[Abramowitz M., Stegun I.A.] Handbook of Mathemati(BookZa.org)

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Preface’ 
The present volume is an outgrowth of a Conference on Mathematical Tables 
held at Cambridge, Mass., on September 15-16, 1954, under the auspices of the 
National Science Foundation and the Massachusetts Institute of Technology. The 
purpose of the meeting was to evaluate the need for mathematical tables in the light 
of the availability of large scale computing machines. It was the consensus of 
opinion that in spite of the increasing use of the new machines the basic need for 
tables would continue to exist. 
Numerical tables of mathematical functions are in continual demand by scien- 
tists and engineers. A greater variety of functions and higher accuracy of tabula- 
tion are now required as a result of scientific advances and, especially, of the in- 
creasing use of automatic computers. In the latter connection, the tables serve 
mainly forpreliminarysurveys of problems before programming for machine operation. 
For those without easy access to machines, such tables are, of course, indispensable. 
Consequently, the Conference recognized that there was a pressing need for a 
modernized version of the classical tables of functions of Jahnke-Emde. To imple- 
ment the project, the National Science Foundation requested the National Bureau 
of Standards to prepare such a volume and established an Ad Hoc Advisory Com- 
mittee, with Professor Philip M. Morse of the Massachusetts Institute of Technology 
as chairman, to advise the staff of the National Bureau of Standards during the 
~course of its preparation. In addition to the Chairman, the Committee consisted 
of A. Erdelyi, M. C. Gray, N. Metropolis, J. B. Rosser, H. C. Thacher, Jr., John 
Todd, C. B. Tompkins, and J. W. Tukey. 
The primary aim has been to include a maximum of useful information within 
the limits of a moderately large volume, with particular attention to the needs of 
scientists in all fields. An attempt has been made to cover the entire field of special 
functions. To carry out the goal set forth by tbe Ad Hoc Committee, it has been 
necessary to supplement the tables by including the mathematical properties that 
are important in computation work, as well as by providing numerical methods 
which demonstrate the use and extension of the tables. 
The Handbook was prepared under the direction of the late Milton Abramowitz, 
and Irene A. Stegun. Its success has depended greatly upon the cooperation of 
many mathematicians. Their efforts together with the cooperation of the Ad HOC 
Committee are greatly appreciated. The particular contributions of these and 
other individuals are acknowledged at appropriate places in the text. The sponsor- 
ship of the National Science Foundation for the preparation of the material is 
gratefully recognized. 
It is hoped that this volume will not only meet the needs of all table users but 
will in many cases acquaint its users with new functions. 
ALLEN V. ASTIN, L?imctor. 
Washington, D.C. 
Preface to the Ninth Printing 
The enthusiastic reception accorded the “Handbook of Mathematical 
Functions” is little short of unprecedented in the long history of mathe- 
matical tables that began when John Napier published his tables of loga- 
rithms in 1614. Only four and one-half years after the first copy came 
from the press in 1964, Myron Tribus, the Assistant Secretary of Com- 
merce for Science and Technology, presented the 100,OOOth copy of the 
Handbook to Lee A. DuBridge, then Science Advisor to the President. 
Today, total distribution is approaching the 150,000 mark at a scarcely 
diminished rate. 
The success of the Handbook has not ended our interest in the subject. 
On the contrary, we continue our close watch over the growing and chang- 
ing world of computation and to discuss with outside experts and among 
ourselves the various proposals for possible extension or supplementation 
of the formulas, methods and tables that make up the Handbook. 
In keeping with previous policy, a number of errors discovered since 
the last printing have been corrected. Aside from this, the mathematical 
tables and accompanying text are unaltered. However, some noteworthy 
changes have been made in Chapter 2: Physical Constants and Conversion 
Factors, pp. 6-8. The table on page 7 has been revised to give the values 
of physical constants obtained in a recent reevaluation; and pages 6 and 8 
have been modified to reflect changes in definition and nomenclature of 
physical units and in the values adopted for the acceleration due to gravity 
in the revised Potsdam system. 
The record of continuing acceptance of the Handbook, the praise that 
has come from all quarters, and the fact that it is one of the most-quoted 
scientific publications in recent years are evidence that the hope expressed 
by Dr. Astin in his Preface is being amply fulfilled. 
LEWIS M. BRANSCOMB, Director 
National Bureau of Standards 
November 1970 
Foreword 
This volume is the result of the cooperative effort of many persons and a number 
of organizations. The National Bureau of Standards has long been turning out 
mathematical tables and has had under consideration, for at least IO years, the 
production of a compendium like the present one. During a Conference on Tables, 
called by the NBS Applied Mathematics Division on May 15, 19.52, Dr. Abramo- 
witz of t,hat Division mentioned preliminary plans for such an undertaking, but 
indicated the need for technical advice and financial support. 
The Mathematics Division of the National Research Council has also had an 
active interest in tables; since 1943 it has published the quarterly journal, “Mathe- 
matical Tables and Aids to Computation” (MTAC),, editorial supervision being 
exercised by a Committee of the Division. 
Subsequent to the NBS Conference on Tables in 1952 the attention of the 
National Science Foundation was drawn to the desirability of financing activity in 
table production. With its support a z-day Conference on Tables was called at the 
Massachusetts Institute of Technology on September 15-16, 1954, to discuss the 
needs for tables of various kinds. Twenty-eight persons attended, representing 
scientists and engineers using tables as well as table producers. This conference 
reached consensus on several cpnclusions and recomlmendations, which were set 
forth in tbe published Report of the Conference. There was general agreement, 
for example, “that the advent of high-speed cornputting equipment changed the 
task of table making but definitely did not remove the need for tables”. It was 
also agreed that “an outstanding need is for a Handbook of Tables for the Occasional 
Computer, with tables of usually encountered functions and a set of formulas and 
tables for interpolation and other techniques useful to the occasional computer”. 
The Report suggested that the NBS undertake the production of such a Handbook 
and that the NSF contribute financial assistance. The Conference elected, from its 
participants, the following Committee: P. M. Morse (Chairman), M. Abramowitz, 
J. H. Curtiss, R. W. Hamming, D. H. Lehmer, C. B. Tompkins, J. W. Tukey, to 
help implement these and other recommendations. 
The Bureau of Standards undertook to produce the recommended tables and the 
National Science Foundation made funds available. To provide technical guidance 
to the Mathematics Division of the Bureau, which carried out the work, and to pro- 
vide the NSF with independent judgments on grants ffor the work, the Conference 
Committee was reconstituted as the Committee on Revision of Mathematical 
Tables of the Mathematics Division of the National Research Council. This, after 
some changes of membership, became the Committee which is signing this Foreword. 
The present volume is evidence that Conferences can sometimes reach conclusions 
and that their recommendations sometimes get acted on. 
V 
,/” 
VI FOREWORDActive work was started at the Bureau in 1956. The overall plan, the selection 
of authors for the various chapters, and the enthusiasm required to begin the task 
were contributions of Dr. Abramowitz. Since his untimely death, the effort has 
continued under the general direction of Irene A. Stegun. The workers at the 
Bureau and the members of the Committee have had many discussions about 
content, style and layout. Though many details have had t’o be argued out as they 
came up, the basic specifications of the volume have remained the same as were 
outlined by the Massachusetts Institute of Technology Conference of 1954. 
The Committee wishes here to register its commendation of the magnitude and 
quality of the task carried out by the staff of the NBS Computing Section and their 
expert collaborators in planning, collecting and editing these Tables, and its appre- 
ciation of the willingness with which its various suggestions were incorporated into 
the plans. We hope this resulting volume will be judged by its users to be a worthy 
memorial to the vision and industry of its chief architect, Milton Abramowitz. 
We regret he did not live to see its publication. 
P. M. MORSE, Chairman. 
A. ERD~LYI 
M. C. GRAY 
N. C. METROPOLIS 
J. B. ROSSER 
H. C. THACHER. Jr. 
JOHN TODD 
‘C. B. TOMPKINS 
J. W. TUKEY. 
Handbook of Mathematical Functions 
with 
Formulas, Graphs, and Mathematical Tables 
Edited by Milton Abramowitz and Irene A. Stegun 
1. Introduction 
The present Handbook has been designed to 
provide scientific investigators with a compre- 
hensive and self-contained summary of the mathe- 
matical functions that arise in physical and engi- 
neering problems. The well-known Tables of 
Funct.ions by E. Jahnke and F. Emde has been 
invaluable to workers in these fields in its many 
editions’ during the past half-century. The 
present volume ext,ends the work of these authors 
by giving more extensive and more accurate 
numerical tables, and by giving larger collections 
of mathematical properties of the tabulated 
functions. The number of functions covered has 
also been increased. 
The classification of functions and organization 
of the chapters in this Handbook is similar to 
that of An Index of Mathematical Tables by 
A. Fletcher, J. C. P. Miller, and L. Rosenhead. 
In general, the chapters contain numerical tables, 
graphs, polynomial or rational approximations 
for automatic computers, and statements of the 
principal mathematical properties of the tabu- 
lated functions, particularly those of computa- 
tional importance. Many numerical examples 
are given to illustrate the use of the tables and 
also the computation of function values which lie 
outside their range. At the end of the text in 
each chapter there is a short bibliography giving 
books and papers in which proofs of the mathe- 
matical properties stated in the chapter may be 
found. Also listed in the bibliographies are the 
more important numerical tables. Comprehen- 
sive lists of tables are given in the Index men- 
tioned above, and current information on new 
tables is to be found in the National Research 
Council quarterly Mathematics of Computation 
(formerly Mathematical Tables and Other Aids 
to Computation). 
The ma.thematical notations used in this Hand- 
book are those commonly adopted in standard 
texts, particularly Higher Transcendental Func- 
tions, Volumes 1-3, by A. ErdBlyi, W. Magnus, 
F. Oberhettinger and F. G. Tricomi (McGraw- 
Hill, 1953-55). Some alternative notations have 
also been listed. The introduction of new symbols 
has been kept to a minimum, and an effort has 
been made to avoid the use of conflicting notation. 
2. Accuracy of the Tables 
The number of significant figures given in each 
table has depended to some extent on the number 
available in existing tabulations. There has been 
no attempt to make it uniform throughout the 
Handbook, which would have been a costly and 
laborious undertaking. In most tables at least 
five significant figures have been provided, and 
the tabular’ intervals have generally been chosen 
to ensure that linear interpolation will yield. four- 
or five-figure accuracy, which suffices in most 
physical applications. Users requiring higher 
1 The most recent, the sixth, with F. Loesch added as cc-author, was 
published in 1960 by McGraw-Hill, U.S.A., and Teubner, Germany. 
2 The second edition, with L. J. Comrie added as co-author, was published 
in two volumes in 1962 by Addison-Wesley, U.S.A., and Scientific Com- 
puting Service Ltd., Great Britain. 
precision in their interpolates may obtain them 
by use of higher-order interpolation procedures, 
described below. 
In certain tables many-figured function values 
are given at irregular intervals in the argument. 
An example is provided by Table 9.4. The pur- 
pose of these tables is to furnish “key values” for 
the checking of programs for automatic computers; 
no question of interpolation arises. 
The maximum end-figure error, or “tolerance” 
in the tables in this Handbook is 6/& of 1 unit 
everywhere in the case of the elementary func- 
tions, and 1 unit in the case of the higher functions 
except in a few cases where it has been permitted 
to rise to 2 units. 
IX /- 
. 
X INTRODUCTION 
3. Auxiliary Functions and Arguments 
One of the objects of this Handbook is to pro- 
vide tables or computing methods which enable 
the user to evaluate the tabulated functions over 
complete ranges of real values of their parameters. 
