11(0105,11—01 2. ?oi? (1,4/Il-fl 9.1(0411,1—u? 4.7i106,’4,l:1—0?
-s.119355220-l0
1—I TOOl F
o P
.51641)1)40-u,)
1?
~
TORI F
9. 71c959420—u7
:~:~:~
—1410554600—lI1, 551’,36(?v—lu
)XPUN(-l,T
—5
0
(0.,)
1) 1
5.57462’.4?l)—11 0.429355lllO-U1
2
3.9747u4(50-Il1
S
—4.61 7060560—01
4 5 6
—1 .9?(ilu160<0 2.931497010-110
7
1.267617060—42 1.R94040510—04
3,71045777)1—06
0 9
—2.11)7850710—04 —1.831115~60—35
8.4S7105905—ij7 4.59047003—59
1)) 11
3.699610750—07 1.39131722ll—07
12 1—is
9.2113(9390—oN
EXP0NEN1~ —1
TABlE
C(M) -4.371090800-01 —7. 1 FF869 6490—01 —2.41)1741810—01 2.043083090—01 8.069878060—02 8. 345443o80—03 —9.717786950—04 —3.523225870—04 —4.056133370—05 —6.438346420—06 1 .633947870-08 1.750094650—06 1.913255636—09
FXP0NF6T~ —0
~12
99000FNT
-6
CO.?)
1.34191146111 0)) 2. o594/ b?’— 01 1 208305521 0)1 73029 11 —01 1’ 1332233’ —01 6~0031004303-03 3:8236155Nll~O1 1.323983501—04
I) 1 2 3 4 5 6 7 13
—1.0943720100 00 —3. 374706400—1(1 —6 9159990.0—uT 1 .1 o3023o00 OIl 5 007169(00—01 —1 1)06115370—01 301075(00-02 8.824103670—04 10.890800190—04
—3.984106211—05 —3.82642022~—06 1,16210024u—09 1.854140100—08 1.1791Q59))1—uN
9 10 11 12 13
4.175212710—05 —5.045646770—06 —6.349257090—07 —7.498172690—09 2.033923780—09
641’901105111—01
9 1)3 11 12 13
FOPUN6NTO -7
F’
0 1 ? 4 5 6 7 3
6013
C)??)
—
CO.’) 0 1 2 3 4 N 6 7
S.43586704D01 4.252667750—01 0.31 081 7110—0! —2 . 3 320 75 1)00—01 —8.705820s10—u2 7.018834550—06 3•u56?44064—IJ5 4.104354400—04
N~14 F’ 0 1
EXPONENTO —7 CO..?) —2.639151920—01 —6.410579750—02
N~15
E0P0NENT~ —0
F’
CC??)
0 1
2.230149320—01 1.266412010—00
FIB. Sheorey / Double chebysher expansions for ware junctions 2 3
—2.2328(8560—01 —1 .357724420—01 6.419723790—02 2.467142516—02 —2.558252500—03 —1.091209975—83 —3.075635850—06 1.6383705100—05 9.1035198)0—07 —7.0~87~~9o008 —8.594181945—09
4
5 6 7 8 0
10 11 12
2 3 4 5 6 7 8 9 10 11 1? 13
1.490980150—01 —2.131053910—01 —9.574569030—06 2.296693320—02 8.474378310—03 —4.517450690—04 —2.383333910—04 —4.487987040—06 2.446347930—06 1.5259054ti11-07 —6.561425850—69 —8.563703600—10
TABLE 1—V l’,16
E0PIDNENT
—9
4—17
EXPOFdENT—10
II
CO.?)
H
CO.?)
0
3.915102920—01 1.15329M011D—01 3.140491290—01 2.145449970—01 —I .1076166:SD—01 —4.306491500—02 S.83625~os0—03 2.2124(6950—03 —5.071576730—05
0 1 2 3 4 S 6 7 8
—2.9105417640—01 3.254106460—02 —1.914560130—01 2.941523070—01 1 .4112761480—01 —3.~o947833D—02 —1.383941460—02 1.081168440—05 4.533805110—04
1 3 4 5 F,
7 13 9 11l
—4.1724217s0—05 —1.234303930—06 45OS2ioIS—O?
9 10 11
—1.527746120—66 —5.983673850—06 —2.240435330—07
11 12
3.OI 2.0721094s5—08
12 13 14
3.1780684511-00 1.524643490—09 —6.230661 580—10
N18
EXPONE4T—11
)4~1c
C(M)
(1
5 1, 7
0 1 2 3 4 5 6 7
10
1.503761161=04
10
1 2 3 4
C(F’(
F’
—4. 5o851)/960—01 —1.606770961’—Ol —3.46316250)l—O1 —2.673592441—01 1.474976130—01 5.9169960711—42 —9.SuO?5710i’—OS —3.4521396555—03
O
9 10
7. 570520134))— (15 7.4131006900—07
9 10
11 12
—7 - 1 982o6300—0 7 —2.59365705o—0l1
11 12
EXPLlNENT~—12
3.873391,60—01 —4.241111100—06 7.499147640—01 —2.612701460—01 —1.87(000960—01 3.456315(00—00 1.929469970—06 —9.1115531610—06 —6.7109036550—06 —6. 33561)5090—06 9.905(l((520—0o
2
—1.316337500—07
Table 2 Expansion coefficients i3iirn ofeq. (10), here labeled C(M) lO,p~ 8.O,—1~T)~I —
0
EXPONE4T~
0
for
C(M)
0 1 2 3 4 S 6 7 8 9 10 11 12 13 14
1.257894920 01 1.250382570 01 8.897288650 00 6.009628620 00 3.183397150 00 9.065242090—01 9.087610760—02 —1.982463730—02 —8.902221240—03 —1 .707600610—03 —2.169151850—04 —2.047384420—OS —1.518130940—06 —9.169438090—08 —4.603893040—09
1
CCII)
H
O 1 2 3 4 5 6 7 8
4.01971 2930 00 4.517129320 00 1.90721 5960 00 1.045745430 00 8.370769490—02 —5.238068210—01 —3.566611770—01 —1.169399680—01 —2. 420601620—02
9
.3.534320680—03
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 13
N. 4
‘3.87832547004 —3.335703400—05 —2.317871820—06 —1.331189840—07 —6.450434690—09
EXPONENT.
0
CUlt
M
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
—3.942320090 00 —1.005614990 00 —7.025337740—02 7.48206093D—01 1.432373970 00 7.586192580—01 1.819840810—01 1.922966110—02 —6.884676660—04 —5,516046000—04 —9.642124510—05 —1.066989160—OS —8.736808050—07 —5.650338170—08 —2.990661380—09
EXPONENT—
0
N. 5
EXPONENT.
0
F’
CUll
H
CCII)
10 2 3 4 5 6 7 8 9 10
—5.68441665D 00 —6.07077440000 —3.090021170 00 —2.083539240 00 5281633840—o1 1.219446100—01 8.918916150—02 1.949693050—02 2.001517460—03 2.848069700’OS —2. 211922940—05
10 2 3 ~ 5 6 7 8 9 10
1.305628880—01 —1.14325535000 —6.827893580—01 —6.656295920—01 —5.498834250—01 —1,41330548D—O1 —4.405318430—04 6.11018991D—03 1.303436920—03 1.284617190—04 4.213666610—06
11 12 13 14
—3.800381530—06 —3.801593940—07 —2.778182300—08 —1.600962530—09
11 12 13 14
—5.963731660—07 —1.103907620—07 —1.026924670—08 —6.8491 39920—10
EXPONENT— —1
N
N
6
F’
CU?)
0 1 2 3 4
5
1
EXPONENT
—1
CCII)
10 11 12 13 14
10 11 12 13 14
N. 3
TABLE 2—Il
6 7 8
2—1 N
0
F’
o
4
FXPOP~T
7
EXPONENT. —1
H
CCII)
6.55726862000 9.349862910 00 2.577304910 00 2.480915480 00 —2. 254386695—01 —6.452428160—01
0 1 2 3 4 5
4.629640760—01 1.925767450 00 1.186857470 00 8.236200050—01 6.834141620—01 6.258543030—02
—1.63243269)1—01 —8.724677340—03 2.638353015—03 5949899370—04 5.664181420—05 2.300128230—06 —1.045810290—07 —2.445520830—08 —2. 191555830—09
6
4 . 0 5409 36 0—Fl 7
__________
TABLE
N.
129
1.129857320 02 1.152774550 02 7.656417820 01 4.502889270 01 1.734783770 01 8.550554160—01 —1.913695900 00 —8.410850920—01 —1,932039160—01 —2.973592860—02 3.36830408D03 —2.959344710—04 —2.08808362D—0S —1.21355018D—06 —5.914827050—08 —2.466086450—09
7 8 9 10 11 12 13
—4.307235070—02
—1.124659620—02 —7.972584330—04 7.266915630—05 1.965233310—05 1.830905250—06 8.059731490—08 —9.252391390—10
TABLE 2—Ill 8
0400N)Nro
—?
No 9
F0004),N1’
—/
ri
((Fl)
I’
((0)
‘I 1
(1 1 2
—5~5944T0600—l)1 —1 .3517/3110 1)0 —9.111770100-01
3
—4.8 39999(00—Ill
1.
—3.309Qu01?i (IS —7.111530)1071’ ill —4.9.’2s29020—u1 —1.4552091 ?t’ OIl 7.157434000—01 7.5953541511—01 9.448817550—02
4 5 6
—4.397057695—Ill 1.7)15944510—07 4.974110080—O(
7
—1./844904,10—0?
