Accelerating the convergence of the zonal harmonic series representation in the Schumann resonance problem

jonmd ofzitmorphwic and TE-EIICZ~~~ Phyricr,Vol.

4% PP. 977-980

Q papsnon Press Ltd., 1978. Rinted in Northem Ireland

ACCELERATING TBE CGNVRRGENCE OF TlBl ZONALNARlX)NICSERIES ~P~SN~ATION

IN TSR So

J.N.L.

RBSONANCR PROBLEM

Connor and D.C. Mackay

Department of Chemietry, University of Mauchester, Manchester Ml3 9PL U.K. (Received 3 May 1978)

A formula of Niclcolaenho and Rabinowitz for the calculation the first

hind of complex degree is corrected.

A numerical

of Legendre’s

function

test of the corrected

of

formula

is preseoted. INTRODUCTION

1. In conxection have discussed

with the Schumann resonance problem,

wethods for calculating

Legendre’e

when the degree v is a complex number and 0 < harmonic series suited

representation

for Pv(-co6

for numerical computation.

technique

to accelerate

different

series

Nicholaenho and Rabinowitx

function

of the first

They point

e 6 II.

6) is very slowly

However, using a clever

‘eingularity eerier,

for Pv(-COB 61, which is better

(1974)

Pv(-car

S),

out that the xonal

convergent

the convergence of the zonal hanmuic

representation

kind,

and therefore

badly

extraction’ they obtain

suited

a

for computational

purposea. We have also been interested theories

of elastic

scattering

and Jakubetz 1977,1978).

in methods for calculating

in atomic and molecular

Pv(-toe

collisions

e),

(see,

in comection

with

for example, Conuor

However, we have found that the formula of Nicholaenlco and

Rabinowitx cannot be used directly,

because it contains

errors.

The purpose of this

short

paper is (a)

to correct

(b)

to report

the errore

in their

a numerical test 2.

Gut starting

point

fonmla

of the (corrected)

P,(-COBe) 5 i

equation

representation

m=o v (v+l)-m(m+l)

where Pwkos 6) is the Legendre polynomial differ8

formula - see Section

3.

for Pv(-coa

6):

(2x+1)

II

(1953),

2,

SERIES REPRESENTATION

is the eonal harmonic eeries

sin(m)

formulae given by Erd&yi

- see Section

of degree m.

Hansen (1975),

~~(~08e),

(1)

Equation (11 agrees with similar

MacRobert (1967) and Wait (19701, but

by a minus sign from the formula quated by Nickolaenho and Rabinowitz - see their (4).

ShortPapers

978

By following we obtain,

representation n

the singularity

after

a considerable for Pv(-co8

extraction

method described

awunt of algebraic

by Nickolaenko

manipulation,

and Rabinowitz,

the following

series

6):

= (4) P (‘COB e) = -2Rl-R2-2 [v (v+l) +l] R3-3 (3” (v+l) +2] R4+Joar,, Pw(CoS 6) ,

-

(2)

‘.

sin(m) where

a(x) ,

R1’ R2 =

$(3x2-4x+,),(x)-

R3 -

i

R4 -

$[2(1-x)3]*-

(l-x) (5x2-4sl)a(x)-

+x + 1,

$2(l-x)3]

1+ &

(1-x2) [2(1-x)]

‘+ &(15x2 - 27x + 14),

with [(l-x)*+2* L(x)

- hl

-

X

1

~7zi-v

COB

e,

and

at4) m

((w+l) (x+2) (x+3) (x+4) [v(v+l)-m(m+l)l)-1

I

x~2m[v2(v+1)2+22v(v+l)+l2]+l7v2(v+l)2+74v~v+l)+24). Equation (2) differs of its

from the correepouding

eigns and in the expression

NIJMRRICALTEST

3. We have carried

formula of Nickolaenko and Rabinowite in some

for R4.

out a numerical teat of the series

comparing the results

from equation

subroutine

(1975),

procedures

in the CERNeubroutine

representation

(2).

We did this by

(2) with those obtained

which calculates

Pv(-cos

from a CRRNprogram library -6 6) to a relative accuracy of 10 . Thenumerical

are not based on equation

(2).

80 this

allows UB to carry

out an independent test of formula (2). Results

for v - (0.1,O.l) In all

1 and 2 respectively. have aleo carried obtained evaluated

out calculation.9

very close

by recursion

Finally,

by 68 cot8).

figures.