In order to achieve this object, frequent use has 
been made of auxiliary functions to remove the 
infinite part of the original functions at their 
singularities, and auxiliary arguments to co e with 
infinite ranges. An example will make t fi e pro- 
cedure clear. 
The exponential integral of positive argument 
is given by 
The logarithmic singularity recludes direct inter- 
polation near x=0. The unctions Ei(x)-In x P 
and x-liEi(ln x-r], however, are well- 
behaved and readily interpolable in this region. 
Either will do as an auxiliary function; the latter 
was in fact selected as it yields slightly higher 
accuracy when Ei(x) is recovered. The function 
x-‘[Ei(x)-ln x-r] has been tabulated to nine 
decimals for the range 05x<+. For +<x12, 
Ei(x) is sufficiently well-behaved to admit direct 
tabulation, but for larger values of x, its expo- 
nential character predominates. A smoother and 
more readily interpolable function for large x is 
xe-“Ei(x); this has been tabulated for 2 <x510. 
Finally, the range 10 <x_<m is covered by use of 
the inverse argument x-l. Twenty-one entries of 
xe-“Ei(x), corresponding to x-l = .l(- .005)0, suf- 
fice to produce an interpolable table. 
4. Interpolation 
The tables in this Handbook are not provided 
with differences or other aids to interpolation, be- 
cause it was felt that the space they require could 
be better employed by the tabulation of additional 
functions. Admittedly aids could have been given 
without consuming extra space by increasing the 
intervals of tabulation, but this would have con- 
flicted with the requirement that linear interpola- 
tion is accurate to four or five figures. 
For applications in which linear interpolation 
is insufficiently accurate it is intended that 
Lagrange’s formula or Aitken’s method of itera- 
tive linear interpolation3 be used. To help the 
user, there is a statement at the foot of most tables 
of the maximum error in a linear interpolate, 
and the number of function values needed in 
Lagrange’s formula or Aitken’s method to inter- 
polate to full tabular accuracy. 
As an example, consider the following extract 
from Table 5.1. 
775 . 
zez El (2) 
89268 7854 d0 
ze*El (z) 
. 89823 7113 
;:; : 89384 89497 6312 9666 g. I .89927 90029 7306 7888 
E : 89608 89717 8737 4302 ix8: 4 : .90227 90129 60”3 4695 
[ 1 ‘453 
The numbers in the square brackets mean that 
the maximum error in a linear interpolate is 
3X10m6, and that to interpolate to the full tabular 
accuracy five points must be used in Lagrange’s 
and Aitken’s methods. 
8 A. C. Aitken On inte elation b iteration of 
out the use of diherences, ‘B i: 8 
roportional parts, with. 
rot Edin urgh Math. oc. 3.6676 (1932). 
Let us suppose that we wish to compute the 
value of xeZ&(x) for x=7.9527 from this table. 
We describe in turn the application of the methods 
of linear interpolation, Lagrange and Aitken, and 
of alternative methods based on differences and 
Taylor’s series. 
(1) Linear interpolation. The formula for this 
process is given by 
jp= (1 -P)joSPfi 
where jO, ji are consecutive tabular values of the 
function, corresponding to arguments x0, x1, re- 
spectively; p is the given fraction of the argument 
interval 
p= (x--x0>/(x1-~0> 
and jP the required interpolate. In the present 
instance, we have 
jo=.89717 4302 ji=.89823 7113 p=.527 
The most convenient way to evaluate the formula 
on a desk calculating machine is.to set o and ji 
in turn on the keyboard, and carry out t d e multi- 
plications by l-p and p cumulatively; a partial 
check is then provided by the multiplier dial 
reading unity. We obtain 
j.6z,E.‘;9;72;&39717 4302)+.527(.89823 7113) 
Since it is known that there is a possible error 
of 3 X 10 -6 in the linear formula, we round off this 
result to .89773. The maximum possible error in 
this answer is composed of the error committed 
INTRODUCTION XI 
by the last roundingJ that is, .4403X 10m5, plus The numbers in the third and fourth columns are 
3 X lo-‘, and so certainly cannot exceed .8X lo-‘. the first and second differences of the values of 
(2) Lagrange’s formula. In this example, the xezEl(x) (see below) ; the smallness of the second 
relevant formula is the 5-point one, given by difference provides a check on the three interpola- 
f=A-,(p)f_z+A-,(p)f-1+Ao(p>fo+A,(p)fi 
tions. The required value is now obtained by 
+A&)fa 
linear interpolation : 
Tables of the coefficients An(p) are given in chapter 
25 for the range p=O(.Ol)l. We evaluate the 
formula for p=.52, .53 and .54 in turn. Again, 
in each evaluation we accumulate the An(p) in the 
multiplier register since their sum is unity. We 
now have the following subtable. 
x m=&(x) 
7.952 .89772 9757 
10622 
7.953 .89774 0379 -2 
10620 
7.954 .89775 0999 
fn=.3(.89772 9757)+.7(.89774 0379) 
= 239773 7192. 
In cases where the correct order of the Lagrange 
polynomial is not known, one of the prelimina 
T interpolations may have to be performed wit 
polynomials of two or more different orders as a 
check on their adequacy. 
(3) Aitken’s method of iterative linear interpola- 
tion. The scheme for carrying out this process 
in the present example is as follows: 
.; & Yn=ze”G@) Yo. I Yo. 1, (I Yo, 1.2. I Yo.1.a.s.n X,-X 
1 7.9 : 
89823 7113 .0473 
89717 4302 89773 44034 
2 8.1 89927 7888 :89774 48264 .89773 71499 
-. 0527 
3 7.8 : 89608 8737 
. 1473 
2 90220 2394 . 89773 71938 -. 1527 
4 8.2 . 90029 7306 4 98773 1216 
ii 
89773 71930 . 2473 
5 7.7 . 89497 9666 2 35221 2706 30 -. 2527 
Here 
1 Yo 
yo,n=- 
20-x 
x.--20 Yn x,-x 
Yo.1 
1 Yo.1 x,-x ,n=- G--z1 l/O.” x,-x 
1 l/0.1. . . Yo. ., n-1.98 x,-x 
1. 
. . 
., m--l.m.n-- ~n-%n Yo.1. . . -, m-1.n x,-x 
1 
If the quantities Z.-X and x~--5 are used as 
multipliers when forming the cross-product on a 
desk machine, their accumulation (~~-2) -(x,-x) 
in the multiplier register is the divisor to be used 
at that stage. An extra decimal place is usually 
carried in the intermediate interpolates to safe- 
guard against accumulation of rounding errors. 
The order in which the tabular values are used 
is immaterial to some extent, but to achieve the 
maximum rate of convergence and at the same 
time minimize accumulation of rounding errors, 
we begin, as in this example, with the tabular 
argument nearest to the given argument, then 
take the nearest of the remaining tabular argu- 
ments, and so on. 
The number of tabular values required to 
achieve a given precision emerges naturally in 
the course of the iterations. Thus in the present 
example six values were used, even though it was 
known in advance that five would suffice. The 
extra row confirms the convergence and provides 
a valuable check. 
(4) Difference formulas. We use the central 
difference notation (chapter 25), 
S2fl 
safz 
wa 
Here 
Sf1l2=f1-f0, 8f3/a=fz-f1, . . . ,, 
a2/1=sf3ia-afiia=fa-2fi+fo 
~af3~~=~aja-~aj~=fa-3j2+3fi-k 
8'fa=~aj~fsla-6~3~2=f4-~f~+~ja-4f~+fo 
and so on. 
In the present example the relevant part of the 
difference table is as follows, the differences being 
written in units of the last decimal 
B 
lace of the 
function, as is customary. The sma ness of the 
high differences provides a check on the function 
values 
xe=El(x) 
7:9 .89717 4302 
SY S4f 
-2 2754 -34 
8.0 . 89823 7113 -2 2036 -39 
Applying, for example, Everett’s interpolation 
formula 
. j~=(l-P)fo+E2(P)~*jo+E4(P)~4jo+ . . . 
+Pfl+F2(P)~afl+F4(P)~4fl+ . * * 
and takin the numerical values of the interpola- 
tion toe flf cients Es(p), E4 
!l 
), F,(p) and F,(p) 
from Table 25.1, we find t at 
,,/ 
XII INTRODUCTION 
10Qf.6,= .473(89717 4302) + .061196(2 2754) - .012(34) 
+ .527(89823 7113) + .063439(2 2036) - .012(39) 
= 89773 7193. 
We may notice in passing that Everett’s 
formula shows that the error in a linear interpolate 
is approximately 
mPwfo+ F2(P)wl= m(P) + ~2(P)lk?f0+wJ 
Since the maximum value of IEz(p)+Fz(p)I in the 
range O<p<l is fd, the maximum error in a linear 
interpolate is approximately 
can be used. We first compute as many of the 
derivatives ftn) (~0) as are significant, and then 
evaluate the series for the given value of 2. 
An advisable check on the computed values of the 
derivatives is to reproduce the adjacent tabular 
values by evaluating the series for z=zl and x1. 
In the present example, we have 
f(x) =xeZEt(x) 
f’(z)=(l+Z-‘)f(Z)-1 
f”(2)=(1+2-‘)f’(Z)--Z-Qf(2) 
f”‘(X) = (1 -i-z-y’(2) -22~Qf’(5) +22-y(2). 
With x0=7.9 and x-x0= .0527 our computations 
are as follows: an extra decimal has been retained 
in the values of the terms in the series to safeguard 
against accumulation of rounding errors. 
i 
p:‘(xo)/k! (x--so) y’k’(x0)/k! 
.89717 4302 .89717 4302 
; - .01074 .00113 0669 7621 -.ooooo .00056 6033 3159 3 5 
3 .00012 1987 .ooooo .a9773 0017 7194 9 
(5) Taylor’s series. In cases where the succes- 
sive derivatives of the tabulated function can be 
computed fairly easily, Taylor’s expansion 
~(x,=~(xo)+(x-x,,~~+(x-xo,~~~ 
+(~-x,)q$+ . . . 
5. Inverse Interpolation 
With linear interpolation there is no difference 
in principle between direct and inverse interpola- 
tion. In cases where the linear formula rovides 
an insufficiently accurate answer, two met fl ods are 
available. We may interpolate directly, for 
example, by Lagrange’s formula to prepare a new 
table at a fine interval in the neighborhood of the 
approximate value, and then apply accurate 
inverse linear interpolation to the subtabulated 
values. Alternatively, we may use Aitken’s 
method or even possibly the Taylor’s series 
method, with the roles of function and argument 
interchanged. 
It is important to realize that the accuracy of 
an inverse interpolate may be very different from 
that of a direct interpolate. This is particularly 
true in regions where the function is slowly 
varying, for example, near a maximum or mini- 
mum. The maximum precision attainable in an 
inverse interpolate can be estimated with the aid of 
the formula 
AxmAj/df 
dx 
in whichAj is the maximum possible error in the 
function values. 
Example. Given xe”Ei(z) = .9, find 2 from the 
table on page X. 
(i) Inverse linear interpolation. The formula 
for v is 
In the present example, we have 
.9 
‘=.90029 
- .89927 7888 72 2112 
7306- .89927 7888=101=‘708357’ 
The desired z is therefore 
z=zQ+p(z,--2,,)=8.1+.708357(.1)=8.17083 57 
To estimate the possible error in this answer, 
we recall that the maximum error of direct linear 
interpolation in this table is Aj=3X lOwe. An 
approximate value for dj/dx is the ratio of the 
first difference to the argument interval (chapter 
25), in this case .OlO. Hence the maximum error 
in x is approximately 3XlO-e/(.OlO), that is, .0003. 