7
6./40817260—1l3
7
3
FIB. S/wore)’ / Double Gebrsher
130 O i~’ 11 1,1 11
—5.2201111351—03 —4. 094111871)1’— 04 1.119705695—70 4.0960803(11—06
0 0 11) 11 12 15
4.5336974511_07
2.107282411,—OF’
(‘.V/.laFlOions
—4.569204130—04 —1 .74664941(1—06 —1 .44293oT?t,—oS —9.755441560—00 9.397377560—uP 8.8345050 (0—119
CIM)
6
((oil
1) 1 2
1,109S46310 00 —3.27387056100 1.048779640 (10 1 . 724 5u84? 0—01 9.6465213?7r—Ol 8.113365411—01 —1 - 11(76397,11—01 —8.2o751128?o—u7 —3 - 80029 756 0—04 2.317955270—03 1.3433846311—04 —2.118416150—05 —2.253870961=06 1.75410(1l4I1—09 1 O5453337)1~09
0 1 2 3
—3.931086020 00 —4.14469690000 —4.146396440 110 —6.219431610—Ill
4
—1.352046240
3
5
0
3 4 5 1,
/ 3 9 11)
11 12 13 14
FOPIINFNTO —1
N11
..4
M
C(,i)
7.9968276? 111)l 3.2147444111 011 —3,5274253/11—01 4.S72806?40—0l ~5,71l2036PSr—0l —4 . 5437024411—01 —1.745285090—02 2.178366820—02 3.096981640—03 —5.631269310—05
II 1 2 3 4 5 6 7 8 9
3.0364S061T1 110 5.4924145010011 4.095425090 011 1.66402153050 1.739681790 00 —2 • 592 S 7,400— 01 —2./65209440—u1 —1.839360620—02 o.812273u00—03 1.019195600—03 7.008399110—06 —8.617848050—06 —7.737315410—07 —1.83075(730—00 1.708545030—09
—4.403736390—OS
10
—3.78458496)1—06 —5.753771175l1—09 1.4227410011—08 1.410039930—09
11 12 13 14 2—10
F0PONFNT= —5
N~13
E%PONENT
7
0 S 10 11
roll
TABlE
s=12
ERPONF-l,T
~
17
13 14
F,’18
EOPONFBT—b0
1’
COO.
0 1 7 3 4 5 31 7~ 8 9 10 lb lo
—2.16625620000 4.111220683000 —3.068144051100 —5.370908500—01 —1.349964540 00 —1.195079891, 00 2. 041 6570(O’—Ol 1.1797628611—01 —3.118717370—03 —4,684018000—03 —1.520559860—04 5.966673610—05 3,932o60971I—015
M
CO.?)
F’
CU?)
31 1
—4.49900231)1—01 —9.64211081000)1 2.694150350 00
0 1 2
—1.010385920 00 —1.4(9t1440000 -1 .671838390 05
Table 3
3 4 5 6 7 8 0’ 10 11 12 13 16
—7.863071690—01 2.0o1144090 00 1.60029964000 —4. 134191 320)0—02 —1.544767400—01 —9.152.050810—03 1.580241780—01 2.544375920—04 5.387770770—06 —1 .3359/0660—06 —1.272568100—07 —3.6492508)0—09
3 4 5 6 7 8 9 10 11 12 13 14
—3. 73i1t48l 10—01 —4.685711520—01 1.155058630—01 9 . 5361 789S0—02 1.512694720—03 —3.464291620—03 —3.042395770—04 2.771252620—uS 5.001607650—06 1 .505114040—07 —1.658759980—08 —1.717931)200—09
1 = 0, P~ 8.0,
FplInsion
EXPONENT’
—1)
4.15
EAPONENT. ..7
F’
CCII)
I?
CU?)
(I
—3.023641860—01 2.0561005711 OIl —9.00811105u—01 3.558395230—02 —5.21095181111—01 —4. 37606489 11—01 3.638893810—02 3.683260t30—02 1.3882864111—05 —B.01134819u—04 —7.500549430—UP 3.664522615—0? 7.051400740—u7 2.8730589o0—08 —1.6665(117(1—00
0
2.320327000 0(1 7.851767961, ~ 2.61,753333000 5.759023(20—01 9.143469530—01 —3. 074007~20—o1 —2.275340180—01 5.670093160—03 1.05081366o—0~ 5.7636206F9D—04 —1.659230(50—04 —1.445066870—u5 3.456551(40—07 1.643673940—U/ 4.217596580—09
2 3 4 5 11
7 8 ‘2 10 11 1? 13 14
1
2 3 4
5 F, 7 8 0 10
11 17 13 14 TAO) B
~.1o
EOPCINF9T
—6
5 3, 7 8 9 10 11 12 13 14 15 16 N’19
F’ 0 1 2 3 4 5 6 7 8 9 10 11 12
UI)
5.71I4(3544O—01 6.004710°50-u 1 —7.37(1(1920—02 —2. 200882250—00 —4.75559(690—04 4.080240660—04 2.589473950—05 —2.552S09 470—06 —2.814809580—07 6.131421 40D09 4.537165810—09 —7.753213500—TO EXPON060’—il
C)??) 5.59679618000 3.68261644008 5.86474489000 6.210811470—01 1.830146740 00 —4.738473500—01 —5 . 840894 (40—01 8.382041060—03 3.461139060—02 1.440650680—03 —7.190563630—06 —5.0.14060080—05 6.85064070—06
coeffwien)s c 00, of eq. (17), herclabcled (‘till) for — I °ui07 ~2 C
~
--
TABLE 3—I
4. 0
EXPONENT’
0
N’ 1
EXPONENTS —1
II
CC??)
I?
CCI?)
0 1 2
3.942307770 00 2.817083350—02 6.267078200—04
0 1 2
3.688251 450—01 1.994583920—01 1.776868320—02
3
—5.024827330—05 —1.62148717006 5.338951555—08 —1 .424884010—09
4
4.14
ful(CtiOIIS
I,
I,
,,~10
for ware
5 6
4’ 2
F’
EXPONENT’ —3
CU?)
0 1 2 3 4 5 6 7 8
7.502180080—01 4.476733390—01 8.905088580—02 1.672707780—02 3.090874170—04 6.271432490—05 —5.759711330—06 1.019811390—0? 1.952586900—08
9
—1.164944540—09
2—V
3 4 5 6 7 8 9 10 4’ 3
N 0 1 2 3 4 5 6 7 8
2.359724330—03 —3.540555970—04 1.438163450—05 4.398841230—06 —9.198415900—01 1.349683720—08 2.482666680—08 —4.003214520—09 EXPONENT. —4
CU?) 2.443218510—01 1.497075120—01 3.455370140—02 7.442129190—03 5.066950180—06 7.036997870—OS —1.123861580—06 1.734515960—07 —1.197367770—08
TABLE 3—Il
0=17
EXPONENT’
—9
N~
4
EXPONENT— —8
4. 5
EXPONENTU
7
yB. Sheorey F’
CCII)
N
0 1 2 3 4 5
1.107166230 00 6.849792050—01 1.666827360—01 3.812297160—02 3.328298500—03 4.970460210—04 7.394074650—06 2.215293150—06 —9.088699350—08 5.631115100—09
0 1 2 3 4 5 6 7 8 9
6
7 8 9 N
6
EXPONENT’ —9
4’
/ Double C’hebvsher expansions for scale functions CCII)
9 10
6.402346480—01 3.974747580—01 9.882564520—02 2.336658400—02 2.245994410—03 3.55500706004 1.005550640—05 2.120656020—06 —4.550831490—08 6.821940960—09 7
EXPONENT.—10
F’
C)??)
H
CC??)
0
4.471 54971 0—01 2.7780504511—01 6.962o237°0—02 1.681391240—0? 1.686861070—03 2.792922761’—O4 9.663055886—0? 1.968887250—06
0 1
3.635106400 00 2.256661240 00 5.659950570—01 1.387196200—01 1.415496380—02 2.435063280—03 9.110972570—05 1.919226040—OS
2 3 4 5 6 7
2
3 4 5 6 7
1.79596?9?D—07 .3.991630300—08
4
EXPONENT.
II 0 1 2 3 4 5 6 7 8 9 10
II
8
FXPOOENT’lA
0 1 2
3 6
S 6 7
N~
5
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
0
EXPONENT—
0
CCII)
H
5.86058110D 00 —1.153229680 00 4.528614180—02 2.862755730—03 —1.190049030—03 9.818049650—05 2.001660690—OS —6.374465440—06 3.215650276—07 1.803805360—07 —3.96160504s—0l(
EXPONENT’
1.801924~3o—o7 —3.9A1129i,60—00
4—Il
0
CU?)
N— 6 TAELE
9 10 TABLE
4
131
1.431276700 01 —1.362764000 00 4.642026140—02 3.093237970—03 —1.194783460—03 9.778085700—OS 2.003131620—05 —6.373484100—06 3.218535550—07 1.820524310—07 —4.596455420—06 2.834862320—09 9.748021000—18 6.307117940—10 —7.216742340—10
#~ 7
EXPONENT’
U
3—Ill N’ 9
EXPONENT’—ll
CC??)
F’
CCI?)
3.310673730—01 2.055202060—01 5.1696178s1’—O? 1.28067440L’—02 3.3298135911—03 2.3412851)10—04 9.49271548)1—06 2.112852590—06
0 1 2 3 4 5 6 7
3.506886120—01 2.167353(40—01 5.371724650—02 1.344011750—82 1.358101250—03 2.509225050—04 9.125637200—06 2.406712080—06
F’
CCII)
C, 1 2 3 4 5 6 7
1.019488440 01 —1.531462730 00 4.6932715o0—02 3.221926750—03 —1.196519420—03 9.764233800—05 2.002298570—05 —6.183964710—06
1.1
CC??)
0 1 2 3 4 5 6 7
6.052416680 00 —1.6719115100 00 4.717868650—02 3.300253,240—03 —1.197151470—53 9.738527620—05 2.007491200—05 —6.064848440—0?
TABLE 4—Ill N~ 8
Table 4 6,orn of eq. (18), here labeled C(M) for Expansion coefficients
EXPONENT—
61
1
0
9
EXPONENT•
6
N
C (F’)
0 1 2 3 4 5 6 7 10. 9 10
1.011383110 01 —1.894693940 UI) 4.7171079190—02 3.386031900—03 1.19920067003 1.003135430—04 1.834711730—05 —4.450213090—06 —4.O’0A720510—O( —3.316304280—010 4.717520270—05
(5)
1= O,p
0
=
—e
8.0, --1 ~
0 1 2 3 4 5 6 7 8
-—
TAPLE 4—I N
0
F’
EXPoNENT—
0
CCII)
N’ 1
EXPONENT—
II
CC??)
F’
0 1 2 3
1.10912453000 5.163604970—01 —4.305551760—02 —3.563677840—03
0 1 2 3
6.42302342000 1.655324770—01 —7. 75413422D—03 —1.616881380—03
~ 6 7 8 9 10
_:1~:~ —2.003661520—05 6.373105060—06 —3.214072360—07 —1.801963190—07 3.982915390—08
‘ 6 7 0 9
1:0370769006 —1.070272050—46 6.01 719(300—09 1.049635680—08 —S.937S19360—10
EXPONENT’
N’
1.44898T6S0 01 —1.70989461011 0(1 4.711763140—02 1.353493910—03 1.19795504003 9.738773430—05 1.992850510—05 —7.192465330—06 1.649339470—06
-—
Table 5 Expansion coefficients cnm ofeq. (17). here labeled C(M), for = 0, po = 8.0,0 V~1) ‘~ I
1
—-——~—
TABLE 5i
0 4. 2
0
3
EXPONENT
———
EXPII1)ENT
0
?1
1
EXPONENT’ —1
0 C(S)
M
CC??)
