Table Of ContentOTHER TITLES IN THE MATHEMATICAL TABLES SERIES
Vol. 1. ATTWOOD—Six-figure Logarithms, Antilogarithms and Logarithmic
Trigonometrical Functions.
Vol. 2. ATTWOOD—Six-figure Trigonometrical Tables and Formulae.
Vol. 3. FLUGGE—Four-place Tables of Transcendental Functions.
Vol. 4. CHISTOVA—Tables of Bessel Functions.
Vol. 5. LEBEDEV and FEDOROVA—A Guide to Mathematical Tables.
Vol. 6. BURUNOVA—A Guide to Mathematical Tables Supplement No. 1.
Vol. 7. SMIRNOV—Tables of Airy Functions and Special Confluent Hyper-
geometric Functions.
Vol. 8. DYNKIN—Theory of Markov Processes.
Vol. 9. DEKANOSIDZE—Tables of Lommel's Functions of Two Variables.
Vol. 10. ABRAMOV—Tables of lnr(z) for Complex Argument.
Vol. 11. FADDEYEVA and TERENT'EV—Tables of Values of the Probability
Integral for Complex Argument.
Vol. 12. ZHONGOLOVICH and AMELIN—Tables and Nomograms for the
Processing of Observations Made on Artificial Earth Satellites.
Vol. 13. NOSOVA—Tables of Thomson Functions and Their First Derivatives.
Vol. 14. WARMUS—Tables of Elementary Functions.
Vol. 15. KIREYEVA and KARPOV—Tables of Weber Functions.
Vol. 16. SMIRNOV—Tables of Students T' Distribution.
Vol. 17. ARSCOTT and KHABAZA—Tables of Lame Polynomials.
C. tf. EEJIOVCOB
TABJIHIJ;H
HOPMHPOBAHHLIX
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JIEJKAH^PA
TABLES OF
NORMALIZED ASSOCIATED
LEGENDRE POLYNOMIALS
by
S. L. BELOUSOV
Translated by
D. E. BROWN, M.A.
PERGAMON PRESS
OXFORD LONDON • NEW YORK • PARIS
1962
PERGAMON PRESS LTD.
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4 & 5 Fitzroy Square, London W.l
PERGAMON PRESS INC.
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PERGAMON PRESS G.m.b.H.
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Copyright
©
1962
PERGAMON PRESS LTD.
This translation has been made from S. L. Belousov's book
entitled Tablitsy normirovannykh prisoyedinennykh polinomov
Lezhandra, published in Moscow 1956 by the Academy of
Sciences U.S.S.R.
Library of Congress Card Number: 62-17650
Printed in Great Britain at the Villafield Press, Bishopbriggs, Glasgow
INTRODUCTION
1. Spherical Functions and their Application to the Solution of Physical
Problems. The need arises in many problems in which a role is played by
functions defined on the surface of a sphere, to write the functions as series in
an orthogonal system of functions. A closed system of orthogonal functions on
the surface of the unit sphere is formed by spherical functions of the type:
Y m (0, k) = P™ (cos 0) cos mk, Y„'m (0, k) = P™ (cos 0) sin mk, (1)
n
where 0 and k are spherical co-ordinates: 0 is the colatitude, k the longitude
(0 ^ 0 ^ 7i, 0 ^ k ^ 2n); P m(cos0) is the associated Legendre polynomial, whose
n
indices m and n take the values: m = 0, 1, ...; n = m, m +1, ...
Associated Legendre polynomials are solutions bounded on the segment
-1 ^ x ^ 1 of the differential equation:
( +l, w=o (2)
£K'>^FH" " -^h
(x = cos 9).
An expansion in spherical functions can be written as the following series:
oo oo
F (M) = E I F: Y? (6, X) + F'm y„'m (6, X), (3)
n
m = 0 n = m
where F(0,X) is a function defined on sphere, and the coefficients F„m and F'™
are given by:
2n n
f fF(9,A)Y?(9,k)sin9d9dk
rm _ 0 0
f f[Y?(9,A)Ysm9d9dk
0 0
2n n
j fF(0, k) Y„'m (9, k) sin 9 d9 dk
n In n ' V }
f f [Y,;m (9, k)f sin 9 d9dk
0 0
The infinite series.(3) is uniformly convergent and its sum is equal to F(9,k)
with very wide assumptions regarding F(9,k). It is in fact sufficient for the
integral of F over any circle of latitude (with the pole of the sphere at the point
9,k) to be a function with continuous first derivative with respect to its argument.