(ii) Subtabulation method. To improve the 
ap roximate value of x just obtained, we inter- 
po ate directly for p=.70, .7l and .72 with the aid P 
of Lagrange’s 5-point formula, 
X xe=El (x) 6 QQ 
8. 170 . 89999 
8.171 . 90000 
8. 172 90001 
-.-_ 
1 0151 
3834 -2 
1 0149 
3983 
Inverse linear interpolation in the new table 
gives 
Hencex=8.17062 23. 
An estimate of the maximum error in this result 
is 
Ajl z df~1x10-8_1x10-7 .OlO 
(iii) Aitken’s method. This is carried out in the 
same manner as in direct interpolation. 
INTRODUCTION XIII 
n yn=xeeZE1(x) 2, Z0.n QJ.98 a.1 2.n zo.l.2.3.74 l/n-u 
0 . 90029 7306 8. 2 .00029 7306 
4 : 89927 90129 6033 7888 8. 8. 3 1 8. 8. 17083 17023 5712 1505 8. 1706,l 9521 -. 00072 00129 . 2112 6033 
3 . 89823 7113 8. 0 8. 17113 8043 2 5948 8. 17062 2244 -. 00176 2887 
% : 90227 89717 4302 4695 8. 7. 4 9 8. 8. 16992 17144 9437 0382 2 1 8142 7335 415 231 8. 17062 2318 265 -. .00227 00282 4695 5G98 
The estimate of the maximum error in this 
result is the same as in the subtabulation method. 
I 
discrepancy in the highest interpolates, in this 
case xo and .I ,2 .3 A, ZLI .2 .8 .s. 
An indication of the error is also provided by the 
6. Bivariate Interpolation 
Bivariate interpolation is generally most simply 
performed as a sequence of univariate interpola- 
tions. We carry out the interpolation in one 
direction, by one of the methods already described, 
for several tabular values of the second argument 
in the neighborhood of its given value. The 
interpolates are differenced as a check, and 
interpolation is then carried out in the second 
direction. 
An alternative procedure in the case of functions 
of a complex variable is to use the Taylor’s series 
expansion, provided that successive derivatives 
of the function can be computed without much 
difficulty. 
7. Generation of Functions from Recurrence Relations 
Many of the special mathematical functions 
which depend on a parameter, called their index, 
order or degree, satisfy a linear difference equa- 
tion (or recurrence relation) with respect to this 
parameter. Examples are furnished by the Le- 
gendre function P,(z), the Bessel function Jn(z) 
and the exponential integral E,(x), for which we 
have the respective recurrence relations 
J n+*-~Jn+J.-l=O 
nE,+,+xE,,=e-=. 
Particularly for automatic work, recurrence re- 
lations provide an important and powerful com- 
puting tool. If the values of P&r) or Jn(z) are 
known for two consecutive values of n, or E',(z) 
is known for one value of n, then the function may 
be computed for other values of n by successive 
applications of the relation. Since generation is 
carried out perforce with rounded values, it is 
vital to know how errors may be propagated in 
the recurrence process. If the errors do not grow 
relative to the size of the wanted function, the 
process is said to be stable. If, however, the 
relative errors grow and will eventually over- 
whelm the wanted function, the process is unstable. 
It is important to realize that st,ability may 
depend on (i) the particular solution of the differ- 
ence equation being computed; (ii) the values of 
x or other parameters in the difference equation; 
(iii) the direction in which the recurrence is being 
applied. Examples are as follows. 
Stability-increasing n 
Pm(x), p:(2) 
Qnb), Q:(x) (x<l) 
y&9, KG) 
J-n-&), z-t44 
&Cd (n<d 
Stability-decreasing 7t 
P”(X), P.,(z) @<l) 
Qnh), Q:(x) 
J&4, Z.@) 
Jn+Hcd , Zn+&) 
Em(z) (n >r) 
F,,(t, p) (Coulomb wave function) 
Illustrations of the generation of functions from 
their recurrence relations are given in the pertinent 
chapters. It is also shown that even in cases 
where the recurrence process is unstable, it may 
still be used when the starting values are known 
to sufficient accuracy. 
Mention must also be made here of a refinement, 
due to J. C. P. Miller, which enables a recurrence 
process which is stable for decreasing n to be 
applied without any knowledge of starting values 
for large n. Miller’s algorithm, which is well- 
suited to automatic work, is described in 19.28, 
Example 1. 
XIV INTRODUCTION 
8. Acknowledgments 
The production of this volume has been the 
result of the unrelenting efforts of many persons, 
all of whose contributions have been instrumental 
in accomplishing the task. The Editor expresses 
his thanks to each and every one. 
The Ad Hoc Advisory Committee individually 
and together were instrumental in establishing 
the basic tenets that served as a guide in the forma- 
tion of the entire work. In particular, special 
thanks are due to Professor Philip M. Morse for 
his continuous encouragement and support. 
Professors J. Todd and A. Erdelyi, panel members 
of the Conferences on Tables and members of the 
Advisory Committee have maintained an un- 
diminished interest, offered many suggestions and 
carefully read all the chapters. 
Irene A. Stegun has served eff ectively as associate 
editor, sharing in each stage of the planning of 
the volume. Without her untiring efforts, com- 
pletion would never have been possible. 
Appreciation is expressed for the generous 
cooperation of publishers and authors in granting 
permission for the use of their source material. 
Acknowledgments for tabular material taken 
wholly or in part from published works are iven 
on the first page of each table. Myrtle R. Ke ling- Yi 
ton corresponded with authors and publishers 
to obtain formal permission for including their 
material, maintained uniformity throughout the 
bibliographic references and assisted in preparing 
the introductory material. 
Valuable assistance in the preparation, checkin 
and editing of the tabular material was receive IFi 
from Ruth E. Capuano, Elizabeth F. Godefroy, 
David S. Liepman, Kermit Nelson, Bertha H. 
Walter and Ruth Zucker. 
Equally important has been the untiring 
cooperation, assistance, and patience of the 
members of the NBS staff in handling the myriad 
of detail necessarily attending the publication 
of a volume of this magnitude. Especially 
appreciated have been the helpful discussions and 
services from the members of the Office of Techni- 
cal Information in the areas of editorial format, 
graphic art layout, printing detail, preprinting 
reproduction needs, as well as attention to pro- 
motional detail and financial support. In addition, 
the clerical and typing stafI of the Applied Mathe- 
matics Division merit commendation for their 
efficient and patient production of manuscript 
copy involving complicated technical notation. 
Finally, the continued support of Dr. E. W. 
Cannon, chief of the Applied Mathematics 
Division, and the advice of Dr. F. L. Alt, assistant 
chief, as well as of the many mathematicians in 
the Division, is gratefully acknowledged. 
M. ABRAMOWITZ. 
1. Mathematical Constants 
DAVID S. LIEPMAN ’ 
Contents 
Page 
Table 1.1. Mathematical Constants ............... 2 
+i,nprime <lOO, 20s. .................. 2 
Some roots of 2, 3, 5, 10, 100, 1000, e, 20s ..........2 
e *n, n=l(l)lO, 25s .................... 2 
e *tns, n=l(l)lO, 20s .................... 2 
eas , e*‘, 20s ....................... 2 
ln n, log,, n, n=2(1)10, primes <lOO, 26, 25s ........ 2 
In 7~, In&, logI, ?r, log,, e, 25s ............... 3 
n In 10, n=1(1)9, 25s ................... 3 
na, n=1(1)9, 25s ..................... 3 
a*“, n=l(l)lO, 25s .................... 3 
Fractions of T, powers and roots involving T, 25s ....... 3 
1 radian in degrees, 26s .................. 3 
lo, l’, 1” in radians, 24D. ................. 3 
~,lny, 24D ....................... 3 
r(4), l/r($), 15D ..................... 3 
r(2),l/r(z),lnr(2),2~3,a,g,q,~,g,g,~, 15D. ........ 3 
1 National Bureau of Standards. 
MATHEMATICAL CONSTANTS 
TABLE 1.1. MATHEMATICAL CONSTANTS 
1 4142 13562 37;: 
i7320 50807 56887 
50488 
72935 
2.2360 67977 49978 96964 
2..6457 51311 06459 05905 
3.3166 24790 35539 98491 
3.6055 51275 46398 92931 
4.1231 05625 61766 05498 
4.3588 98943 54067 35522 
4.7958 31523 31271 
5.3851 64807 13450 :E: 
5.5677 64362 83002 19221 
6.0827 62530 29821 96890 
6.4031 24237 43284 86865 
1O’fi 3.1622 77660 16837 93320 
1O’fl 2.1544 34690 03188 10"' 1.7782 79410 03892 %X 
101’5 1.5848 93192 46111 34853 
1OOlfl 4.6415 88833 61277 loo”5 2.5118 86431 50958 FKd * 
1000"' 5.6234 13251 90349 08040 
lOOO"5 3.9810 71705 53497 25077 * 
2lB 1.2599 21049 89487 31648 
3’B 1.4422 49570 30740 83823 
2114 1.1892 07115 06275 20667 
3114 
2-m 
3-m 
5-‘fi (-- lj 4.4721 35954 99957 93928 
1.3160 74012 95249 24608 * 
- 1) 7.0710 67811 86547 52440 
- 1) 5.7735 02691 89625 76451 6.5574 38524 30200 06523 
6.8556 54600 40104 41249 
7.2801 09889 28051 82711 
7.6811 45747 86860 81758 
7.8102 49675 90665 43941 
8.1853 52771 87244 99700 
8.4261 49773 i7635 
8.5440 03745 31753 
8.8881 94417 31558 
9.1104 33579 14429 
9.4339 81132 05660 
9.8488 57801 79610 
2.7182 81828 45:04 
7.3890 56098 93065 
1) 2.0085 53692 31876 
lj 5.4598 15003 31442 
2) 1.4841 31591 02576 
2) 4.0342 87934 92735 
3) 1.0966 33158 42845 
3) 2.9809 57987 04172 
3) 8.1030 83927 57538 
4) 2.2026 46579 48067 
86306 
11679 
88501 
88819 
38113 
47217 
52353 60287 
02272 30427 
67740 92853 
39078 11026 
60342 11156 
12260 83872 
85992 63720 
82747 43592 
40077 09997 
16516 95790 
enr 
1) 2.3140 69263 27792 69006 
2) 5.3549 16555 24764 73650 
4) 1.2391 64780 79166 97482 
5) 2.8675 13131 36653 29975 
6) 6.6356 23999 34113 42333 
8j 1.5355 29353 95446 69392 
9) 3.5533 21280 84704 43597 
LO) 8.2226 31558 55949 95275 
12) 1.9027 73895 29216 12917 
13) 4.4031 50586 06320 29011 
0.6931 47180 
1.0986 12288 
1.3862 94361 
1.6094 37912 
1.7917 59469 
1.9459 10149 
2.0794 41541 
2.1972 24577 
2.3025 85092 
2.3978 95272 
2.5649 49357 
2.8332 13344 
2.9444 38979 
3.1354 94215 
3.3672 95829 
3.4339 
3.6109 
I: % 
26224 
72417 
87204 
17912 
72066 
00115 
14792 64190 
99019 79852 
In n 
55994 53094 
66810 96913 
11989 
43410 xz::: 
22805 50008 
05531 33051 
67983 59282 
33621 93827 
99404 56840 
79837 05440 
46153 67360 
05621 60802 
16644 04600 
92914 96908 
98647 40271 
48514 62459 
64422 44443 
70430 78038 
69356 24234 
172321 
952452 
344642 
007593 
124774 
053527 
516964 
904905 
179915 
619436 
534874 
495346 
090274 
067528 
832720 
291643 
680957 
667634 
728425 
4.8104 77380 96535 16555 
2.1932 80050 73801 54566 
(-- 11 2.0787 95763 50761 90855 
i- li 4.5593 81277 65996 23677 
. - I ~- ~--~ - -~- - - - - - - - - - - 
1.6487 21270 70012 81468 
(- 1) 6.0653 06597 12633 42360 
1.3956 12425 08608 95286 
(- 1) 7.1653 13105 73789 25043 
- 1) 3.6787 94411e-“71442 32159 55238 
- 1) 1.3533 52832 36612 69189 39995 
- 2) 4.9787 06836 78639 42979 34242 
- 2j 1.8315 63888 87341 80293 71802 
- 3) 6.7379 46999 08546 70966 36048 
- 3) 2.4787 52176 66635 84230 45167 
- 4) 9.1188 19655 54516 20800 31361 
- 4) 3.3546 26279 02511 83882 13891 
- 4) 1.2340 98040 86679 54949 76367 
- 5) 4.5399 92976 24848 51535 59152 
e--nr 
- 2) 4.3213 91826 37722 49774 
- 3) 1.8674 42731 70798 88144 
- 5) 8.0699 51757 03045 99239 
- 6) 3.4873 42356 20899 54918 
- 7) 1.5070 17275 39006 46107 
- 9i 6. 5124 12136 07990 07282 
-1oj 2.8142 68457 48555 27211 
-11) 1.2161 55670 94093 08397 
-13) 5.2554 85176 00644 85552 
-14j 2.2711 01068 32409 38387 
- 2) 6.5988 03584 53125 37077 
- 1) 5.6145 94835 66885 16982 
log10 12 
1. 