F’
CC?.?)
O 1 2 3 4 5 6 7 0
1.722684100 00 —4.951691 550—01 3.405172960—02 1.1957199o0—03 —1.075422870—03 1.041014710—04 1.886830950—05 —6.370881140—06 3.324333010—07
0 1 2 3 4 5 6 7 8
1.000989090 01 —8.8007534A0—01 4. 24978A560—I2 7.388303090—03 —1.172921720—03 9.946778170—05 1.992228750—05 —6.377763230—06, 3.219843480—07
0 1 2 3 4 5 0
4.068089330 05 3.5050’577l11—02 1.0313431140—03 1. 33752347—OS —3.160304300—06 —1.074931140—07 —3,21 124110110—09
F’ 0 1 2 3 4 5 6 7 8 9 10
CC’?) 4.255518440—01 2.390455100—01 3.026388?SD—02 3.764343120—03 —2.332434010—04 2.583066150—O6 5.211389970—06 —7.01584451)0—07 —2.433369930—08 2.678004680—08 —3.185377030—09
132 II’
F lB. 2
FXPONPNT=
NI
—T
N— 3
CC??) 9. 547211X1?tN—01 5.9756220?’lOl 1.470.0414100—01 2.8993742211—0? 2.701480420—03 1.727964550—04 2.674930839—06 —1.38646033,10—OX 2.518S35400—0R
II
2 3 4 5 6
1)
Sheoic’y
/ Double C’hebo’shec expansons EXPONENT. —5
II
CCII)
0 1 2 3 4 5 6 7 5 9
3.400S67140 00 2.210401900 no 6.403169930—01 1.463889(70—01 1.837218100—02 2.107537280—03 1.332139930—04 7.811890180—06 2.491999010—07 9.020729270—09
2 3 4 5 6 ~ 11 9 10
N
2
for wa~’~filnction.V
5.0841239.00—02 3.5107027480—03 —1.195932410—03 9.755518170—05 2.0040342611—OS —6.373029270—06 3.21412758007 1.801964660—07 —3.98291 55711—00
2 3 4 5 6 7 8 9
EXPONENT’
N
0
CCII) TABLE EXPoNENT’ —5
I~
5
EXPONOIIT’
I
CO.?)
II
LOll
(1 1 2 3 4 5
1.67175,’O?D—Ob 1.109100920—01 3,4721 A18?6—02
0 1
1.04055845000 6.996930400—01 2.298006060—01
8.0.79404711—03
2 3
1,
1.7744540(0—03 1.690700250—04 1.491924890—05
4 5 6
7 1
1 .194258660—06 6.706924250—08
7 10
O
3 . 602 2 7 72411—09
EOPIINONT
6
—7
NO
1’
7.7o405044,.—U?
0
1 2 3 4
5.27261241011—0? 1. 7006o497U—O? 4. 713,1(1141’1.—33 X.21567955’,—04 1. ?378704”11—04
1 2 3 4
6.710602840—01 4.591925660—01 1.599353660—01 6. 30030701 0—07 (.019224010—03 1.219973680—03
7“
l.957ll7734l1~O5 1. Tl8A’,575”—0A
6 7 1)
1.448605090—04 1.509944940—05 1.315085920—06
2 3 4
5 F’
7 N
6=9
9.045774900—01 —3.63631 246D—02 —2.383037730—03 1173575600—03 —9.946918860—05
6 7 8 ‘2 10
—1.886800530—05 6.3708770811—06 —3.324353470—07 —1.795971230—07 3.99178700(1—08
6 7 8 9 10
—1.992241120—05 6.377788320—06 —3.21976741 0—01 —1.802000140—07 3.982577810—08
EXPONENT—
0
N
5
EXPONENTU
0
CU?)
F’
CCII)
N
0 1 2 3 4 5 7 8
4.387745340 OX 1.175088001) 00 —3.961829090—02 —2.860847280—03 1.190278590—03 —9.818275930—05 —2.0(11683710—05 6.374458610—06 —3.215686000—07
0 1 2 3 4 5 6 7 8
1.367792960 01 1.382094490 00 —4.159585200—02 —3.093369060—03 1.190.762650—03 ..9.778865590—OS —2.003020730—05 6.371092240—06 —3.21731 511 0—07
109
—1.802608590—07 3.985618720—08
109 11
—1.757206230—07 5.951879280—08 —1.108776110—09
12 13
—1.071737370—10 —4.240672550—09
14
3.027982380—09
Cli”)
6.356=69909—111 4,37216/510—01 1 . 540710111, [—01 4, 2,1 51’ (‘1411—U,’ 7.=’l’lo’175,,,—03 1.71211/261’ —50 19’,7’lo’,5’,’—34 1.7h12,~51,,_0S 1 .01321318 ‘11 —06
II 1
7.7138599/70—FJl 5.1l493731000l
2 3
1 .00094031 0—01 5.57724751 0—02
CC??)
M
4 S
0.X0735169D—03 1.3,11847420(l—03
II 1
1.0232(801901 1.548432679 00
0 1
F,
7.060067520—06
0
—4.269243430—02
7
7.4776811910—09
3
—3.223131030—03
A
2.50050122(1—06
4 5 1
1 .196415230—03 —9.767809870—05 —1.999802610(1—05
2 3 4 5 6
1.686999060 00 —4.340957450—02 —3.301334190—03 1 .197030330—03 —9.763305110—05 —1.995692360—05
7
6.1o5367430—06
7
6.267404660—06
N’
--
6
ExPONENT’
0
C(M), for
N
TABLE
7
EXPONENT
0
C U?) 6.64995529000
6—Ill
8O,p~ ~ ~ I TABCE
-
6-I N’
NI
00
N~
Table 6
0
7.439213400
CIII)
Expansion coefficients O,~ of eq. (18), here labeled
II
4
p,
—-
~ oo
5
12 3 4 5
EXpONENT——il
-
=
0
5—Ill
EXPI100NT=—1l1
II 1
01
(I’
EXPONENT’ —9 C(S)
8
1.035397330
TABLE 6—Il
2.8847303,50—00 7
1?
o’
CCII)
(4
9.217977330—07 5 . 7281 55680—08
9
T0BIE
0
1.306838450—05
(IF’)
I
E0PONENT
5.222609590—01 —2.7260838411—02 —1.185452050—03 1.07671875003 —1.040976390—04
12 3
5.878893000—02 9.657074210—03 1.384853560—03 1.423228490—04
10 I.
3
5—il 0
6’ 4
1.512397(70—02 1.633666230—03 —1.015S83770—X4 —5.0997087X0—06 1.071701530—06 —6.004024080—09 1.04955794D.08 5.933213030—10
EXPONENT’
0
CCI?)
0
1.15036667001
1
—4.85365491 0—01
N’
1
EXPONENT—
0
F’
CC??)
0 1
1.22239342001 —9.617012000—02
8
EXPONFNT
0
N’ 9
II
CIII)
H
O 1 2
0
-~
1.55S410639 01 1,XV18077T0 00 —4. 41 21 4)26 D —02 —1.35421X050—03
4
1.195618389—03
4
1 2 3
EXPONENT—
0
CCII) 1.159555600 01 1.905411870 00 —4 . 449262 750—02 —3.391598950—03 1.103372090—03
yB. Sheorey / Double C’hebyshei’ expansions for wave functions 5
—9.67915884U—05 —2.014481650—OS 7 786551 36l,~t2, —3620769850—07 —7:25614124x~t7 4.1331 70910—0?
5
7
‘0
10
5
—9.523617380—05 —2.061929100—OS 7.384064920—06 ~..560945180—06 1 .848804920.06
6
7 A 9
L
59467611—UT 9 93484811—01
U 1
—6
2
3
3
—1
5
—-
7606ATD—02
133
F’
0
—7.9132120—01 .2.50088011—01
—5.9448020—01 —4.607051 0—01
8.0103500—01 —0.878030 11—CT
1.0322870 00
9,9906580—01
—? .85085111.01
1.7005440—01
—6.531 874D—02
.9.9786010—01
1.0183420 00 —1. 20582)0—01 —1.0073009 011
9,9393510—01
—7.5807800—00
1.13321 4D—U1
0.900)1701—01
—9.8306620—01
1.7498411—01
~
~
1 0131740 00
6
—15904950—UT
7 0
—1 ‘021 77UD 00 5:0810880—02
TAIL?
Table 7 Regular and irregular Coulomb functions and their derivatives —
—
—
1991)
81A’-,I.5l,
7—I
9.
81*L
11,9
3 4 O 6 O A
5,~ 1
o
To, 4
1’
-
U
8,4575,,oo—’,1 , 69,4511I,—flT
4 5 O 7 A
—1
O,171,Il5I1—IIT
Ol,02o7’lfl—l’
1,’i9.’01011—l’1 1.o7’,ô’,21,—U2 1,94”972003 t.024’~420U4 1,4337141’111 0?)’,3~10—’1Z
I ?.9,’310l0’O1 O.311’15?fl”II? 4.5(01,111813 1•,16517,-U.’13 9,61763311—00 7.14443116’,’ 11
5,1777870—’’’
ST171S%O’~’7
—i
0’
. 16011041
Ii
—T .1097701
1
O
—7.7745570—01
1 2 I 4 0 6 7 S
—3.9613250—4)1 7,430T140’II 5.05571116 00 6.78414011—01 3.0910430—01 1.1006360-01 1,1119211102 0,49402311-103
40.’