We

for many other values of v and 6 and have similarly

used up to 100 to point

from equation

(2) and those from the CRRN

terms in the suummtion (2),with

out that when Ivlal,

for P,,(-COB 6) ie the uniform asymptotic replaced

values of 6 are shown in Tables

the Pu(cos 6)

(Magnue et c&1966).

we would like

however, a misprint

at various

agreement between the results

Our calculation8

subroutine.

and v - (3.0,3.0)

cases there is agreement to 5 or 6 significant

in this paper.

a very useful

approximate formula

expansion derived by Chulchrukidze (1965)

In the equation

defining

A,(6),

60 cot6

(Note,

should he

979

Short Papers

TABLE 1:

Valuesof Pv(-COBa) for v = (0.1,O.l).The numberin parentheses indicatesthe power of 10 by which the entrymust be multiplied. 1

equation (2) aldeg IP~(-COS

0) I

CRRN

arg Pv(-co8e)

pv(-~O~

arg Pv(-cose)

e) t

20

.743105

-.167482n

.743105

40

.816989

-.934672n(-11

.816989

-.934b72n

60

.a73902

-.573736v(-1)

.a73902

7.573736~(-1)

a0

.916564

-.354491vf-1)

.916565

-.354491n(-1)

100

.948468

-.209456v(-1)

.948468

-.209456*f-l)

120

.971742

-.111534v(-1)

.971742

-.111534v(-1)

140

.987654

-.4782Olv(-2)

.987654

-.478204nf-2)

160

.996944

-.117131v(-2)

.996944

-.117231n(-2)

TARIJI 2:

-.167482n (-1)

Valuesof Pv(-co8e) for v = (3.0,3.0). The number in parentheses indicatesthe power of 10 by which the entrymust be multiplied.

equation (2)

CRRR

aldeg {~v(-cose)l

arg Pv(-co89)

lgt-co5

6) I

arg

Pv(-co8a)

20

.131a59 (4)

-.986330n

.13ias9 (4)

-.98633On

40

.346269 (3)

-.5909llr

.346269 (3)

-.590911n

60

.105804 (3)

-,198762*

.105804 (3)

-.198762n

80

.350599 (2)

.192534v

.350599 (2)

.192534n

100

.123764 (2)

.5s3a47+

.123764 (2)

583847%

120

-466920 (1)

.975305v

.466920 (1)

.975305v

140

.188772 (1)

-.63106lr

.188772 (1)

-.63106lv

160

.101260 (1)

-.20454Yv

.I.01260(1)

-.20454av

4.

CONUUSIONS

A formulaof Nickolaenko and Rebinowitzfor Pv(-cos0) has been corrected. Tbe correctedformulacan be used to evaluatePv(-co8Of to high accuracyon a digitalcorrtputer. A~~O~RD~NTS Supportof this researchby the ScienceResearchCouncilin the form of a Research Studentship to DCM ie gratefullyacknowledged.The numericalcalculations were carriedout on the CDC 7600 computerof the Universityof ManchesterRegionalComputerGentre.

980

Short Papers

REFERENCES CRl.lK.RRLlKIDZE N.K. (English translation :

1965

Zh.Vt/chieZ.Mat.mat.Fiar.l 742.

1965

USSR Comput.Math.Math.Php. 2(4)222)

CONNOR J.N.L. and JAKUBETZ W.

1977

Mutec.Phy8.33, 1619.

CONNOR J.N.L. and JAKUBETZ W.

1978

MoZec.Phye.2,

ERDRLYI A.

1953

Higher%ammmdentuZ

949. Functions.

McGraw-Hill, New York, Vol.1, p.167,equation (6). HANSEN E.R.

1975

A Table of Seriesand Products. Prentice-Rail,Englewood Cliffs, p.301, equation (46.2.20).

1967

MacROBERT T.M.

SpherbaZ

Eannonica.

Pergamon, Oxford,

3rd edition, revised with the assistance of I.N. Sneddon, p.300, equation (34). MAGNDS W. OBERHETTINGERF. and SON1 R.P.

1966

&IUVRdaS and !rheomn8 for Func-tti

th@ Specdut

of ldmYmatti1

P~8icS.

Springer, Berlin, 3rd enlarged edition, p.232. NICKOLAENKO A.P. and RABINOWITZ L.M.

1974

J. Akno8.TePT.Phy8.

WAIT J.R.

1970

Ete&wmpetic

2,

979.

Wavesin Stratified

Media.Pergamon, Oxford, 2nd edition including supplementedmaterial, p.164,equation (8.3).

Referewe

it, aho

sradg to the fottooaCng unpublished nut&t:

CERN Computer Centre Program Library

1975

subroutine C311.