0102 
7712 
0205 
9897 
7815 
4509 
0308 
5424 
0000 
0413 
1139 
2304 
2787 
3617 
4623 
4913 
5682 
99956 
12547 
99913 
00043 
12503 
80400 
99869 
25094 
00000 
92685 
43352 
48921 
53600 
27836 
97997 
61693 
01724 
63981 
l!id62 
27962 
36018 
83643 
14256 
91943 
39324 
00000 
15822 
30683 
37827 
95282 
01759 
89895 
83427 
06699 
19521 
43729 
39042 
80478 
63250 
83071 
58564 
87459 
00000 
50407 
67692 
39285 
89615 
%Ei 
26796 
49968 
37389 
50279 
74778 
62611 
87668 
22163 
12167 
~:~:: 
50200 
::%i 
X%f 
32847 
66704 
08451 
1.6127 83856 71973 54945 09412 
1.6334 68455 57958 65264 05088 
*See page xx. 
MATHEMATICAL CONSTANTS 
TABLE 1.1. MATHEMATICAL CONSTANTS-Continued 
In n 
3.8501 47601 71005 85868 209507 
3. 9702 91913 55212 18341 444691 
4.0775 37443 90.571 94506 160.504 
log10 n 
97857 93571 
75869 60078 
52011 64214 
1.6720 
1.7242 
1.7708 
1.7853 
1.8260 
1.. 8512 
1. 8633 
I.. 8976 
74644 14219 
90456 32992 
29835 01076 
74802 70082 
58348 71907 
22860 12045 
27091 29044 
1. 9190 78092 37607 
1. 9493 90006 64491 
1. 9867 71734 f ?624 
41902 60656 
70338 85749 
64341 49132 
52860 92829 
59010 74387 
14279 94821 
39038 32760 
27847 23543 
48517 84362 
loglog (-1) 4.9714 98726 94133 85435 12683 
logl0e (-1) 4.3429 44819 03251 82765 11289 
3. 1415 
6. 2831 
!a. 4247 
( 1) 1. 2566 
( 1) 1.5707 
( 1) 1. 8849 
( 1) 2. 1991 
( 1)2.5132 
( 1) 2.8274 
92653 5s”9”19 32384 62643 
85307 17958 64769 25287 
77960 76937 97153 87930 
37061 43591 72953 85057 
96326 79489 66192 31322 
55592 15387 59430 77586 
14857 51285 52669 23850 
74122 87183 45907 70115 
33388 23081 39146 16379 
n 
; 
3 
4 
x 
7 
i 
10 
*-” 
-1) 3. 1830 98861 83790 67153 77675 
-1) 1.. 0132 11836 42337 77144 38795 
-2j 3.2251 53443 31994 89184 42205 
-2) 1. 0265 98225 46843 35189 15278 
-3) 3.2677 63643 05338 54726 28250 
-31 1. 0401 61473 29585 22960 89838 
s-ii 
;h; J2 
r-1 13 
?r-l/4 
+f3 
,-%I4 
*-3/Z 
r--c 
(2r)-'12 
(a/,)"2 
2'f2/?r 
1’ 
1” 
In Y 
l/~U/‘4 
l/Ul/3) 
-4j 3. 3109 36801 77566 76432 59528 
-4) 1. 0539 03916 53493 66633 17287 
-5) 3.3546 80357 20886 91287 39854 
-5) 1. 0678 27922 68615 33662 04078 
4.7123 88980 38468 98576 93965 
90204 78639 09846 
82938 15836 62470 
95835 47756 28694 
40632 55295 68146 
55444 64942 48285 
40770 35411 61438 
4.1887 
4.4428 
(-1) 5. 6418 
(- 1) 6. 8278 
(-1) 7. 5112 
(-1) 4. 6619 
(- 1) 4. 2377 
(-1) 1. 7958 
( -2) 4.. 4525 
(-1) 3. 9894 
(-1) 7. 9788 
(-1) 4.5015 
72081 23757 59679 
71221 25166 56168 
26726 69229 06151 
22804 01432 67793 
16858 
15881 * 
80795 
70208 
87030 
19885 
10077 
90820 
35273 
99461 
98921 
75996 
45608 02865 35587 
81580 78533 03477 
0. 0002 90888 20866 57215 96154r 
0. 0000 04848 13681 10953 59936r i 
-0. 5495 39312 98164 48223 37662 
0. 5641 89583 
0. 3732 82173 
547756 
907395 
621648 l/r(2/3) 0. 7384 88111 
mm) 0. 2758 15662 830209 
mw 0. 8160 48939 098263 
m4/3) :l. 1198 46521 722186 
mw) l. 1077 32167 432472 
mw l. 1032 62651 320837 
uw4 :l. 0880 65252 131017 
In r(4/3) -0. 1131 91641 740343 
In r(5/3) -0. 1023 14832 960640 
In r(5/4) -0. 0982 71836 421813 
in r(7/4) -0. 0844 01121 020486 
4.1108 73864 17331 12487513891 
4. 2046 92619 39096 60596 700720 
4.2626 79877 04131 54213 294545 
4. 2904 59441 14839 11290 921089 
4.3694 47852 46702 14941 729455 
4.4188 40607 79659 79234 754722 
4.4886 36369 73213 98383 178155 
4. 5747 10978 50338 28221 167216 
1. 1447 29885 84940 01741 43427 
(-1) 9. 1893 85332 04672 74178 03296 
nln 10 
2.3025 85092 99404 56840 17991 
4.6051 70185 98809 13680 35983 
6.9077 55278 98213 70520 53974 
9.2103 40371 97618 27360 71966 
( 1) 1. 1512 92546 49702 28420 08996 
( 1) 1. 3815 51055 79642 74104 10795 
( 1) 1. 6118 09565 09583 19788 12594 
( 1) 1. 8420 68074 39523 65472 14393 
( 1) 2. 0723 26583 69464 11156 16192 
7P 
3.1415 92653 58979 32384 62643 
9.8696 04401 08935 86188 34491 
( 1) 3. 1006 27668 02998 20175 47632 
( 1) 9. 7409 09103 40024 37236 44033 
C 
( 
2) 
2j 
3.0601 96847 85281 
75304 
45326 27413 
9.6138 91935 43703 02194 
( 3) 3.0202 93227 77679 20675 14206 
( 3) 9.4885 31016 07057 40071 28576 
( 4) 2.9809 09933 34462 11666 50940 
( 4) 9.3648 04747 60830 20973 71669 
1. 5707 96326 79489 66192 31322 
1. 0471 97551 19659 77461 54214 
(-1) 7.8539 81633 97448 30961 56608 
1. 7724 53850 90551 60272 98167 
1.4645 91887 56152 32630 20143 
1.3313 
2. 1450 
2.3597 
5. 5683 
t: 1) 2. 2459 
2. 5066 
1. 2533 
2. 2214 
35363 
29397 
30492 
27996 
15771 
28274 63100 05024 15765 
14137 31550 02512 07883 
41469 07918 31235 07940 
57. 2957 
0. 0174 
79513 
53292 
0. 5772 15664 
1. 7724 53850 
2. 6789 38534 
1. 3541 17939 
3. 6256 09908 
1. 2254 16702 
0.8929 79511 
80038 97127 97535 
11102 56000 77444 
41469 68875 78474 
83170 78452 84818 
83610 45473 42715 
08232 08767 98155’ 
51994 32957 69237r 
90153 28606 06512 
905516 
707748 
426400 
221908 
465178 
569249 
0. 9027 45292 950934 
0. 9064 02477 055477 
0. 9190 62526 848883 
0. 9854 20646 927767 
0.3031 50275 147523 
1.2880 22524 698077 
0.2032 80951 431296 
I-, -, 
r (714) 
In r(i/3) 
In r(2/3) 
In r(lj4) 
ln r(3/4) 
. 
*See page II. 
2. Physical Constants and Conversion Factors 
A. G. MCNISH 1 
Contents 
Table 2.1. Common Units and Conversion Factors . . . . . . . . . 
Table 2.2. Names and Conversion Factors for Electric and Magnetic 
Units . . . . . . . . . . . . . . . . . . . . . . . 
Table 2.3. Adjusted Values of Constants . . . . . . . . . . . . . 
Table 2.4. Miscellaneous Conversion Factors. . . . . . . . . . . . 
Table 2.5. Conversion Factors for Customary U.S. Units to Metric 
Units . . . . . . . . . . . . . . . . . . . . . . . 
Table 2.6. Geodetic Constants . . . . . . . . . , . . . . . . . . 
Page 
6 
6 
7 
8 
8 
8 
* National Bureau of Standards. 
2. Physical Constants and Conversion Factors 
The tables in this chapter supply some of 
the more commonly needed physical con- 
stants and conversion factors. 
All scientific measurements in the fields of 
mechanics and heat are based upon four in- 
ternational arbitrarily adopted units, the 
magnitudes of which are fixed by four agreed 
on standards: 
Length- the meter -fixed by the vacuum 
wavelength of radiation corresponding to the 
transition 2Plu-5Da of krypton 86 
(1 meter - 1650763.73h). 
Mass-the kilogram -fixed by the interna- 
tional kilogram at S&vres, France. 
Time-the second- fixed as l/31,556,925.9747 
of the tropical year 1900 at 12” ephemeris 
time, or the duration of 9,19‘2,631,770 cycles 
of the hyperfine transition frequency of cesi- 
urn 133. 
Temperature-the degree-fixed on a ther- 
modynamic basis by taking the temperature 
for the triple point of natural water as 273.16 
“K. (The Celsius scale is obtained by adding 
-273.15 to the Kelvin scale.) 
Other units are defined in terms of them by 
assigning the value unity to the proportion- 
ality constant in each defining equation. The 
entire system, including electricity units, is 
called the Systi.?me International d’unitds 
(SI). Taking the l/100 part of the meter as 
the unit of length and the l/1000 part of the 
kilogram as the unit of mass, similarly, gives 
rise to the CGS system, often used in physics 
and chemistry. 