1’
2 3
1,65)3470—00
1.0357240—02 9.840967(1—04
5.7108680 Il
1.2012510—04
—0,4680280 02 —3,5956990 03
1,0730900—57
3.7269050—00
7
1.1850850-1,8 3,9005820—10
1.8720950-07 6.37)29211—09
5,7724640 02 1,1084340 03 1,4011200 05 3.215=170 06 0,369571007
A
1.16=5920-IT
2,1192230-10
7.5070060
6,1250120-06
6.8334180-05
81*1
1.00
X
L
2.2757620—Ut 1
7,5170000-00
7 0 4 0 0 7 0
1.471510—I’2 ?.08~,4990—03 2.29=2930—04 2,0701770—05 1.97=4480—06 1,0506000—1’? ‘.1935630—00
L
F
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yB. Sheorev / Double C’hehvshev expansions for
134
wale finctions
angular momentum, 1, is taken IC) be zero and that of
wave (‘oulonib functions for any value of the indepen-
Po to be 8.0. These tables have all been printed iii the sanle form. Thus, for example, the function A, 1(77) of eq. (6)is given by
dent variable and for ~iin the range
51 =
lO~ ~ IS
a,,,11T(i~),
(22)
0
—
I <
< 1. For
attractive and repulsive Coulomb t’ie]ds with charge, z = ±I , this corresponds to all particle energies greater than or eqLtal to 13.6 eV. The coefficients for expansbus of the derivatives of the Coulomb functions are obtained using methods discussed in [1]. For I > 0 the
svhere F, the exponent, is also given in the tables. It should be noted that the expansion coefficients a,1180.
functions may be obtained by recursion using the method also discussed in [1]. 1-or low energies, that is,
cm0 and °IOI7C [cf. eqs. (6), (10), (17) and (18)1 have all been labeled 6’(M) in tables [-—-6. Finally in table 7 we give values of the radial regular and irregular Coulomb functions and their derivatives for the partial waves, I = 0, I, 2 8 and for some values of the independent variable, p, which has been labeled x in this table. For x <8.0 we have used coefficients from tables I and 2 and for x ~ 8.0 we have calculated the functions using appropriate coefficients ~ ii tables 3-—6
for very large values of i~, this method would probably be uneconomical. Some methods for such a case are discussed by Frhherg [41.
~nI1l’
roi
5. Conclusions It is clear that the double Chebyshev expansion technique enables us to rapidly evaluate the radial s-
References [I[ yB. Sheorey, Conoput. Phys. Commun. 7 (1974) 1. [21 NI. Abraniowitz and IA. Stegun, Handbook of MathemathoOt Functions, NBS Appt. Math. Series NI). 55 (US Govt. Printing Office, Washington D.C.. 1968). [31 C.W. Clenshaw, Mathematical Tables, Vol. 5, National Physical Laboratory (H.M. Stationary Office. London, 1962 [4[ (‘.l. l”robL’rg, Rev. Mod. Ploys. 27 (1955) 399.
DOUBLE CHEBYSHEV EXPANSIONS FOR WAVE FUNCTIONS yB. SHEOREY Physical Research Laboratory, Narrangpura, Ah,nedahad-380 009, India Received 26 May 1976
A double Chebyshev expansion technique is suggested for evaluating positive energy solutions of the radial Coulomb equation. It is shown that, for a particle with a wide range of energies and in either an attractive or a repulsive Coulomb field, the two linearly independent radial solutions can he rapidly evaluated.
I. Introduction
2. The radial Coulomb equation
In a previous paper [11we discussed a method for evaluating Chebyshev expansion coefficients in terms of which the two linearly independent positive energy solutions of the radial Schrbdinger equation for an electron irs a Coulomb field could be expressed. It was shown that, for a fixed value of the particle energy and of the charge, the s-wave regular and irregular solutions could be evaluated to a very high accuracy for the entire range of the independent variable. The solutions br other pattial waves could be obtained from recursion relations. It was pointed out that once these two solutions were evaluated everywhere the continuum Green’s function could also be constructed. These functions are required for solving various problems in atomic physics such as photoionization, charged particle impact excitation and ionization of ions. in such applications the continuum Coulomb functions or the Green’s function are generally required for several values of the particle energy. We propose a double Chebyshev expansion in terms of which the radial s-wave regular and irregular solutions. for a wide range of particle energies and for both attractive and repulsive Coulomb potentials, may be expressed. In this paper we follow the notation of [11and use the results obtained therein.
The radial Schrhdinger equation for a particle in a Coulomb field may be written as [21 2
~
1~
+ 1 -p (1) where us a non negative integer and denotes the angular momentum; t~= z/k, where z is the charge (z <0 for an attractive potential); k, the wave number and k2 is numerically equal to the energy in rydbergs. The
H
Ll~2
1(1 + 1) ~
solution, F(r~IIp),and the irregular solution G(r~lJp), are determined by their asymptotic behaviour ~~@ilIp)
sin 01 ;
G(r~lIp)
p
cos 01,
(2)
p
where 01
=
~ in 2p
--
~lir
+
~
(3)
,
and where 01, the Coulomb phase shift equals arg F(l + 1 + in). As in [11 we consider solutions of eq. (1) in two regions, namely, the inner region, 0
,
(4)
tlB. S/iyoi-c’t’ / Douhic L~he/,~‘s/,yy
1 26
where C(i
7, /)is the normalization. It was shown in ]j that the function h(s~l~ p) could be expanded in terms of shifted Chebyshev polynomials of order n, T,1 . In particular, for /
=
0, we have
A
~A,,(77)T~(p/p0),
=
(5)
nO
where we have now explicitly written the expansion coefficients as functions of 77 and the prime on the summation symbol indicates, here and elsewhere, that the coefficient of the r = 0 term is to be halved. Consider the range of r~ as - 1 <17 < I, then, for any particular value of n, if function A0(77) is continuous and of bounded variation in this interval, it may be expanded as
As before, then. A
n0,,, T(77).
=
‘l’(ril = OIp)
F
~o
~,,,,,T,0(77)T~p/p0)
in= 0
I
2.2. Solutions in the outer region in ill the solutions of eq. (1) in the outer region, Po
+
iF = e’~I)(l~p),
(6)
(12)
A
{A,,(77)
~
N
W,,(77)~T~(p0/p).
A
R,,(77) FFO
a,,,,,T,,,(77)T,,(p/p0).
(7)
X
,i=0 1,1—0
The irregular solution of eq. (1), G(77l~p), may be cx-
[cos
e0(77) + i sin O~(77)1T,~(p0/p) 2
F(~l~p)In 2p
+
Q(771)1
and
=
[A~(77)+ B~(77)]’/
=
tan
(15)
C7~(~ 0)
+D(77l)p~’(77l~p), where Q(pl) is a known function =
(14)
,
where
pressed as
D(pI)
(13)
Tins equation may be written as
M
=
+
0
n
(I)(77l~p)
=
(1])
The expansion coefficients for derivatives of 1 and sji with respect to p are easily obtained using eq. (6) of
~1)(77lI p) =
G(7711p)
.
--0
Thus, from a knowledge of the coefficients a,,,,,, the coefficients A0(77), and hence the function rI~(i7,Ot,o) can rapidly be evaluated for any ~iin the interval. The function r1 can then be written as
(1)(77tp)
.11
=
and the function (I)(77l~p) was expanded as
ti F?!
for nary Jiictioiis
expa/isiolls
-
——-~
(8) of
p and of / and
.
We had shown that the function ‘1J(pl~p) could be cx-
punded as a convergent Chehyshev series, namely, A
B,,(p)T~(p/p0).
0(77)
1 {B 11(77)/A,,(77)~.
(16)
It is evident from eq. (14) that the real and imaginary parts of the expansion coefficients, as functions of p, have a sinusoidal modulation and consequently their Chebyshev expansion would be expected to converge
(2/ + 1) C(p/)
‘I’(pllp) = ~
e
(9)
slowly. There is also one other difficulty. It is clear from eqs. (1), (12) and (13) that for the case of / = 0 and p = 0 the value ofA0 = 2 and A0 = 0 for n >0. The imaginary part, B,, = 0 for n >0. Thus the function e,, for n >0 is indeterminate. In fact, the phase
n0
Agair each of the coefficients B,,() may be expressed as a Chebyshev series in the variable 17. u =
,fl
0
a,,,,, T(p).
(10)
function (—),, has a finite discontinuity of absolute magnitude ~r radians at p = 0. This is evident from the differeutial equation satisfied by b(pl~p)(cf, eqs. (37). (38) and (39) of [11). In fig. I we give a plot of phase function in the interval - 1
=
1. We
FIB. Sheorey
/ Double chebysher
expansions for ,yave jimcrions
127
I
panded as N
f(x) =
3,
~
anTr,~x),
(19)
where N
3.
-
6.1
a
0
=
N1~0 f(cos ~i/~ cos(~nf /~
where the double prime on the stimmation symbol in-
-
dicates that the! = 0 and the
I
N terms are to be
halved. The error, ha0, in a0 is given by
1
Fig. I. Phase function O,,(~)(radians) for n = I and P0 = 8, = 0. Note that scales are different for the ranges --1 © © 0 and 0< ,~< 1.
therefore obtain Chebyshev series expansions for the two functions R11(p) and e0(77) in two ranges of p, namely, —1
R0(p)
(20)
(17)
CnmTm(p)
=
ha0 = a2N
fl
+ a2N+,, + a4N - ,~+ a4N+fl +
(21)
and which is negligible ifNis sufficiently large. In our calculation we have fitted the functions defined by the equations mentioned above to a high accuracy. For most values of it, the expansions in p give the expansion coefficients in eqs. (5), (9) and (14) correct to eight significant digits. For the last few values of n, that is, for n = N, ii = N~ I ,etc. the coefficients are correct to six significant digits. However, the cxpansion coefficients for these values of n are themselves very small and this decrease in accuracy does
- 0
not lead to any significant differences in the final resuit.
and in’
0,,,nTm(P).
=
(18)
0
4. Results
Thus the regular and irregular solutions of eq. (1), for any p in the ranges mentioned above, may be evaluated from a knowledge of the expansion coefficients for the functions R0(p) and The expansion enefficients for the derivative of 1 with respect to p are obtained as before.
e0(77).
3. Numerical method The calculation of Chebyshev expansion coefficients for the functions defined by eqs. (6), (10), (17) and (18) is straightforward [3]. The method is based on the use of the orthogonal property of summation of the Chebyshev polynomials. If the range is
In the set of tables 1 --6 we give Chebyshev cxpansion coefficients for functions A0(p), B0(p), R0(p) and O,,(p) defined by eqs. (6), (10), (17) and (18), respectively. Tables 1 and 2 are for expansion coefficients of functions A,,(p) and B~,(p), respectively, and are valid for any p in the range --1
JIB. Sheorer / Doubt,’ Chebv,vhc,’ e.vpans,ons Jb- ware junctions
128 Table I
o
Expansion coefficients o,,,,, of eq. (6), here labeled C(M) for
n~1 -
-
—1 ~?39Q430
Y —us
10
i~
:l~~
~
11
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11
F N
)J
=
=
1
F~ 8
1 /
1,0, 11’
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711—0)?