For the proof of the convergence of the series, and for further information on
3
4 Tables of Normalized Associated Legendre Polynomials
spherical functions, see, e.g. V. I. Smirnov's Course of Higher Mathematics [5],
A convenient method of evaluating the coefficients F„m and F' m, in the case when
n
the function F(0,X) is given at the vertices of a regular polyhedron, is due to
V. A. Ditkin and L. A. Lyusternik [2].
The number of terms of series (3) needed for approximating F(0,X) can be
roughly estimated from the number and distribution of extrema of the function
F(6,X). We must always write in series (3) the terms corresponding to all the
spherical functions Y™(6,X) and Y™(6,X), the number of extrema of which with
respect to 6 and X is not greater than the number of extrema of the function
F(6,X). The functions Y?(6,X) and Y™(0,X) have n-m and 2m-1 zeros with
respect to 6 and X respectively inside the interval 0 < 0 < n 0 < X < 2n. A satis-
9
factory approximation by series (3) of certain functions F(Q,X) encountered in
physical problems is rendered difficult by the absence of tables of associated
Legendre polynomials P„m(cos0) with sufficiently large indices m and n.1
Of the existing tables of associated Legendre polynomials, the most complete
as regards the number of tabulated functions P„m(cos0) are the tables of Ya. M.
Kheifets [6], where the values of /^(cosfl) are given every 5° for indices
1 <; m ^ 12, m ^ n ^ 20. Let us estimate the minimum extent of perturbations
of the function F(0,X) that can be reflected with the aid of series (3) with m ^ 12
and n ^ 20. We shall assume that the perturbation has the same dimensions
with respect to 6 and X. On equating m and n — m +1, we now find that (m) =
max
(n — w+l) » 10, so that the minimum size of perturbation of a given sign
max
which can be satisfactorily represented by series (3) amounts to 180°: 10 = 18°
along a meridian and along a latitude.
The present tables contain the values of the associated Legendre polynomials
for indices 0 ^ m ^ 36 and m ^ n ^ 56 with intervals of 6 equal to 2.5° (the
domain of indices m and n corresponding to the tabulated functions is repres-
ented schematically in Fig. 1 by the trapezoid ACDE). On repeating the above
estimation of the minimum size of perturbation, we find that the present tables
enable a perturbation of the order 180°: -^-^ 6.5° with respect to 6 and X to
be mapped with the aid of series (3). With a 2.5° step in the tables, the per-
turbation in question will be represented by two to three points along a meridian.
It may be mentioned that the presence in series (3) of the terms with indices
m and n corresponding to the triangular domain DEF (Fig. 1) does not make
the representation of functions substantially more accurate, provided we leave
out of account perturbations of small extent with respect to X (less than 5°) and
"stretched" along the meridian (by more than 9°).
Functions of spherical co-ordinates 9 and X figure in numerous geophysical
applications. The meteorological characteristics of the atmosphere (fields of
temperature, pressure, etc.), the intensity of the Earth's magnetic field, the
components of the electromagnetic field of radio waves produced on dif-
fraction round the Earth's surface, etc., are referred to these co-ordinates.
Functions of spherical co-ordinates also play a part in problems of the scat-
tering of light by particles, in the wave functions of quantum mechanics, and
so on.
The possibility of representing with the aid of series (3) small pertur-
bations of function F(6,X) amounting to a few degrees is important, in
particular, for numerous meteorological problems, in which an expansion in
1 See [10] for a bibliography of tables of Legendre polynomials.
Tables of Normalized Associated Legendre Polynomials 5
spherical functions is a very widespread mathematical method. Some works may
be mentioned, where this method has been used for solving the problem of the
pressure distribution in the Earth's atmosphere. By solving a system of equa-
tions of hydrodynamics as applied
to the general circulation of the
atmosphere, E. N. Blinova [1] F
found an explanation of the exis-
tence and disposition of stationary
pressure waves in the atmosphere,
considered as a rotating spherical D
gas layer. The pressure distribu-
tion over the terrestrial sphere was
- G
obtained in this work as a series
in spherical functions. Further
precision was introduced into
these calculations by A. M. /& 8
Mkhitaryan [3],
We also find problems allied
Fig. 1
to the above — as regards the
methods of solution — in astro-
physical studies on the circulation of the atmospheres of planets [12].