Table 2.1. Common Units and Conversion 
Factors 
~ 
The SI unit of electric current is the ampere 
defined by the equation 2r,,,Z1ZJ4~= F giving 
the force in vacua per unit length between 
two infinitely long parallel conductors of in- 
finitesimal cross-section. If F is in newtons, 
and rrn has the numerical value 477 X lo-‘, 
then I1 and Zr are in amperes. The custom- 
ary equations define the other electric and 
magnetic units of SI such as the volt, ohm, 
farad, henry, etc. The force between elec- 
tric charges in a vacuum in this system is 
given by Q, Qn/4nrerg= F, re having the nu- 
merical value 10r/4nc2 where c is the speed 
of light in meters per second (r,= 8.854 
x 10-12). 
The CGS unrationalized system is obtained 
by deleting 4n in the denominators in these 
equations and expressing F in dynes, and r 
in centimeters. Setting r,,, equal to unity de- 
fines the CGS unrationalized electromagnetic 
system (emu), re then taking the numerical 
value of 1/c2. Setting re equal to unity de- 
fines the CGS unrationalized electrostatic 
system (esu), r,,, then taking the numerical 
value of l/cz. 
Table 2.2. Names and Conversion Factors for Electric and Magnetic Units 
Quantity 
Current 
Charge 
Potential 
Resistance 
Inductance 
Capacitance 
Magnetizing force 
Magnetomotive force 
Magnetic flu* 
Magnetic flux density 
Electric displacement 
= 
SI 
name 
ampere 
coulomb 
volt 
ohm 
henry 
farad 
amp. turns/ 
meter 
amp. turns 
weber 
tesla 
= 
- 
I 
1 
- 
emu 
I 
esu 
name name 
tbampere 
tbcoulomb 
abvolt 
abohm 
centimeter 
statampere 
statcoulomb 
statvolt 
statohm 
centimeter 
oersted 
gil bert 
maxwell __---___----_- 
gauss _-______-_____ 
--._-_-______ I_-..____..____ 
= 
- 
10-l 
LO-’ 
108 
100 
100 
10-g 
4*x IO-3* 
4rX lo-I* 
108 
10’ 
10-J* 
- 
SI unit/ 
emu unit 
= 
- 
SI unit/ 
esu unit 
-3x 100 
-3 x 109 
-(1/3)X 10-Z 
-(1/9)X 10-u 
%(1/9)X 10-l’ 
-9x 10” 
-3 x loo* 
-3/10** 
-(1/3)X 10-z 
-(1/3)X 10-B 
-3x 105* 
Example: If the value assigned to a current is 100 amperes its value in abamperes is 100X10-‘=lO. 
*Divide this number by 4?r if unrationalized system is involved; other numbers are unchanged. 
6 
3. Elementary Analytical Methods 
MILTON ABRAMOWITZ l 
Contents 
Elementary Analytical Methods ................. 
3.1. Binomial Theorem and Binomial Coefficients; Arithmetic and 
Geometric Progressions; Arithmetic, Geometric, Harmonic 
and Generalized Means ............... 
3.2. Inequalities ...................... 
3.3. Rules for Differentiation and Integration ......... 
3.4. Limits, Maxima and Minima .............. 
3.5. Absolute and Relative Errors .............. 
3.6. Infinite Series ..................... 
3.7. Complex Numbers and Functions ............ 
3.3. Algebraic Equations .................. 
3.9. Successive Approximation Methods ........... 
3.10. Theorems on Continued Fractions ............ 
Page 
10 
10 
10 
11 
13 
14 
14 
16 
17 
18 
19 
Numerical Methods ....................... 
3.11. Use and Extension of the Tables ............ 
3.12. Computing Techniques. ................ 
References ............................ 
19 
19 
19 
23 
Table 3.1. Powers and Roots . . . . . . . . . . . . . . . . . . 
n’“, k=l(l)lO, 24, l/2, l/3, l/4, l/5 
n=2(1)999, Exact or 10s 
24 
The author acknowledges the assistance of Peter J. U’Hara and Kermit C. Nelson in 
the preparation and checking of the table of powers and roots. 
1 National Bureau of Standards. (Deceased.) 
3. Elementary Analytical Methods 
3.1. Binomial Theorem and Binomial Coeffi-cients; Arithmetic and Geometric Progres- 
sions; Arithmetic, Geometric, Harmonic and 
Generalized Means 
Binomial Theorem 
3.1.1 
(a+b)“=a”+c) a”-lb+@ an-%2 
+C) a”-w+ . . . +b” 
(n a positive integer) 
Binomial Coefficients (see chapter 24) 
3.1.2 
n * 
0 k 
=nc7t=n(n-l) . .k., (n--k+l)= n! 
(n-k)!k! 
3.1.3 @=(n:k)=(-l)k (k-;-l) 
3.1.4 
3.1.5 
3.1.6 1+c)+@+. . . +c)=2” 
3.1.7 1-G)+@- . . . +(--I)“@=0 
Table of Binomial Coefficients 
0 
z 
3.1.8 
2---I 1 2 
3---- 1 3 
4---- 1 4 
5..--- 1 5 
6-e-m 1 6 
7---- 1 7 
8---- 1 8 
9---- 1 9 
lo_--_ 1 10 
ll---- 1 11 
12---- 1 12 
; 1 
6 4’ 1 
10 10 5 1 
15 20 15 6 1 
21 35 35 21 7 1 
28 56 70 56 28 8 1 
36 84126126 84 36 9 1 
45120210252210120 45 10 
55165330462462330165 55 
66220495792924792495220 
For a more extensive table see chapter 24. 3.2.2 min. a<M(t)<mnx. a 
3.1.9 
Sum of Arithmetic Progression to n Terms 
a-t-b+d)+b+24+ . . . +(a++-114 
=na+; 7+-l)&; (a+z)) 
last term in series=Z=a+(n-1)d 
Sum of Geometric Progreamion to n Terms 
3.1.10 
a(l-P) s,=a+ar+a?+ . . . +a+‘=--- 
l--P 
lim s.=a/(l-r) (--l<r<l) 
n-t- 
Arithmetic Mean of n Quantities A 
3.1.11 &h+az+ . - . +a. 
n 
Geometric Mean of n Quantities G 
3.1.12 G= (a,&. . . a,,)l’” (at>O,k=1,2,. . .,n) 
Harmonic Mean of n Quantities H 
3.1.13 
.+$) (ah>O,k=1,2,. . .,n) 
General&d Mean 
3.1.14 M(t) ==(i g a: yf 
3.1.15 M(t)=O(t<O, some ak zero) 
3.1.16 lim M(t) =max. @I, a2, * * *, a,) =mtLx. a 
t+m 
3.1.17 t&M(t)=min. (a,,%,. . .,a,)=min.a 
3.1.18 liiM(t)=G 
3.1.19 M(l)=A 
3.1.20 M(-l)=H 
3.2. Inequalities 
Relation Between Arithmetic, Geometric, Harmonic 
and Generalized Means 
3.2.1 
A> G>H, equality if and only if al=az= . . . =a,, 
*See page II. 
10 
ELEMENTARY ANALYTICAL METHODS 11 
min. a<G<max. a Minkowski’s Inequality for Sums 
If p>l and &, bk>O for all x:, 
3.2.12 
3.2.3 
equality holds if all ak are equal, or t<O 
and an an is zero 
3.2.4 M(t)<&!(s) if t<s unless all ak are equal, 
or s<O and an an is zero. 
3.2.5 
3.2,6 
Triangle Inequalities 
Iall-la21_<lal+~2111all+la21 
Chehyshev’s Inequality 
If alla2>a,z . . . >a, 
b,>b,>b,> . . . >bn 
3.2.7 n 5 akbk> 2 a 
k=l -(kc, ‘> (& “) 
Hiilder’s Inequality for Sums 
If ;++,p>1, fj>l 
equality holds if and only if jbkl=cIuEIP-’ (c=con- 
stant>O). If p=q=2 we get 
Cauchy’s Inequality 
3.2.9 
[& akb,]2<& a; & b: (equality for &=Cbk, 
c constant). 
Hiilder’s Inequality for Integrals 
4 , 
1r;++=1,p>1, q>l 
3.2.10 
equality holds if and only if jgCs)I=clflr)Ip-’ 
(c=constant>O). 
If p=p=2 we get 
3.2.11 
Schwa&s Inequality 
(& (ak+bk)p~‘p<(& a;)“‘+($ bi$“j 
t?qUdity holds if and only i-f bk=C& (c=con- 
stant>O). 
Minkowski’s Inequality 5or Integrals 
If P>l, 
3.2.13 
(Jb I?(~)+9(z)I~~s)llp~(~b lJ~(z)l%iz)l’p 
a a 
+(Jb Iscd lqp 
a 
equality holds if and only if g(z) =cf(x) (c=con- 
stant>O). 
3.3. Rules for Differentiation and Integration 
Derivatives 
3.3.1 
3.3.2 
3.3.3 
3.3.4 
$; (cu) =c $9 c constant 
-g (u+v)++g 
& (uv) =u E+v 2 
& (u/v)= 
vdu/dx-udv/dx 
v2 - 
3.3.5 .g u(v) =g gi 
3.3.6 & (u”) =uD e %+1.n U 2) 
Leibniz’s Theorem for Differentiation of an Integral 
3.3.7 
d 
s 
b(c) 
& 
a(c) f (2, wx 
b(c) b 
3 
s 
a(cj ,J(x,c)dx+f@, 4 $--f (a, 4 2 
12 ELEMENTARY ANALYTICAL METHODS 
3.3.9 
Leibniz’s Theorem for Differentiation of a Product 
3.3.8 
s (t&g u+(y) g g +(g ds g 
+...+~)d~rg+...+cg 
dX 
-&=1g a 
3.3.10 
#x -d2y dy -3 
dy2=dr2 zc 0 
3.3.11 $= -B g-3 (g--j ($)-" 
Integration by Parts 
3.3.12 jidu=w+dti 
3.8.13 jkudx=(jidx) v-s(judx) 2 dx 
Integrals of Rational Algebraic Functions 
(Integration const,ants are omitted) 
3.3.14 S (ux+f,)"dx=(ux+6)"+1 4n+1> (n#-1) 
3.3.15 S $&In )ax+b) 
3.3.26 
3.3.27 
The following formulas are useful for evaluating 
S P(x)dx (ux”+ fJx+cy where P(x) is a polynomial and 
n>l is an integer. 
3.3.16 
S dx 2 (ax2+br+c)=(4c&c-bz)~ 
(b2--4uc<O) 
3.3.17 
I 
2az+b- (b2--4uc)t 
2azfbf (P-4acy 
(b2-4m>O) 
3.3.18 
-2 =2az+b (P-4ac=O) 
3.3.19 
S S 
3.3.20 
S c+dx (a+ bx$c+dxj=k bc In - I I a+bx (ad # bc) 
3.3.21 ___ S dx 1 =- arctan E! u2+b2;C2 ub U 
3.3.22 S ln b2+ b2x21 
3.3.23 S 
3.3.24 S (x2;1-“a2)2=& arctan ~+20~(xf+U2) 
3.3.25 S 
Integrals of Irrational Algebraic Functions 
S dx -d(a+ bx) 1’2 t(u+bx) (c+dx)11’2 =h2 arctan C b(c+dx) 1 W<O) 
=+ arcsin 
2bdx+ad+ bc 
bc-ad > (b>O, d<O) 
3.3.28 =h2 ln J[bd(n+ bx)]1/2+ b(c+dx)1/21 (bd>O) 
3.3.29 S dx (a+bx)“P(c+dx)=[d(bc~ud)]1~2 arctan ~~~~-J” (d(u&-bc)<O) 
=[d(ad&]1/2 In 
d(u+bx)1’2-[d(ad-bc)]1’2 
d(a+bx)1’2+[d(ud-bc)]“2 I 
@b-J--4>O) 3.3.30 
ELEMENTARY ANALYTICAL METHODS 17 
If zn=un+ivn, then ~~+l:=u,+,+iv~+~ where 
3.7.23 u,+~=xu~-~v,; v,+,=xv,+yu, 
9?z” and 92” are called harmonic polynomials. 