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5
r7
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17
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3 4 0 7 0 ‘5 10 11 12
6
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07
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1 2 3 4 5 7
5
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TORI F
9. 71c959420—u7
:~:~:~
—1410554600—lI1, 551’,36(?v—lu
)XPUN(-l,T
—5
0
(0.,)
1) 1
5.57462’.4?l)—11 0.429355lllO-U1
2
3.9747u4(50-Il1
S
—4.61 7060560—01
4 5 6
—1 .9?(ilu160<0 2.931497010-110
7
1.267617060—42 1.R94040510—04
3,71045777)1—06
0 9
—2.11)7850710—04 —1.831115~60—35
8.4S7105905—ij7 4.59047003—59
1)) 11
3.699610750—07 1.39131722ll—07
12 1—is
9.2113(9390—oN
EXP0NEN1~ —1
TABlE
C(M) -4.371090800-01 —7. 1 FF869 6490—01 —2.41)1741810—01 2.043083090—01 8.069878060—02 8. 345443o80—03 —9.717786950—04 —3.523225870—04 —4.056133370—05 —6.438346420—06 1 .633947870-08 1.750094650—06 1.913255636—09
FXP0NF6T~ —0
~12
99000FNT
-6
CO.?)
1.34191146111 0)) 2. o594/ b?’— 01 1 208305521 0)1 73029 11 —01 1’ 1332233’ —01 6~0031004303-03 3:8236155Nll~O1 1.323983501—04
I) 1 2 3 4 5 6 7 13
—1.0943720100 00 —3. 374706400—1(1 —6 9159990.0—uT 1 .1 o3023o00 OIl 5 007169(00—01 —1 1)06115370—01 301075(00-02 8.824103670—04 10.890800190—04
—3.984106211—05 —3.82642022~—06 1,16210024u—09 1.854140100—08 1.1791Q59))1—uN
9 10 11 12 13
4.175212710—05 —5.045646770—06 —6.349257090—07 —7.498172690—09 2.033923780—09
641’901105111—01
9 1)3 11 12 13
FOPUN6NTO -7
F’
0 1 ? 4 5 6 7 3
6013
C)??)
—
CO.’) 0 1 2 3 4 N 6 7
S.43586704D01 4.252667750—01 0.31 081 7110—0! —2 . 3 320 75 1)00—01 —8.705820s10—u2 7.018834550—06 3•u56?44064—IJ5 4.104354400—04
N~14 F’ 0 1
EXPONENTO —7 CO..?) —2.639151920—01 —6.410579750—02
N~15
E0P0NENT~ —0
F’
CC??)
0 1
2.230149320—01 1.266412010—00
FIB. Sheorey / Double chebysher expansions for ware junctions 2 3
—2.2328(8560—01 —1 .357724420—01 6.419723790—02 2.467142516—02 —2.558252500—03 —1.091209975—83 —3.075635850—06 1.6383705100—05 9.1035198)0—07 —7.0~87~~9o008 —8.594181945—09
4
5 6 7 8 0
10 11 12
2 3 4 5 6 7 8 9 10 11 1? 13
1.490980150—01 —2.131053910—01 —9.574569030—06 2.296693320—02 8.474378310—03 —4.517450690—04 —2.383333910—04 —4.487987040—06 2.446347930—06 1.5259054ti11-07 —6.561425850—69 —8.563703600—10
TABLE 1—V l’,16
E0PIDNENT
—9
4—17
EXPOFdENT—10
II
CO.?)
H
CO.?)
0
3.915102920—01 1.15329M011D—01 3.140491290—01 2.145449970—01 —I .1076166:SD—01 —4.306491500—02 S.83625~os0—03 2.2124(6950—03 —5.071576730—05
0 1 2 3 4 S 6 7 8
—2.9105417640—01 3.254106460—02 —1.914560130—01 2.941523070—01 1 .4112761480—01 —3.~o947833D—02 —1.383941460—02 1.081168440—05 4.533805110—04
1 3 4 5 F,
7 13 9 11l
—4.1724217s0—05 —1.234303930—06 45OS2ioIS—O?
9 10 11
—1.527746120—66 —5.983673850—06 —2.240435330—07
11 12
3.OI 2.0721094s5—08
12 13 14
3.1780684511-00 1.524643490—09 —6.230661 580—10
N18
EXPONE4T—11
)4~1c
C(M)
(1
5 1, 7
0 1 2 3 4 5 6 7
10
1.503761161=04
10
1 2 3 4
C(F’(
F’
—4. 5o851)/960—01 —1.606770961’—Ol —3.46316250)l—O1 —2.673592441—01 1.474976130—01 5.9169960711—42 —9.SuO?5710i’—OS —3.4521396555—03
O
9 10
7. 570520134))— (15 7.4131006900—07
9 10
11 12
—7 - 1 982o6300—0 7 —2.59365705o—0l1
11 12
EXPLlNENT~—12
3.873391,60—01 —4.241111100—06 7.499147640—01 —2.612701460—01 —1.87(000960—01 3.456315(00—00 1.929469970—06 —9.1115531610—06 —6.7109036550—06 —6. 33561)5090—06 9.905(l((520—0o
2
—1.316337500—07
Table 2 Expansion coefficients i3iirn ofeq. (10), here labeled C(M) lO,p~ 8.O,—1~T)~I —
0
EXPONE4T~
0
for
C(M)
0 1 2 3 4 S 6 7 8 9 10 11 12 13 14
1.257894920 01 1.250382570 01 8.897288650 00 6.009628620 00 3.183397150 00 9.065242090—01 9.087610760—02 —1.982463730—02 —8.902221240—03 —1 .707600610—03 —2.169151850—04 —2.047384420—OS —1.518130940—06 —9.169438090—08 —4.603893040—09
1
CCII)
H
O 1 2 3 4 5 6 7 8
4.01971 2930 00 4.517129320 00 1.90721 5960 00 1.045745430 00 8.370769490—02 —5.238068210—01 —3.566611770—01 —1.169399680—01 —2. 420601620—02
9
.3.534320680—03
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 13
N. 4
‘3.87832547004 —3.335703400—05 —2.317871820—06 —1.331189840—07 —6.450434690—09
EXPONENT.
0
CUlt
M
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
—3.942320090 00 —1.005614990 00 —7.025337740—02 7.48206093D—01 1.432373970 00 7.586192580—01 1.819840810—01 1.922966110—02 —6.884676660—04 —5,516046000—04 —9.642124510—05 —1.066989160—OS —8.736808050—07 —5.650338170—08 —2.990661380—09
EXPONENT—
0
N. 5
EXPONENT.
0
F’
CUll
H
CCII)
10 2 3 4 5 6 7 8 9 10
—5.68441665D 00 —6.07077440000 —3.090021170 00 —2.083539240 00 5281633840—o1 1.219446100—01 8.918916150—02 1.949693050—02 2.001517460—03 2.848069700’OS —2. 211922940—05
10 2 3 ~ 5 6 7 8 9 10
1.305628880—01 —1.14325535000 —6.827893580—01 —6.656295920—01 —5.498834250—01 —1,41330548D—O1 —4.405318430—04 6.11018991D—03 1.303436920—03 1.284617190—04 4.213666610—06
11 12 13 14
—3.800381530—06 —3.801593940—07 —2.778182300—08 —1.600962530—09
11 12 13 14
—5.963731660—07 —1.103907620—07 —1.026924670—08 —6.8491 39920—10
EXPONENT— —1
N
N
6
F’
CU?)
0 1 2 3 4
5
1
EXPONENT
—1
CCII)
10 11 12 13 14
10 11 12 13 14
N. 3
TABLE 2—Il
6 7 8
2—1 N
0
F’
o
4
FXPOP~T
7
EXPONENT. —1
H
CCII)
6.55726862000 9.349862910 00 2.577304910 00 2.480915480 00 —2. 254386695—01 —6.452428160—01
0 1 2 3 4 5
4.629640760—01 1.925767450 00 1.186857470 00 8.236200050—01 6.834141620—01 6.258543030—02
—1.63243269)1—01 —8.724677340—03 2.638353015—03 5949899370—04 5.664181420—05 2.300128230—06 —1.045810290—07 —2.445520830—08 —2. 191555830—09
6
4 . 0 5409 36 0—Fl 7
__________
TABLE
N.
129
1.129857320 02 1.152774550 02 7.656417820 01 4.502889270 01 1.734783770 01 8.550554160—01 —1.913695900 00 —8.410850920—01 —1,932039160—01 —2.973592860—02 3.36830408D03 —2.959344710—04 —2.08808362D—0S —1.21355018D—06 —5.914827050—08 —2.466086450—09
7 8 9 10 11 12 13
—4.307235070—02
—1.124659620—02 —7.972584330—04 7.266915630—05 1.965233310—05 1.830905250—06 8.059731490—08 —9.252391390—10
TABLE 2—Ill 8
0400N)Nro
—?
No 9
F0004),N1’
—/
ri
((Fl)
I’
((0)
‘I 1
(1 1 2
—5~5944T0600—l)1 —1 .3517/3110 1)0 —9.111770100-01
3
—4.8 39999(00—Ill
1.
—3.309Qu01?i (IS —7.111530)1071’ ill —4.9.’2s29020—u1 —1.4552091 ?t’ OIl 7.157434000—01 7.5953541511—01 9.448817550—02
4 5 6
—4.397057695—Ill 1.7)15944510—07 4.974110080—O(
7
—1./844904,10—0?