2. Definition of P™(cos0). The published tables contain the values of the
normalized associated Legendre polynomials P™(cos0), defined by
^(cosfl) = J ^ ^ ^ P : ( c o s e ), (5)
2 (n + m)!
where dmP (cosO)
P™(cos0) = (sin0) n (6)
d(cos0)"
Here, JP„(COS0) is the ordinary Legendre polynomial, given by
P:(x) = -^1- ^-(dx"2 -l)n; (x = cos0). (7)
2" n dxn
It may be remarked that a different normalization of the Legendre polynomials
is taken in [8], viz., the normalizing factor in formula (5) is equal to
2n+l(n — m)\
In (n + m)!
We have in our normalization:
a
/KW] dx= 1. (8)
3. Computation of P™(cos0). Of the various representations of the normalized
associated Legendre polynomials, the following can be used for purposes of
tabulation:
1. The representation of P™(cos6) as a series in trigonometric functions of
multiple arcs.
6 Tables of Normalized Associated Legendre Polynomials
2. Recurrence relationships between the polynomials P™(cos6) of different
indices n and m.
The first of these representations was used for making up the tables of
Legendre polynomials of A. Schmidt [9], Tallquist [11], and also [10]. One
of the possible methods of computing the coefficients of the trigonometric series
representing the normalized Legendre polynomials is discussed in detail in a
work already mentioned [8]. The coefficients of the trigonometric series are
computed in the same work to fourteen places after the decimal for normalized
Legendre polynomials with indices m = 0, 1,... 20 and n = m, m +1,... 20
with all the intermediate working.
The method described in [8] for computing the coefficients of the trigono-
metric series consists in the following. Starting from the definition given above
of P™(cos9), the coefficients of the series for
dmP (cos6)
n
d(cos9)m
in powers of cos0 are computed. These series have the following form:
dm P (cos 6) ^, , * /m
n w m
" ' = I M*> "0 (cos 0)\ (9)
d(cos6)m k
Series (9) is then multiplied by (sin0)m, and the products of powers of trigono-
metric functions are replaced by trigonometric series. After regrouping the
terms, the following trigonometric series are obtained for the normalized
Legendre polynomials:
P;(cos0) = Xa*(«,m)^°V (10)
Here, cos q6 corresponds to even values of m, and sin qO to odd values of m.
With n even, q = 2k0 and with n odd, q = 2(k+ 1)0. The maximum value of
9
the argument qO in no case exceeds nO.
All the coefficients a (n,m) in formula (10) are less than unity. They are
k
formed, however, as a result of computing algebraic sums, the terms of which
are as a rule much greater than unity. The number of terms of a sum is of the
order
/ n-m\
max(
The computation of a (n,m) as a small difference between large numbers
k
requires that the intermediate calculations be carried out with a sufficiently
great absolute accuracy. Thus, the terms of the sums defining the coefficients
a (n,m) are computed in [8] to twenty-eight figures. This enabled the co-
k
efficients a (n,m) for n = 20, m > 5 to be obtained with only twenty-two correct
k
places after the decimal.
It may be mentioned that, when computing the coefficients a (n,m) the loss
k
in accuracy increases with increase of n and m,.
The other possible method of tabulating normalized associated Legendre
polynomials is based on the use of various recurrence formulae.
Both the methods mentioned for computing P^(cos0) were used for making
Tables of Normalized Associated Legendre Polynomials 7
up the present tables. The polynomials P™(cos6) for which the coefficients of
the trigonometric series are given in [8] were computed from formulae of type
(10), using the indicated coefficients a (n, m). This part of the tables corresponds
k
to the triangle ABG of Fig. 1. The remaining part of the tables was computed
from recurrence formulae, and corresponds to the area BCDEG of Fig. 1.
The fundamental difficulty that arises when tabulating P™(cos6) by means of
repeated application of recurrence formulae, is bound up with the possibility of
the error in the initial magnitudes accumulating in the computational process.