3.7.24 
3.7.25 
Roots 
3.7.26 z*=&=rte+rs=r+ cos @+iri sin $0 
If --?r<~< ?r this is the principal root. The 
other root has the opposite sign. The principa: 
root is given by 
3.7.27 d=[+(r+x)]+&-i[$(r--x)]*=ufiv where 
2uv=y and where the ambiguous sign is taken tc 
be the same as the sign of y. 
3 7 28 . . Zl/n,Tl/nefe/n , (principal root if - ?r<0 5 7r) 
Other roots are Pet(B+2rn’ln (k=l,2,3, a . ., n-1) 
Inequalities 
3.7.29 
I I 
l&l-I221 _<1z1~~2111211+I~21 
Complex Functions, Catwhy-Riemann Equations 
f(z)=f(x+iy)=u(x,yy)+iv(x,y)whereu(x,y),v(z,y 
aA real, is unaly& at those points z=z+$ a 
which 
3.7.30 
au av au av -=-, -=-- 
a~ by by ax 
If z=Tefff, 
3.7.31 g=; ;, ; f$=-$ 
Laplace’s Equation 
The functions U(X, y) and v(x, y) are callec 
harmonic functions and satisfy Laplace’s equation 
Cartesian Coordinates r 
3.7.32 i!?+$E~2+g2=o 
Polar Coordinates 
3.7.33 r ; (r g)+$=r ; (r g+g=o 
3.8. Algebraic Equations 
Solution of Quadratic Equations 
3.8.1 Given az2+ bz+c=O, 
21,2=- - 0 2”, j-k gf, p= b2-4ac, 
.zi+,za= -b/u, Z~Z~=C/U 
If a>O, two real roots, 
p=O, two equal roots, 
a<O, pair of complex conjugate roots. 
Solution of Cubic Equations 
3.8.2 Given Z3+a2z2+ulz+a0==0, let 
If $+P>O, one real root and a pair of complex 
c.onjugate roots, 
$+9=0, all roots real and at least two are 
equal, . 
p3+r2<0, all roots real (irreducible case). 
Let 
sl=[r-+(q3+r2)q+, sz=[T’-((p3+?3*]* 
then 
If zl, z2, z3 are the roots of the cubic equation 
Z~+Z~+Z~=-CIC~ 
~~~2+~$,+&~,:=~~ 
Solution of Quartic Equations 
3.8.3 Given 24+~323+a3~~+~1l~+ag=O, find the 
real root u1 of the cubic equation 
U3 - a2U2 i- (~23 - 4ao)U - (a; j- Uoa$ - kW3) = 0 
and determine the four roots of the quartic as 
solutions of the two quadratic equations 
18 ELEMENTARY ANALYTICAL METHODS 
If all roots of the cubic equation arc real, USC 
the value of U, which gives real coefficients in the 
*quadratic equation and select signs so that if 
then 
pl+p2=a2,plp2+pl+q2=a2,p~q2+p2q~=al, 4142==0. 
If zl, z2, z3, z4 are the roots, 
z2 j= -a3, z2 j2,2t= -&, 
Czj2,=u2, z1z2z3z4=ao. 
3.9. Successive Approximation Methods 
General Comments 
3.9.1 Let z=zl be an approximation to x=[ 
where f(t) =0 and both x1 and [ are in the interval 
a$r<b. We define 
GI+1=G+C&n) (n=l, 2, . . .). 
Then, if f’(z)>0 and the constants cn are 
negative and bounded, the sequence x,, converges 
monotonically to the root [. 
If c,,=c=constant<O and f’(z)>O, then the 
process converges but not necessarily monotoni- 
cally. 
Degree of Convergence of an Approximution Process 
3.9.2 Let zl, z2, x3, . . . be an infinite sequence 
of approximations to a number f. Then, if 
1% n+~-~I<&n-tlk, (n=l, 2, . . .) 
where A and k areindependent of n, the sequence 
is said to have convergence of at most the kth 
degree (or order or index) to [. If k=l and 
A<1 the convergence is linear; if k=2 the con- 
vergence is quadratic. 
Regula Falsi (False Position) 
3.9.3 Given y=f(z) to find 5 such that f(.$)=o, 
choose ~0 and x1 such that f(rO) and f(zl) have 
opposite signs and compute 
x*=x, 
f 1~o-JoX, 
-Hi f,= jlVfO * 
Then continue with x2 and either of x0 or x1 for 
which f(;ro) or j(zl) is of opposite sign to f(zl). 
Regula falsi is equivalent to inverse linear inter- 
polation, 
Method of Iteration (Successive Substitution) 
3.9.4 The iteration scheme Q+~=F(z~) will 
converge to a zero of z=F(z) if 
(1) IF’(s)J<q<l for aLzSb, 
(2) a<xo &‘F(~)~xo’~ b. - 
Newton’s Method of Successive Approximations 
3.9.5 
Newton’s Rule 
If z=zk is an approximation to the solution 
z= I of f(z) =0 then the sequence 
xk+l= xk 
fcxk) 
f’ bk) 
will converge quadratically to x=5: (if instead of 
the condition (2) above), 
(1) Monotonic convergence, f(zO)r’(zo) >0 
and f’(s), j”(z) do not change sign in the 
interval (Q, t), or 
(2) Osdato y conwgence, f(xJf” (x0) <0 
and f’(s), f”(z) do not change sign in the 
interval (x0, x1), xo<E<xl. 
Newton’s Method Applied to Real nth Roots 
3.9.6 Given x”=N, if zk is an approximation 
x=N’l” then the sequence 
xk+l=- ; [$i+(n-l)xk] 
will converge quadratically to a. 
Aitken’e G-Process for Acceleration of Sequences 
3.9.7 If 2k, &+I, zri+2 are three successive iterates 
in p, sequence converging with an error which is 
approximately in geometric progression, then 
&=xk- 
(5k--k+1)*=;tk~k+2-2:+1. 
A*& A*Xk ’ 
is an improved estimate of x. In fact, if zk”x+* 
OGtk) then Z=s+O(P), Ix\<~. 
ELEMENTARY ANALYTICAL METHODS 
3.10. Theorems on Continued Fractions 
Definitions 
=b,,+&e&. . . 
If the number of terms is finite, j is called a 
ternlinating continued fraction. If the number 
of ternls is infinite, j is called an infinite cont’inued 
fraction and the terminating fraction 
is called the nth convergent of j. 
A 
(2) If lim -A exists, the infinite continued frac- 
It-+- 88 
tion j is said to be convergent. If uf= 1 and the 
bt are integers there is always convergence. 
Theorems 
(1) If at and br are positive then j2n<j2n+2, 
fin-1 >f*n+, . 
(2) If j.=+ 
n 
A,=b,A,-~+a,A,-2 
Bn=bnBn-l+anBn-2 
where A-1=l, A,,=bo, B-1=0, B,=l. 
A,B,_l-A,-lB,=(-l)n-’ kiI al; 
* 
(4) 
(5) For every n>O, 
j,=b, 1 claI ClC& c2c3a3 &I-lW% 
c,bl+ czbz+ caba+ ’ * * c,b,’ 
(6) l+b,+b,b,+ . . . +bzb3. . . b, 
1 bz b3 =-- _- b, 
l- b,+l- b,+l- * ’ ‘--b,+l 
d+$+ . . . +;=-& --& . . . $yu 
I 1 2 n1 n 
1 --- x +A . . . l t(-1,n----5 
a0 aof aoGa2 _ . . . a, 
1 aox =- ___ a12 %-1X _- 
uo+ al-x+ I12-xf * . . +un-2 
0 .2 .4 .6 .8 
FIGURE 3.1 
1 i y:=xn* *n=0,,5t 29 1, 2, 5. 
Numerical Methods 
3.11. Use and Extension of the Tables I Linear interpolation in Table 3.1 gives 
Example 1. Computti xl9 and x4’ for x=29 
(919.826)“4-5.507144. 
using Table 3.1. 
By Newton’s method for fourth roots with 
N=919.826, 
3p=x9. x10 
1 
= (1.45071 4598. 1013)(4.20707 2333. 1014) 
4 ~7~3+3(5.507144)-]=5.50714 3845 
[ . 
=6.10326 1248. 102’ 
x4’= (x*4)2/x 
= (1.25184 9008. 1036)2/29 
=5.40388 2547. lO6* 
Example 2. Compute x-3’4 for x=9.19826. 
(9.19826)“‘= (919.826/100)1’4= (919.826)1’4/10t 
Repetition yields the same result. Thus, 
~“~=5.50714 3845/10$=1.74151. 1796, 
~-~“=zt/x=.18983 05683. 
3.12. Computing Techniques 
Example 3. Solve the quadratic equation 
x2- 18.2x+.056 given the coeflicients as 18.2 f .l, 
*see page II. 
. . 
20 ELEMENTARY ANALYTICAL METHODS 
< 
.056f .OOI. From 3.8.1 the solution is 
z=4(18.2f-[(18.2)2-4(.05B)]:) 
=3(18.2~[:J31]t)=3(18.2~18.~) 
= 18.1969, .OOJ 
The smaller root may be obtained more accurately 
from 
* .05fi/18.1969= .0031& .OOOl. 
Example 4. Compute (-3 + .0076i)i. 
From 3.7.26, (-3+.0976i)~=u+iv where 
Y 
u=2G? I,-= ( > 
r!y *, j”= (t”+y’)t 
Thus 
r=[(-3)2+(.0076)2]~=(9.00005776)~=3.00000 9627 
Ij= 3.00000 9627- (-3) f= 
2 1 .73205 2196 
.0076 
u=&=2(1.73205 21g6)=.00219 392926 
We note that the principal square root has been 
computed. - 
Example 6. Solve the quartic equation I 
~‘-2.37752 4922x3+6.07350 5741.x’ 
-11.17938 023s+9.05265 5259=0. 
Resolution Into Quadratic Factors 
(22 + p12 + qd w + p2x + 92) 
by Inverse Interpolation 
Starting with the trial value pI = 1 we compute 
successively 
PI q2=; p1= a’--am pz=an-p1 Y(Qd=ql-t92+p*P2 
42-pll - a2 
_____ 
:: 9. 4. 053 526 - -2. 1. 093 543 - 1. . 284 165 5. 383 
2. 2 4. 115 -3. 106 . 729 - 2: E 
Example 5. Solve the cubic equation x3- 18.12 
-34.8=0. 
To use Newton’s method we first form the 
table of f(z)=23-1S.1r-34.8 
. 4” -43.2 f(x) 
5 - .3 
6 72.6 
7 181.5 
We obtain by linear inverse interpolation: 
x,=5+ 
O-(-.3) 
72.6-(-.3)=5’oo4’ 
Using Newton’s method, f’(x) =3x2- 18.1 we get 
21 =zo-f&J/f’ (d 
=5.004- C--.07215 9936jz5 00526 
57.020048 ' ' 
Repetition yields x1=5.00526 5097. Dividing 
f(x) by x-5.00526 5097 gives x2+5.00526 5097x 
i-6.95267 869 the zeros of which are -2.50263 2549 
f.83036 8OOi. 
We seek that value of y, for which y(nJ =O. 