7
6./40817260—1l3
7
3
FIB. S/wore)’ / Double Gebrsher
130 O i~’ 11 1,1 11
—5.2201111351—03 —4. 094111871)1’— 04 1.119705695—70 4.0960803(11—06
0 0 11) 11 12 15
4.5336974511_07
2.107282411,—OF’
(‘.V/.laFlOions
—4.569204130—04 —1 .74664941(1—06 —1 .44293oT?t,—oS —9.755441560—00 9.397377560—uP 8.8345050 (0—119
CIM)
6
((oil
1) 1 2
1,109S46310 00 —3.27387056100 1.048779640 (10 1 . 724 5u84? 0—01 9.6465213?7r—Ol 8.113365411—01 —1 - 11(76397,11—01 —8.2o751128?o—u7 —3 - 80029 756 0—04 2.317955270—03 1.3433846311—04 —2.118416150—05 —2.253870961=06 1.75410(1l4I1—09 1 O5453337)1~09
0 1 2 3
—3.931086020 00 —4.14469690000 —4.146396440 110 —6.219431610—Ill
4
—1.352046240
3
5
0
3 4 5 1,
/ 3 9 11)
11 12 13 14
FOPIINFNTO —1
N11
..4
M
C(,i)
7.9968276? 111)l 3.2147444111 011 —3,5274253/11—01 4.S72806?40—0l ~5,71l2036PSr—0l —4 . 5437024411—01 —1.745285090—02 2.178366820—02 3.096981640—03 —5.631269310—05
II 1 2 3 4 5 6 7 8 9
3.0364S061T1 110 5.4924145010011 4.095425090 011 1.66402153050 1.739681790 00 —2 • 592 S 7,400— 01 —2./65209440—u1 —1.839360620—02 o.812273u00—03 1.019195600—03 7.008399110—06 —8.617848050—06 —7.737315410—07 —1.83075(730—00 1.708545030—09
—4.403736390—OS
10
—3.78458496)1—06 —5.753771175l1—09 1.4227410011—08 1.410039930—09
11 12 13 14 2—10
F0PONFNT= —5
N~13
E%PONENT
7
0 S 10 11
roll
TABlE
s=12
ERPONF-l,T
~
17
13 14
F,’18
EOPONFBT—b0
1’
COO.
0 1 7 3 4 5 31 7~ 8 9 10 lb lo
—2.16625620000 4.111220683000 —3.068144051100 —5.370908500—01 —1.349964540 00 —1.195079891, 00 2. 041 6570(O’—Ol 1.1797628611—01 —3.118717370—03 —4,684018000—03 —1.520559860—04 5.966673610—05 3,932o60971I—015
M
CO.?)
F’
CU?)
31 1
—4.49900231)1—01 —9.64211081000)1 2.694150350 00
0 1 2
—1.010385920 00 —1.4(9t1440000 -1 .671838390 05
Table 3
3 4 5 6 7 8 0’ 10 11 12 13 16
—7.863071690—01 2.0o1144090 00 1.60029964000 —4. 134191 320)0—02 —1.544767400—01 —9.152.050810—03 1.580241780—01 2.544375920—04 5.387770770—06 —1 .3359/0660—06 —1.272568100—07 —3.6492508)0—09
3 4 5 6 7 8 9 10 11 12 13 14
—3. 73i1t48l 10—01 —4.685711520—01 1.155058630—01 9 . 5361 789S0—02 1.512694720—03 —3.464291620—03 —3.042395770—04 2.771252620—uS 5.001607650—06 1 .505114040—07 —1.658759980—08 —1.717931)200—09
1 = 0, P~ 8.0,
FplInsion
EXPONENT’
—1)
4.15
EAPONENT. ..7
F’
CCII)
I?
CU?)
(I
—3.023641860—01 2.0561005711 OIl —9.00811105u—01 3.558395230—02 —5.21095181111—01 —4. 37606489 11—01 3.638893810—02 3.683260t30—02 1.3882864111—05 —B.01134819u—04 —7.500549430—UP 3.664522615—0? 7.051400740—u7 2.8730589o0—08 —1.6665(117(1—00
0
2.320327000 0(1 7.851767961, ~ 2.61,753333000 5.759023(20—01 9.143469530—01 —3. 074007~20—o1 —2.275340180—01 5.670093160—03 1.05081366o—0~ 5.7636206F9D—04 —1.659230(50—04 —1.445066870—u5 3.456551(40—07 1.643673940—U/ 4.217596580—09
2 3 4 5 11
7 8 ‘2 10 11 1? 13 14
1
2 3 4
5 F, 7 8 0 10
11 17 13 14 TAO) B
~.1o
EOPCINF9T
—6
5 3, 7 8 9 10 11 12 13 14 15 16 N’19
F’ 0 1 2 3 4 5 6 7 8 9 10 11 12
UI)
5.71I4(3544O—01 6.004710°50-u 1 —7.37(1(1920—02 —2. 200882250—00 —4.75559(690—04 4.080240660—04 2.589473950—05 —2.552S09 470—06 —2.814809580—07 6.131421 40D09 4.537165810—09 —7.753213500—TO EXPON060’—il
C)??) 5.59679618000 3.68261644008 5.86474489000 6.210811470—01 1.830146740 00 —4.738473500—01 —5 . 840894 (40—01 8.382041060—03 3.461139060—02 1.440650680—03 —7.190563630—06 —5.0.14060080—05 6.85064070—06
coeffwien)s c 00, of eq. (17), herclabcled (‘till) for — I °ui07 ~2 C
~
--
TABLE 3—I
4. 0
EXPONENT’
0
N’ 1
EXPONENTS —1
II
CC??)
I?
CCI?)
0 1 2
3.942307770 00 2.817083350—02 6.267078200—04
0 1 2
3.688251 450—01 1.994583920—01 1.776868320—02
3
—5.024827330—05 —1.62148717006 5.338951555—08 —1 .424884010—09
4
4.14
ful(CtiOIIS
I,
I,
,,~10
for ware
5 6
4’ 2
F’
EXPONENT’ —3
CU?)
0 1 2 3 4 5 6 7 8
7.502180080—01 4.476733390—01 8.905088580—02 1.672707780—02 3.090874170—04 6.271432490—05 —5.759711330—06 1.019811390—0? 1.952586900—08
9
—1.164944540—09
2—V
3 4 5 6 7 8 9 10 4’ 3
N 0 1 2 3 4 5 6 7 8
2.359724330—03 —3.540555970—04 1.438163450—05 4.398841230—06 —9.198415900—01 1.349683720—08 2.482666680—08 —4.003214520—09 EXPONENT. —4
CU?) 2.443218510—01 1.497075120—01 3.455370140—02 7.442129190—03 5.066950180—06 7.036997870—OS —1.123861580—06 1.734515960—07 —1.197367770—08
TABLE 3—Il
0=17
EXPONENT’
—9
N~
4
EXPONENT— —8
4. 5
EXPONENTU
7
yB. Sheorey F’
CCII)
N
0 1 2 3 4 5
1.107166230 00 6.849792050—01 1.666827360—01 3.812297160—02 3.328298500—03 4.970460210—04 7.394074650—06 2.215293150—06 —9.088699350—08 5.631115100—09
0 1 2 3 4 5 6 7 8 9
6
7 8 9 N
6
EXPONENT’ —9
4’
/ Double C’hebvsher expansions for scale functions CCII)
9 10
6.402346480—01 3.974747580—01 9.882564520—02 2.336658400—02 2.245994410—03 3.55500706004 1.005550640—05 2.120656020—06 —4.550831490—08 6.821940960—09 7
EXPONENT.—10
F’
C)??)
H
CC??)
0
4.471 54971 0—01 2.7780504511—01 6.962o237°0—02 1.681391240—0? 1.686861070—03 2.792922761’—O4 9.663055886—0? 1.968887250—06
0 1
3.635106400 00 2.256661240 00 5.659950570—01 1.387196200—01 1.415496380—02 2.435063280—03 9.110972570—05 1.919226040—OS
2 3 4 5 6 7
2
3 4 5 6 7
1.79596?9?D—07 .3.991630300—08
4
EXPONENT.
II 0 1 2 3 4 5 6 7 8 9 10
II
8
FXPOOENT’lA
0 1 2
3 6
S 6 7
N~
5
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
0
EXPONENT—
0
CCII)
H
5.86058110D 00 —1.153229680 00 4.528614180—02 2.862755730—03 —1.190049030—03 9.818049650—05 2.001660690—OS —6.374465440—06 3.215650276—07 1.803805360—07 —3.96160504s—0l(
EXPONENT’
1.801924~3o—o7 —3.9A1129i,60—00
4—Il
0
CU?)
N— 6 TAELE
9 10 TABLE
4
131
1.431276700 01 —1.362764000 00 4.642026140—02 3.093237970—03 —1.194783460—03 9.778085700—OS 2.003131620—05 —6.373484100—06 3.218535550—07 1.820524310—07 —4.596455420—06 2.834862320—09 9.748021000—18 6.307117940—10 —7.216742340—10
#~ 7
EXPONENT’
U
3—Ill N’ 9
EXPONENT’—ll
CC??)
F’
CCI?)
3.310673730—01 2.055202060—01 5.1696178s1’—O? 1.28067440L’—02 3.3298135911—03 2.3412851)10—04 9.49271548)1—06 2.112852590—06
0 1 2 3 4 5 6 7
3.506886120—01 2.167353(40—01 5.371724650—02 1.344011750—82 1.358101250—03 2.509225050—04 9.125637200—06 2.406712080—06
F’
CCII)
C, 1 2 3 4 5 6 7
1.019488440 01 —1.531462730 00 4.6932715o0—02 3.221926750—03 —1.196519420—03 9.764233800—05 2.002298570—05 —6.183964710—06
1.1
CC??)
0 1 2 3 4 5 6 7
6.052416680 00 —1.6719115100 00 4.717868650—02 3.300253,240—03 —1.197151470—53 9.738527620—05 2.007491200—05 —6.064848440—0?
TABLE 4—Ill N~ 8
Table 4 6,orn of eq. (18), here labeled C(M) for Expansion coefficients
EXPONENT—
61
1
0
9
EXPONENT•
6
N
C (F’)
0 1 2 3 4 5 6 7 10. 9 10
1.011383110 01 —1.894693940 UI) 4.7171079190—02 3.386031900—03 1.19920067003 1.003135430—04 1.834711730—05 —4.450213090—06 —4.O’0A720510—O( —3.316304280—010 4.717520270—05
(5)
1= O,p
0
=
—e
8.0, --1 ~
0 1 2 3 4 5 6 7 8
-—
TAPLE 4—I N
0
F’
EXPoNENT—
0
CCII)
N’ 1
EXPONENT—
II
CC??)
F’
0 1 2 3
1.10912453000 5.163604970—01 —4.305551760—02 —3.563677840—03
0 1 2 3
6.42302342000 1.655324770—01 —7. 75413422D—03 —1.616881380—03
~ 6 7 8 9 10
_:1~:~ —2.003661520—05 6.373105060—06 —3.214072360—07 —1.801963190—07 3.982915390—08
‘ 6 7 0 9
1:0370769006 —1.070272050—46 6.01 719(300—09 1.049635680—08 —S.937S19360—10
EXPONENT’
N’
1.44898T6S0 01 —1.70989461011 0(1 4.711763140—02 1.353493910—03 1.19795504003 9.738773430—05 1.992850510—05 —7.192465330—06 1.649339470—06
-—
Table 5 Expansion coefficients cnm ofeq. (17). here labeled C(M), for = 0, po = 8.0,0 V~1) ‘~ I
1
—-——~—
TABLE 5i
0 4. 2
0
3
EXPONENT
———
EXPII1)ENT
0
?1
1
EXPONENT’ —1
0 C(S)
M
CC??)