Generally speaking, it is possible not to bother about the computational scheme
being stable in the sense of damping of the initial errors, but in this case all the
intermediate calculations have to be carried out with a number of places
substantially exceeding the number of correct places in the result.
Let us compare the degree of accumulation of the errors arising when com-
puting the normalized associated Legendre polynomials with the aid of the
various recurrence formulae. We consider from this point of view the familiar
recurrence formula
P™ (cos 0) = 2a™ cos 0P™_, (cos 6) - b™ P™_ 2 (cos 0), (ID
where
am= jn2~* bm= A2"*1)^-"*-1)^"2-1)
(12)
n Vn2-m2' " \ (2n-3)(n-m)(n + m)
If we take as the initial magnitudes the values of P? (cos0) and P^ofcosfl),
9
obtained say by formula (10) with the coefficients of [8], relationship (11)
enables us in principle to compute P™(cos0) for m=l,2,...20 and «^21.
Inasmuch as each polynomial is now obtained from all the polynomials with
preceding indices n, the errors in the result will correspondingly accumulate.
The absolute error of a given polynomial is connected by a linear relationship
with the error of the initial polynomials. It can be shown that the absolute
error e arising after A>times application of formula (11) is connected with the
k
initial error e for sufficiently large k by the relationship1
0
e = e \X\k, (13)
k 0
where A is the root with the greater modulus of the characteristic equation
l2 = l.2a"cos9-b™. (14)
Relationship (13) holds for the case when a™ and b™ do not vary with increase
of w. In the interval of variation of n of interest to us (21 ^ n ^ 56), the co-
efficients a™ and b™ vary little in comparison with their mean magnitudes,
tending in the limit to unity as «-» oo. To make an approximate estimate of
the error from formula (13), we shall assume these coefficients to be constant
and equal to their mean values. Figure 2 illustrates the coefficients a™ and b™
for m = 10 as functions of n.
As is clear from formula (13), if |A| < 1, the initial error will be damped
in the computational process. Whereas if |A| > 1, the error in P™ will accu-
mulate and may exceed the permissible maximum error.
1 Relationship (13) was indicated to the author by A. A. Abramov.
8 Tables of Normalized Associated Legendre Polynomials
The greater root of equation (14) is
A = <cos0 + V«cos0) 2-^ (15)
With sufficiently large 0, when
(a;cos0)2<fc;r, (16)
\l\=y/l%< '• Thus> if i = 21 and m= 10, we have <#«1.14 and Z>™»0.98,
and | A | « 0.99 at 0 = 30°. We find with smaller values of 0 that | A | > 1.
Thus the initial error in the
values of P™(cos0) does not
accumulate for sufficiently large
0 during the course of compu-
tation using formula (11), and
the formula is suitable for tabu-
lating the polynomials. On the
other hand, when 0 is small (in
our example, when 0 < 30°), the
error can accumulate, and the
computation becomes unstable
when repeated use is made of
formula(ll). However, a direct
check on the stability (see
below) has shown that the re-
Fig. 2 quirements imposed on 0 by
relationship (16) are too rigid.
It may be remarked that the computational stability criterion obtained,
roughly corresponds to the following simple rule: a computation using the
recurrence formula is stable if its coefficients are less than unity and have
different signs.
The coefficients eft increase as m increases (with fixed /i), so that a stable
computation is only possible for large m when cos0 is fairly small.
Formula (11) is therefore not suitable for computing P™(cos0) throughout
the range of variation of the parameters of interest to us.
The most suitable recurrence formula for computing P™(cos0) in the domain
BCDEG (see Fig. 1) has proved to be the following:
P™(cos0) = CPm -l(cos0)-dm cos0P™:1 (cos0) + em cos0P m_ (cos0), (17)
n n n n x
where
2/? + lm + n-lm + n-3
Cn = 2n — 3 m + n m + n —2'
d: = 2n + lm + n — ln — m + l # 2M+ 1 n — m
C =
2n — l m + n m + n — 2* 2n — ln + m
Formula (17) can be obtained say as follows. We write the relationships
6.735.2 and 6.735.4, deduced on p. 385 of [4], in the form
(n-m + l)P;_- 2-(/i + m-3)P^,= - sin 0 i^r/
1
and
cos0 P^_-P^ = -(w + m-1) sin OPjr/.
l