Inverse interpolation in ~(a,) gives ~(a,) =O for 
pl -2.003. Then, 
QI 42 171 P2 Ykll) 
~~---- 
2.003 4. 520 -2. 550 172 . 011 
Inverse interpolation between qI=2.2 and pl= 
2.003 gives ql=2.0041, :md thus, 
QI Qz PI P2 Y h) 
2. 0041 4. 51706 7640 -2. 55259 257 17506 765 .00078 552 
2. 0042 4.,51684 2260 -2. 55282 851 . 17530 358 . 00001 655 
2. 0043 4. 51661 6903 -2. 55306 447 . 17553 955 -. 00075 263 
Inverse interpolation gives q,=2.00420 2152, and we get finally, 
2. 00420 2152 
_. 
- 
Qz PI P2 Y (Ql) 
4. 51683 7410 -2. 55283 358 17530 8659 -. 00000 0011 
4 ELEMENTARY ANALYTICAL METHODS 21 
Double Precision Multiplication and Division on a 
Desk Calculator 
Example 7. MultiplyM=20243 97459 71664 32102 
by m=69732 82428 43662 95023 on a 10X10X20 
desk calculating machine. 
Let MO=20243 97459, Ml=71664 32102, mO= 
69732 82428, ml=43662 95023. Then Mm= 
M0m0102’+ (Mom,+Mlmo) 101o+M~ml. 
(1) Multiply ,W1m1=31290 75681 96300 28346 
and record the digits 96300 28346 appearing in 
positions 1 to 10 of the product dial. 
(2) Transfer the digits 31290 75681 from posi- 
tions 11 to 20 of the product dial to positions 1 to 
10 of the product dial. 
(3) Multiply cumulatively M,mo+Mom,+31290 
75681=58812 67160 12663 25894 and record the 
digits 12663 25894 in positions 1 to 10. 
(4) Transfer the digits 58812 67160 from posi- 
tions 11 to 20 to positions 1 t,0 10. 
(5) Multiply cumulatively Mom,+58812 67160 
=14116 69523 40138 17612. The results as ob- 
tained are shown below, 
9630028346 
1266325894 
14116695234013817612 
141166952340138176121266325894963~?28346 
If the product Mm is wanted to 20 digits, only 
the result obtained in step 5 need be recorded. 
Further, if the allowable error in the 20th place is 
a unit’, the operation MImI may be omitted. 
When either of the factors M or m contains less 
than 20 digits it is convenient to position the 
numbers as if they both had 20 digits. This 
multiplication process may be extended to any 
higher accuracy desired. 
Example 8. Divide N=14116 69523 40138 17612 
by d=20243 97459 71664 32102. 
Method (1 )--linear interpolation. 
N/20243 97459.101’= .69732 82430 90519 39054 
N/20243 97460.10”= .69732 82427 46057 26941 
Difference=3 44462 12113. 
Difference X.71664 32102=24685 64402&10-*O 
(note this is an 11 X 10 multiplication). 
Quotient= 
(69732 82430 90519 39054-246856 44028).10-20 
=.69732 82428 43662 95028 
There is an error of 3 units in the20th place due 
to neglect of the contribution from second differ- 
ences. 
Method @)--If N and d are numbers each not 
more than 19 digits let N=N1+NolOQ, d=dI+ 
dolO where No and do contain 10 digits and N, 
and dl not more than 9 digits. Then 
N NolOQ+N, 1 
d=,lOQ+d, dolO 
zs- [.N-y] 
Here 
N= 14116 69523 40138 1761, 
d=20243# 97459 71664 3210 
No= 14116 69523, do=20243 97459, 
d,=71664 3210 
(1) NodI= 10116 63378 42188 8830 (productdial). 
(2) (Nod,)/do=49973 55504 (quotient dial). 
(3) N- (N&/d,= 14116 69!;22 90164 62106 
(product dial). 
(4) [N- (NodI)/do]/dolOQ= .69732 82428=first 10 
digits of quotient in quotient dial. Remainder 
=r=O8839 11654, in positions 1 to 10 of product 
dial. 
(5) r/(d010Q)=.43662 9502.10-“O=next 9 digits of 
quotient. N/d=.69732 82428 43662 9502. This 
method may be modified to give the quotient of 
20 digit numbers. Method (1) may be extended 
to quotients of numbers containing more than 20 
digits by employing higher order interpolation. 
Example 9. Sum the series S= l-&+*-i 
+ to 5D using the Euler transform. 
The sum of the first 8 terms is .634524 to 6D. 
If u,=ljn we get 
n %7 Au, A*u, A3u, A%,, 
9 . 111111 
-11111 
10 . 100000 2020 
-9091 -505 
11 . 090909 1515 156 
-7576 -349 
12 . 083333 1166 
-6410 
13 . 076923 
From 3.6.27 we then obtain 
SC 634524+.111111 -_ 
2 
(-.011111)+.002020 
22 23 
-(-- .000505) +.000156 
24 26 
= .634524+ .055556+ .002778+ .000253 
-+ .000032+ .000005 
= .693148 
(S=ln 2=.6931472 to 7D). 
22 ELEMENTARY ANALYTICAL METHODS 
Example 10. Evaluate the integral 
s 
m sin 2 
- dx 
Cl J: 
=- G to 4D using the Euler transform. 
s 0 
- F dx=g s,. 
(Icfl)= y dx 
=& s,’ sin;;;;t) dt+% (-l)f g dt. 
Evaluating the integrals in the last sum by 
numerical integration we get 
Ic 
1.85194 
.43379 
.25661 
. 18260 A A2 A3 A4 
. 14180 
-2587 
.11593 799 
- 1788 -321 
.09805 478 153 
-1310 - 168 
.08495 310 
- 1000 
.07495 
The sum to k=3 is 1.49216. Applying the 
Euler transform to the remainder we obtain 
f (.14180)-h (-.02587)+& (.00799) 
-; (-.00321)+$ (.00153) 
= .07090 + .00647 + .00100+ .00020 
+ .00005 
= .07862 
We obtain the value of the integral as 1.57018 as 
compared with 1.57080. 
Example 11. Sum the series $I kep==f using 
P 
the Euler-Maclaurin summation formula. 
From 3.6.28 we have for n= a, 
$?j k-‘=gl k-‘+l& (k+10)-2 I 
1 
+jY&p- . . . 
where f(k) = (k+10)-2. Thus, 
$, k-2=1.54976 7731+.1 
s 
- .005 + .00016 6667 - .OOOOO 0333 
= 1.64493 4065, 
as compared with $=1.64493 4067. 
Example 12. Compute 
x2 4x2 9x2 
arctanx=$3+g7+ . . . 
to 5D for x= .2. Here al=x, an=(n-l)2x2 for 
n>l, &,=O, b,=2n-1, A-l=l, Bdl=O, A,,=O, 
A0 -= 
Bo ’ 
A -r,*g 
Bl 
A=.197368 
B2 
A3 B=.197396 
3 
A4 [II 
3.032 
Bq = 15.36 
Note that in carrying out the recurrence method 
for computing continued fractions the numerators 
A, and the denominators B, must be used as 
originally computed. The numerators and de- 
nominators obtained by reducing An/B, to lower 
terms must not be used. 
ELEMENTARY ANALYTICAL METHODS 
References 
23 
Texts 
[3.1] R. A. Buckingham, Numerical methods (Pitman 
Publishing Corp., New York, N.Y., 1957). 
[3.2] T. Fort, Finite differences (Clarendon Press, Oxford, 
England, 1948). 
[3.3] L. Fox, The use and construction of mathematical 
tables, Mathematical Tables, vol. 1, National 
Physical Laboratory (Her Majesty’s Stationery 
Office, London, England, 1956). 
[3.4] G. H. Hardy, A course of pure mathematics, 9th 
ed. (Cambridge Univ. Press, Cambridge, England, 
and The Macmillan Co., New York, N.Y., 1947). 
[3.5] D. R. Hartree, Numerical analysis (Clarendon 
Press, Oxford, England, 1952). 
[3.6] F. B. Hildebrand, Introduction to numerical analysis 
(McGraw-Hill Book Co., Inc., New York, N.Y., 
1956). 
[3.7] A. S. Householder, Principles of numerical analysis 
(McGraw-Hill Book Co., Inc., New York, N.Y., 
1953). 
[3.8] L. V. Kantorowitsch and V. I. Krylow, Naherungs- 
methoden der Hoheren Analysis (VEB Deutscher 
Verlag der Wissenschaften, Berlin, Germany, 
1956; translated from Russian, Moscow, U.S.S.R., 
1952). 
[3.9] K. Knopp, Theory and application of infinite series 
(Blackie and Son, Ltd., London, England, 1951). 
[3.10] Z. Kopal, Numerical analysis (John Wiley & Sons, 
Inc., New York, N.Y., 1955). 
[3.11] G. Kowalewski, Interpolation und genaherte Quad- 
ratur (B. G. Teubner, Leipzig, Germany, 1932). 
[3.12] K. S. Kuns, Numerical analysis (McGraw-Hill 
Book Co., Inc., New York, N.Y., 1957). 
[3.13] C. Lanczos, Applied analysis (Prentice-Hall, Inc., 
Englewood Cliffs, N.J., 1956). 
[3.14] I. M. Longman, Note on a method for computing 
infinite integrals of oscillatory functions, Proc. 
Cambridge Philos. Sot. 52, 764 (1956). 
[3.15] S. E. Mikeladze, Numerical methods of mathe- 
matical analysis (Russian) (Gos. Izdat. Tehn- 
Teor. Lit., Moscow, U.S.S.R., 1953). 
[3.16] W. E. Milne, Numerical calculus (Princeton Univ. 
Press, Princeton, N.J., 1949). 
[3.17] L. M. Milne-Thomson, The calculus of finite differ- 
ences (Macmillan and Co., Ltd., London, England, 
1951). 
[3.18] H. Mineur, Techniques de calcul numerique 
(Librairie Polytechnique Ch. B&anger, Paris, 
France, 1952). 
[3.19] National Physical Laboratory, Modern computing 
methods, Notes on Applied1 Science No. 16 (Her 
Majesty’s Stationery Office, London, England, 
1957). 
(3.201 J. B. Rosser, Transformations to speed the con- 
vergence of series, J. Research NBS 46, 56-64 
(1951). 
[3.21] J. B. Scarborough, Numerical mathematical anal- 
ysis, 3d ed. (The Johns Hopkins Press, Baltimore, 
Md.; Oxford Univ. Press, London, England, 
1955). 
[3.22] J. F. Steffensen, Interpolation (Chelsea Publishing 
Co., New York, N.Y., 1950). 
[3.23] H. S. Wall, Analytic theory of continued fractions 
(D. Van Nostrand Co., Inc., New York, N.Y., 
1948). 
[3.24] E. T. Whittaker and G. Robinson, The calculus of 
observations, 4th ed. (Blackie and Son, Ltd., 
London, England, 1944). 
[3.25] R. Zurmtihl, Praktische M.athematik (Springer- 
Verlag, Berlin, Germany, 1953). 
Mathematical Tables and Collections of Formulas 
[3.26] E. P. Adams, Smithsonian mathematical formulae 
and tables of elliptic functions, 3d reprint (The 
Smithsonian Institution, Wa,shington, D.C., 1957). 
[3.27] L. J. Comrie, Barlow’s tables of squares, cubes, 
square roots, cube roots a,nd reciprocals of all 
integers up to 12,500 (Chelmical Publishing Co., 
Inc., New York, N.Y., 1954). 
[3.28] H. B. Dwight, Tables of integrals and other mathe- 
matical data, 3d ed. (The Macmillan Co., New 
York, N.Y., 1957). 