F’
CC?.?)
O 1 2 3 4 5 6 7 0
1.722684100 00 —4.951691 550—01 3.405172960—02 1.1957199o0—03 —1.075422870—03 1.041014710—04 1.886830950—05 —6.370881140—06 3.324333010—07
0 1 2 3 4 5 6 7 8
1.000989090 01 —8.8007534A0—01 4. 24978A560—I2 7.388303090—03 —1.172921720—03 9.946778170—05 1.992228750—05 —6.377763230—06, 3.219843480—07
0 1 2 3 4 5 0
4.068089330 05 3.5050’577l11—02 1.0313431140—03 1. 33752347—OS —3.160304300—06 —1.074931140—07 —3,21 124110110—09
F’ 0 1 2 3 4 5 6 7 8 9 10
CC’?) 4.255518440—01 2.390455100—01 3.026388?SD—02 3.764343120—03 —2.332434010—04 2.583066150—O6 5.211389970—06 —7.01584451)0—07 —2.433369930—08 2.678004680—08 —3.185377030—09
132 II’
F lB. 2
FXPONPNT=
NI
—T
N— 3
CC??) 9. 547211X1?tN—01 5.9756220?’lOl 1.470.0414100—01 2.8993742211—0? 2.701480420—03 1.727964550—04 2.674930839—06 —1.38646033,10—OX 2.518S35400—0R
II
2 3 4 5 6
1)
Sheoic’y
/ Double C’hebo’shec expansons EXPONENT. —5
II
CCII)
0 1 2 3 4 5 6 7 5 9
3.400S67140 00 2.210401900 no 6.403169930—01 1.463889(70—01 1.837218100—02 2.107537280—03 1.332139930—04 7.811890180—06 2.491999010—07 9.020729270—09
2 3 4 5 6 ~ 11 9 10
N
2
for wa~’~filnction.V
5.0841239.00—02 3.5107027480—03 —1.195932410—03 9.755518170—05 2.0040342611—OS —6.373029270—06 3.21412758007 1.801964660—07 —3.98291 55711—00
2 3 4 5 6 7 8 9
EXPONENT’
N
0
CCII) TABLE EXPoNENT’ —5
I~
5
EXPONOIIT’
I
CO.?)
II
LOll
(1 1 2 3 4 5
1.67175,’O?D—Ob 1.109100920—01 3,4721 A18?6—02
0 1
1.04055845000 6.996930400—01 2.298006060—01
8.0.79404711—03
2 3
1,
1.7744540(0—03 1.690700250—04 1.491924890—05
4 5 6
7 1
1 .194258660—06 6.706924250—08
7 10
O
3 . 602 2 7 72411—09
EOPIINONT
6
—7
NO
1’
7.7o405044,.—U?
0
1 2 3 4
5.27261241011—0? 1. 7006o497U—O? 4. 713,1(1141’1.—33 X.21567955’,—04 1. ?378704”11—04
1 2 3 4
6.710602840—01 4.591925660—01 1.599353660—01 6. 30030701 0—07 (.019224010—03 1.219973680—03
7“
l.957ll7734l1~O5 1. Tl8A’,575”—0A
6 7 1)
1.448605090—04 1.509944940—05 1.315085920—06
2 3 4
5 F’
7 N
6=9
9.045774900—01 —3.63631 246D—02 —2.383037730—03 1173575600—03 —9.946918860—05
6 7 8 ‘2 10
—1.886800530—05 6.3708770811—06 —3.324353470—07 —1.795971230—07 3.99178700(1—08
6 7 8 9 10
—1.992241120—05 6.377788320—06 —3.21976741 0—01 —1.802000140—07 3.982577810—08
EXPONENT—
0
N
5
EXPONENTU
0
CU?)
F’
CCII)
N
0 1 2 3 4 5 7 8
4.387745340 OX 1.175088001) 00 —3.961829090—02 —2.860847280—03 1.190278590—03 —9.818275930—05 —2.0(11683710—05 6.374458610—06 —3.215686000—07
0 1 2 3 4 5 6 7 8
1.367792960 01 1.382094490 00 —4.159585200—02 —3.093369060—03 1.190.762650—03 ..9.778865590—OS —2.003020730—05 6.371092240—06 —3.21731 511 0—07
109
—1.802608590—07 3.985618720—08
109 11
—1.757206230—07 5.951879280—08 —1.108776110—09
12 13
—1.071737370—10 —4.240672550—09
14
3.027982380—09
Cli”)
6.356=69909—111 4,37216/510—01 1 . 540710111, [—01 4, 2,1 51’ (‘1411—U,’ 7.=’l’lo’175,,,—03 1.71211/261’ —50 19’,7’lo’,5’,’—34 1.7h12,~51,,_0S 1 .01321318 ‘11 —06
II 1
7.7138599/70—FJl 5.1l493731000l
2 3
1 .00094031 0—01 5.57724751 0—02
CC??)
M
4 S
0.X0735169D—03 1.3,11847420(l—03
II 1
1.0232(801901 1.548432679 00
0 1
F,
7.060067520—06
0
—4.269243430—02
7
7.4776811910—09
3
—3.223131030—03
A
2.50050122(1—06
4 5 1
1 .196415230—03 —9.767809870—05 —1.999802610(1—05
2 3 4 5 6
1.686999060 00 —4.340957450—02 —3.301334190—03 1 .197030330—03 —9.763305110—05 —1.995692360—05
7
6.1o5367430—06
7
6.267404660—06
N’
--
6
ExPONENT’
0
C(M), for
N
TABLE
7
EXPONENT
0
C U?) 6.64995529000
6—Ill
8O,p~ ~ ~ I TABCE
-
6-I N’
NI
00
N~
Table 6
0
7.439213400
CIII)
Expansion coefficients O,~ of eq. (18), here labeled
II
4
p,
—-
~ oo
5
12 3 4 5
EXpONENT——il
-
=
0
5—Ill
EXPI100NT=—1l1
II 1
01
(I’
EXPONENT’ —9 C(S)
8
1.035397330
TABLE 6—Il
2.8847303,50—00 7
1?
o’
CCII)
(4
9.217977330—07 5 . 7281 55680—08
9
T0BIE
0
1.306838450—05
(IF’)
I
E0PONENT
5.222609590—01 —2.7260838411—02 —1.185452050—03 1.07671875003 —1.040976390—04
12 3
5.878893000—02 9.657074210—03 1.384853560—03 1.423228490—04
10 I.
3
5—il 0
6’ 4
1.512397(70—02 1.633666230—03 —1.015S83770—X4 —5.0997087X0—06 1.071701530—06 —6.004024080—09 1.04955794D.08 5.933213030—10
EXPONENT’
0
CCI?)
0
1.15036667001
1
—4.85365491 0—01
N’
1
EXPONENT—
0
F’
CC??)
0 1
1.22239342001 —9.617012000—02
8
EXPONFNT
0
N’ 9
II
CIII)
H
O 1 2
0
-~
1.55S410639 01 1,XV18077T0 00 —4. 41 21 4)26 D —02 —1.35421X050—03
4
1.195618389—03
4
1 2 3
EXPONENT—
0
CCII) 1.159555600 01 1.905411870 00 —4 . 449262 750—02 —3.391598950—03 1.103372090—03
yB. Sheorey / Double C’hebyshei’ expansions for wave functions 5
—9.67915884U—05 —2.014481650—OS 7 786551 36l,~t2, —3620769850—07 —7:25614124x~t7 4.1331 70910—0?
5
7
‘0
10
5
—9.523617380—05 —2.061929100—OS 7.384064920—06 ~..560945180—06 1 .848804920.06
6
7 A 9
L
59467611—UT 9 93484811—01
U 1
—6
2
3
3
—1
5
—-
7606ATD—02
133
F’
0
—7.9132120—01 .2.50088011—01
—5.9448020—01 —4.607051 0—01
8.0103500—01 —0.878030 11—CT
1.0322870 00
9,9906580—01
—? .85085111.01
1.7005440—01
—6.531 874D—02
.9.9786010—01
1.0183420 00 —1. 20582)0—01 —1.0073009 011
9,9393510—01
—7.5807800—00
1.13321 4D—U1
0.900)1701—01
—9.8306620—01
1.7498411—01
~
~
1 0131740 00
6
—15904950—UT
7 0
—1 ‘021 77UD 00 5:0810880—02
TAIL?
Table 7 Regular and irregular Coulomb functions and their derivatives —
—
—
1991)
81A’-,I.5l,
7—I
9.
81*L
11,9
3 4 O 6 O A
5,~ 1
o
To, 4
1’
-
U
8,4575,,oo—’,1 , 69,4511I,—flT
4 5 O 7 A
—1
O,171,Il5I1—IIT
Ol,02o7’lfl—l’
1,’i9.’01011—l’1 1.o7’,ô’,21,—U2 1,94”972003 t.024’~420U4 1,4337141’111 0?)’,3~10—’1Z
I ?.9,’310l0’O1 O.311’15?fl”II? 4.5(01,111813 1•,16517,-U.’13 9,61763311—00 7.14443116’,’ 11
5,1777870—’’’
ST171S%O’~’7
—i
0’
. 16011041
Ii
—T .1097701
1
O
—7.7745570—01
1 2 I 4 0 6 7 S
—3.9613250—4)1 7,430T140’II 5.05571116 00 6.78414011—01 3.0910430—01 1.1006360-01 1,1119211102 0,49402311-103
40.’
1’
2 3
1,65)3470—00
1.0357240—02 9.840967(1—04
5.7108680 Il
1.2012510—04
—0,4680280 02 —3,5956990 03
1,0730900—57
3.7269050—00
7
1.1850850-1,8 3,9005820—10
1.8720950-07 6.37)29211—09
5,7724640 02 1,1084340 03 1,4011200 05 3.215=170 06 0,369571007
A
1.16=5920-IT
2,1192230-10
7.5070060
6,1250120-06
6.8334180-05
81*1
1.00
X
L
2.2757620—Ut 1
7,5170000-00
7 0 4 0 0 7 0
1.471510—I’2 ?.08~,4990—03 2.29=2930—04 2,0701770—05 1.97=4480—06 1,0506000—1’? ‘.1935630—00
L
F
—9.5001430”l,3 —5.7904590—Ill 7.0.62170—1 4.11003866—01 1,2666290.111 15300768-01 5,51.549(0.11? 1.67/1800—lI?
~0.U276170~,l ~0.9511670.111 ~.I,,9/1OD—)l 1/7(16660 7.1)20.040 1,4467099 1.016W/C ~.//4.41i4
8’
2 3
—1.2644990—01 —1.058032440
4 5 6 7 8
—4.854’142001 9.2067960’01 1,2304449 ‘10 1.1771940 00 9.1441170-01
0
4.7085990—6,1 8,6307180—03 T .1719390—03 1.21141069—04
1.70414=001 6,9445710 01 4.7722090 02 4.3092700 03
1.1046560—05
4,7839990
8,5151600—07 0.0084940—00
6.26001440, 9,4559714 III
—0.717427001 —7,0640210 02 —1 .9389850 03 —7.147037004 —2.9064490 0” —4.4011270 06
3
1.011
8$
6
—1027,1790—03
6’
4
1.15712)0 3) 1.150)410 00 8.6996720—01 4,81911140—01 2, T 1(40470—91
5.03846911 01’
—1,67,13048 60 —0.21451)0 00
6
7,4199050—32
1.9274810—02
1,4/06470
—1.81059711
7 8
0.4666990—03
1,2705010—02
4.472008601
—0.023533101
1.5246810—00
1,2760950—03
1,539894402
—3.014081007
T
7.05,’44611—02
1.6761750—01 4.2697960—01 3.8730660—01 7.3101130—01
1.0634040—01
1W ‘p1 /4 61 01
1.7141660—01 —7.8476766—01 —0.401,6111,—I1I —.71116611—41 .1.0991/1,’ 00 .15=57376 00 .131’1824[’ 01 -5.3/ 5426” 01
618. 1
F
—1.59001160—UT 4.21 76900—01 1.1187872090 1,7027430 01’ 2.7419=10 00 01
—0,3273199—01 —7.9400056—01 —0.1637050—01 —o.9702039—Q5
01
U
-1.0335100 00
2
—1.0)79140 00 —3.8327110—UT
0
7 9
7.00
5,3079140—61 1.T540.40 00 1.739)120 00 4,65001120-01 22,11366—ol 6.0711010—01 3.
‘7,115650.0—111 —5.1009470—01 1.507/250—01 3.9977270-03 7.4961.750—01 3.5765390.41
4.5521770—01 —1.0002090 011 —1.243857001
0
—30812270—
9
9.211,1299—02
3.2209’llO—Ul
9.5667650—01
1
—8l8404676—01
5:04119116—01
2 1 4 5 6
—9 45704411—111 —9:9577070—02 11,7840500—01 9.6271420—01 5,17301410—01
—1 961735(1—01 ‘9:6021950—01 —5,11666411—01 4.0247100—01 9.1241280-01
7
—8.3591491,—0l
9
—1.T173200
—7.8989410—01 —5.0101910 011 —4.5251270—01 1.9685460—Ill 1.1223770 1,0
16
0.2074)00—01 —7.5278230—01
—7.4542040—01 —7,5253550—01 —0.5459920-01
70
6
6’
0 1 2 3 4 5 6 7
—7, 76781 20—01
6.8234100—01
7,9246090—01
0
—3.0634140—05 1.0007100 00 7.0772610—03 ~0,8507910—01 —5.39317411—01 1.1620080—01 1.0096876 90
—1.0024920 30 —1.8955580—01 1.0193111000 1.5255100—05 —8.7590410—01 —7.5007970—01 3.4640800—01
—11,0892460—05 —4.11114980—01 6.9425740—01 5,1055080—01 —1.1766440—01 —9.9488210—01 1.3111850—01
‘9,3148270—02
9.1990470—01
1.000574000
658
8.110 4’
7.9114760—01 —4.0037520—01 -1.0398150 00 —1,0137/10 OIl
8,91680/0-01 6,0210306—01 1,3214956—01 ‘1,7791820—01
—1.7105416—01 4.515684001 1.14=610090 2.56306)10 IA, 1.7472986 09
—7.6963081)—Ut —(.5094641.—lW -5.9049450-01 —1.123169’ 00 —0.1779950.05
1,01)
0.
—
9
16.110
,
00
71 75
1
6 811
6—
1
5:0.1451=0
2 80 2 —61 0:2231210—01
—4 34556311—01 —1:044,1716 11’, —5,89764110—91 4.6091(90—01 T,0l19920 0”
I’ 7407606—01 856111450—0,1 ‘7.6147870—01 —8,4277690—01 -1,0967(00-01
6.0363711—UI
1,1009600—il
6.0016910—01
—1,4198430—01
—2.1120540—Ill
8,6778190—01
1.00
8
64.00
0’ L
9$ 64.00
0’
0
7.6293120-411 —3.2141856—01 ‘11.39474)11-01
—2.0411980—Ut —l .21167190—81 9,7034560111
9$
7—06
F’
—1.0510680 00 1. 3034060—Ui 1.0467700 OIl
F1800.50
—7.0036390 07
0.7440111—01
1
8
04
6’) 01’
4,1)0
F’
110.
,
0’ —1.2635900 -0.87113730
6’
ETA—0.5)
-4.4350011010
0,11430970 00 3,782874000
0
—3.8468910—01 —5.1692250—01 —1.0144410—01 2.6761900—01 4.1098040-01
07
—1.170084009
1,’ 5.0.45171—Il
19411 70.1 ETA’’II.Sfl 9 I’ll,)
—2.24,5420—lIT 1, CT 55169 00
-3.80727611
I 1.4873440—Ui 1.7141410-01
Fl?.
3 4 5
9 1
0
-0.829957004 —1.4800730 00
1.110
‘
T8FLE 5,926571063
N’
—3.349156000 —2,280285001
5
L
6
3.108007000 0.098468000
0’’
0,846087(1—1 —l,.451’7/41’41 ?./905’06’’1” —I.,,157101’’P, 11.10111170 64 2.4?69181’ 01 0.~111O~~206T (.210411 0” 5,1594110 U? —7.1.6763”6 III 5.5)11/1111’ 1 3.2’’1111’ ‘[4 9.0000)41’-~.’ 4.0.7601” 05 1,,1I?6161’ U16 —.1/61611’ 06
9~
0.50
8.
2.2693870—01 6.9818210—0?
3
618.—I-SI)
1.00 F’
8.3104050—02 1.S9204411—02
6 — ‘~.,,6l”~,9[~’ 0.3,7/SlIl-lIl 4,703I~8UlIII’ -,.61(9741- ~, 1 44,5771=1 ,/73’,?,ll_11 1. 4’’9/o II — .6(401 1,57T5111,I.’ 1 008064,.11/ ,,, 99146 I ll~ _/.4?llI)401 T.10(774113 ‘1.1/3,4=0=’’ ,,,1,(I’Il II -,l, 160=6 , ‘-.63’.Z,’19 , 56.5’/1T11’011.’’1’/4 _II.611111981 3.00T53711_,9 1.0i838’U~05 (6II1/~1l,.._149 10(76118 1.19,7020-”? 1.101’0110,,1 1,069717” _1,8(,,05/1ll 3991 1.’lO—l’’l 6.1,21616—lI 01011161, 11,, .1.11 05371 01’ 1.181509’I—lII 7.1707=7-00 7,4930170/6 -1.9 0934 0
F
7—Hi
10
4 5
O
~
~
F
1’
6
0’
0.91 24070—81 5.1897886—01 ‘.9.1496230—01 —4.2592080—01 8.3452290—01
0 1 2
1.4799450—01 1.0074190 00 4.4711910—01
—6.0014200—01 1.4555840—02 0.8849570—01
—8.7668780—01
—1.3609730’Oi
1,0250800—37 9.0545650—01
—9.91)6826—01 —4.4054990—01
3
—6.0131720—01
7.3109890—01
7.4381870—01
6.8869430—01
4 5 6 7 8
—0.1594260—US 1,5879356—31
—4.1914670—91 —9.7960520—01 —5.01 31660—01 0,2694420—01 5.6224500—UI
—4.2402970—01 —9,9759030—01 —1.0128990—01 9,4651440—01 S,7625A36—)l
—4.5904440—01 —9.45161211—01
—3.0617566—TI
1.0003600
0)
3.5135270—01 —0.3282630—UI
6.6052160—01 6.8921160—01 —1.5502470—TI —9.8425780—01 —1.4550920—01 8.5180580—01
yB. Sheorev / Double C’hehvshev expansions for
134
wale finctions
angular momentum, 1, is taken IC) be zero and that of
wave (‘oulonib functions for any value of the indepen-
Po to be 8.0. These tables have all been printed iii the sanle form. Thus, for example, the function A, 1(77) of eq. (6)is given by
dent variable and for ~iin the range
51 =
lO~ ~ IS
a,,,11T(i~),
(22)
0
—
I <
< 1. For
attractive and repulsive Coulomb t’ie]ds with charge, z = ±I , this corresponds to all particle energies greater than or eqLtal to 13.6 eV. The coefficients for expansbus of the derivatives of the Coulomb functions are obtained using methods discussed in [1]. For I > 0 the
svhere F, the exponent, is also given in the tables. It should be noted that the expansion coefficients a,1180.
functions may be obtained by recursion using the method also discussed in [1]. 1-or low energies, that is,
cm0 and °IOI7C [cf. eqs. (6), (10), (17) and (18)1 have all been labeled 6’(M) in tables [-—-6. Finally in table 7 we give values of the radial regular and irregular Coulomb functions and their derivatives for the partial waves, I = 0, I, 2 8 and for some values of the independent variable, p, which has been labeled x in this table. For x <8.0 we have used coefficients from tables I and 2 and for x ~ 8.0 we have calculated the functions using appropriate coefficients ~ ii tables 3-—6
for very large values of i~, this method would probably be uneconomical. Some methods for such a case are discussed by Frhherg [41.
~nI1l’
roi
5. Conclusions It is clear that the double Chebyshev expansion technique enables us to rapidly evaluate the radial s-
References [I[ yB. Sheorey, Conoput. Phys. Commun. 7 (1974) 1. [21 NI. Abraniowitz and IA. Stegun, Handbook of MathemathoOt Functions, NBS Appt. Math. Series NI). 55 (US Govt. Printing Office, Washington D.C.. 1968). [31 C.W. Clenshaw, Mathematical Tables, Vol. 5, National Physical Laboratory (H.M. Stationary Office. London, 1962 [4[ (‘.l. l”robL’rg, Rev. Mod. Ploys. 27 (1955) 399.