[3.29] Gt. Britain H.M. Nautical Almanac Office, Inter- 
polation and allied tables (Her Majesty’s Sta- 
tionery Office, London, England, 1956). 
[3.30] B. 0. Peirce, A short table of integrals, 4th ed. 
(Ginn and Co., Boston, Mass., 1956). 
[3.31] G. Schulz, Formelsammlung zur praktischen Mathe- 
matik (de Gruyter and Co., Berlin, Germany, 
1945). 
24 ELEMENTARY ANALYTICAL METHODS 
Table 3.1 POWERS AND ROOTS nk 
k 
: 
3 
4 
: 
7 
fz 
10 
24 
l/2 
l/3 
l/4 
l/5 
See Examples 1-5 for use 
of the table. 
Floating decimal notation: 
910=34867 84401 
nL 
ngss= 
= (9)3.4867 84401 
.lO= 
25% 
1024 
$4, 167 77216 
: 
3 
4 
2 
i 
190 
24 
l/2 
l/3 
l/4 
l/5 
2: 
125 
625 
3125 
15625 
78125 
3 90625 
19 53125 
97 65625 
(16)5.9604 64478 
2.2360 61977 
1.7099 75947 
1.4953 48781 
1.3797 29662 
1’0: 
1000 
24 
10000 
1 00000 
10 00000 
100 00000 
1000 00000 
( 9 1 1.0000 00000 
(10 1.0000 00000 
(24) 1.0000 00000 
w 3.1622 77660 
v3 2.1544 34690 
l/4 1.7782 79410 
v5 1.584893192 
2:: 
3375 
50625 
7 59375 
113 90625 
1708 59375 
24 
11)5.7665 03906 
(28)1.6834 11220 
l/2 3.8729 83346 
l/3 2.4662 12074 
l/4 1.9679 89671 
l/5 1.7187 71928 
42000 
8000 
1 60000 
32 00000 
640 00000 
9) 1.2800 00000 
10)2.5600 00000 
(11)5.1200 00000 
(13)1.0240 00000 
24 (31)1.6777 21600 
w 4.4721 35955 
l/3 2.7144 17617 
l/4 2.1147 42527 
l/5 1.8205 64203 
nl= 
T&L s 
n3, 
n‘L 
8 
5 16 12= 
.L 
n7== 
2: 
128 
27 
2:; 
729 
1024 
4096 
2187 16384 
6561 65536 
19683 2 62144 
59049 10 48516 
(11)2.8242 95365 (14)2.8147 49167 
2.0000 00000 
1.5874 01052 
1.4142 13562 
1.3195 07911 
9 
&L 1.4142 13562 
nl/3= 1.2599 21050 
.&4= I.1892 07115 
7$1/5= 1.1486 98355 
1.7320 50808 
1.4422 49570 
1.3160 74013 
1.2457 30940 
6 7 
2:: 3:; 
8 
516; 
4096 
32168 
2 62144 
20 97152 
167 77216 
7% 
6561 
59049 
1296 
7716 
46656 
2 79936 
16 79616 
2401 
16807 
1 17649 
8 23543 
57 64801 
403 53607 
2824 75249 
5 31441 
47 82969 
430 46721 
3874 20489 
( 9)3.4867 84401 
(22)7.9766 44308 
3.0000 00000 
2.0800 83823 
1.7320 50808 
1.5518 45574 
100 77696 
604 66176 
(18)4.7383 81338 
2.4494 89743 
1.8171 20593 
1.5650 84580 
1.4309 69081 
1342 17728 
( 9)1.0737 41824 
(21)4.7223 66483 
2.8284 27125 
2.0000 00000 
1.6017 92831 
1.5157 16567 
(20) 1.9158 12314 
2.6457 51311 
1.9129 31183 
1.6265 76562 
1.4757 73162 
1:: 
1331 
14641 
1 61051 
17 71561 
194 87171 
2143 58881 
12 
1728 
13 
169 1’946 --_ 
2197 
28561 
3 71293 
48 26809 
627 48517 
8157 30721 
2744 
38416 
5 37824 
75 29536 
1054 13504 
20736 
2 48832 
79 85984 -. __._ 
358 31808 
4299 81696 
9)5.1597 80352 
10)6.1917 36422 
( 9)2.3579 47691 
(10)2.5937 42460 
(24)9.8497 32676 
3.3166 24790 
2.2239 80091 
1.8211 60287 
1.6153 94266 
10)1.0604 49937 
11)1.3785 84918 
(26)5.4280 07704 (25)7.9496 84720 
3.4641 01615 
2.2894 28485 
1.8612 09718 
1.6437 51830 
(27)3.2141 99700 
3.6055 51275 
2.3513 34688 
3.7416 57381 
2.4101 42264 
1.9343 36420 
1.6952 18203 
1.8988 28922 
1.6702 71652 
2% 
4913 
16 
4.2426 40687 
3:: 
5832 
2.6207 41394 
1 04976 
18 89568 
2.0597 67144 
340 12224 
6122 20032 
10)1.1019 96058 
1.7826 02458 
11 1.9835 92904 
12 1 3.5704 67227 
(30)1.3382 58845 
256 
4096 
65536 
6859 1 30321 
24 76099 
470 45881 
8938 71739 
83521 
14 19857 
241 37569 
(lOj6;8719 47674 
10 48576 
(12)1.0995 11628 
167 77216 
(28)7.9228 16251 
2684 35456 
I 
4.0000 00000 
914.2949 
2.5198 42100 
67296 
2.0000 00000 
1.7411 01127 
-.- _ -- 
(llj1.1858 78765 
41U3 38673 
I 9j6.9757 57441 
(12)2.0159 93900 
(29)3.3944 86713 
4.1231 05626 
2.5712 81591 
2.0305 43185 
1.7623 40348 
4.3588 98944 
2.6684 01649 
2.0877 97630 
1.8019 83127 
(30)4.8987 62931 
4:: 
10648 
2 34256 
23 
5:: 
13824 
3 31776 
79 62624 
1911 02976 
441 
9261 
1 94481 
40 84101 
857 66121 
( 9)1.8010 88541 
(10)3.7822 85936 
(11)7.9428 00466 
(13)1.6679 88098 
51 53632 
1133 79904 
(32)1.6525 10926 
4.6904 15760 
(31)5.4108 19838 
4.5825 75695 
2.7589 24176 2.8020 39331 
2.1406 95143 2.1657 36771 
1.8384 16287 1.8556 00736 
4.7958 31523 
2.8438 66980 
2.1899 38703 
1.8721 71231 
(32)4.8025 07640 
4.8989 79486 
2.0844 99141 
2.2133 63839 
1.8881 75023 
(33)1.3337 35777 
ELEMENTARY ANALYTICAL METHODS 25 
POWERS AND ROOT!3 nk Table 3.1 
6:: 
15625 
3 90625 
97 65625 
2441 40625 
26 21 
729 7:: 
21952 
6 14656 
172 10368 
4818 90304 
10 1.3492 92851 
11 3.7780 19983 
II 
13 1.0578 45595 
14 2.9619 61661 
(34)5.3925 32264 
5.2915 02622 
3.0365 88972 
2.3003 26634 
1.9472 94361 
10;; 
35937 
11 85921 
391 35393 
9 1.2914'67969 
'10 
'12 
I 
4.2618 44298 
1.4064 08618 
113 4.6411 48440 
15)1.5315 78985 
(36)2.7818 55434 
8:: 
24389 
7 07281 
2105 11149 
5948 23321 
(35)1.2518 49008 
676 
17576 
4 56976 
118 81376 
3089 15776 
9 8.0318 10176 
11 2.0882 70646 
12 5.4295 03679 
14 I 1.4116 70957 
19683 
5 31441 
143 48907 
3674 20489 
10 1.0460 35320 
11 2.8242 95365 
12 
I 
7.6255 97405 
14 2.0589 11321 
(33)3.5527 13679 33)9.1066 85770 
5.0000 00000 5.0990 19514 
2.9240 17738 2.9624 96068 
2.2360 67977 2.2581 00864 
1.9036 53939 1.9186 45192 
34)2.2528 39954 
5.1961 52423 
3.0000 00000 
2.2795 07057 
1.9331 82045 
5.3851 64807 
3.0723 16826 
2.32!05 95787 
1.9610 09057 
32768 
10 48576 
335 54432 
9:: 
27000 
6 10000 
243 00000 
7290 00000 
10 2.1870 00000 
11 6.5610 00000 
Ii 
13 1.9683 00000 
14 5.9049 00000 
31 34 
1156 
39304 
13 36336 
454 35424 
9)1.5448 04416 
961 
29791 
9 23521 
286 29151 
0675 03681 
10 2.7512 61411 
11 8.5289 10374 II 13 2.6439 62216 14 8.1962 82870 
(35)2.8242 95365 
5.4772 25575 
3.1072 32506 
2.3403‘47319 
1.9743 50486 
(35)6.2041 26610 (36)1.3292 27996 (36)5. 6950 03680 
5.8309 51895 
3.2396 11801 
2.41147 36403 
2.0;!43 97459 
5.5677 64363 5.6560 54249 5.7445 62647 
3.1413 80652 3~1748 02104 3.2075 34330 
2.3596 11062 2.3704 14230 2.3967 El727 
1.9873 40755 2.0000 00000 2.0123 46617 
30 
1444 
35 
1225 
42875 
15 00625 
525 21075 
15)2.7585 47354 
24 (37)1.1419 13124 
l/2 
l/3 
l/4 
l/5 
5.9160 79783 
3.2710 66310 
2.4322 99279 
2.0361 68005 
37 
1369 
50653 
18 74161 
693 43957 
19 
15% 
59319 
23 13441 
902 24199 
(38)1.5330 29700 
46656 
16 79616 
604 66176 
54672 
20 85136 
792 35168 
9 3.0109 36384 
11 1.1441 55826 
~12 
I 
4.3477 92138 
14 1.6521 61013 
15 6.2782 11848 
(37)8.2187 60383 
6.1644 14003 
3.3619 75407 
2.4828 23796 
2.0699 35054 
18:; 
79507 
34 18801 
1470 08443 
9 6.3213 63049 
I 11 2.7181 86111 
13 
1 
1.1688 20028 
14 5.0259 26119 
16)2.1611 48231 
39)1.5967 72093 
6.5574 38524 
3.5033 98060 
2.5607 49602 
2.1217 47461 
15)3.6561 58440 
(37)4.3335 25711 (37)2.2452 25771 
6.0827 62530 
3.3322 21852 
2.4663 25715 
2.0589 24137 
6.0000 00000 
3.3019 27249 
2.4494 89743 
2.0476 72511 
6.2449 97998 
3. 3912 11443 
2.4989 99399 
2.0807 16549 
40 
17:: 
74088 
31 11696 
1306 91232 
19;: 
85184 
37 48096 
1649 16224 
(39)2.7'724 53276 
1600 
64000 
25 60000 
1024 00000 
(16)1.0485 76000 
16G 
68921 
28 25761 
1158 56201 
9)4.7501 04241 
11)1.9475 42739 
12)7.9849 25229 
24 (38)2.8147 49767 
6.3245 55320 
3.4199 51893 
2.5148 66859 
2.0912 79105 
45 
(38)5.0911 10945 (38)9.0778 49315 
23:: 
l/2 
l/3 
l/4 
l/5 
6.4031 24237 6.4807 40698 
3.4482 17240 3.4760 26645 
2.5304 39534 2.5457 29895 
2.1016 32478 2.1117 85765 
6.6332 49581 
3.5303 48335 
2.5'755 09577 
2.1'315 25513 
21;: 
97336 
44 77456 
2059 62976 
6.7823 29983 
(39)8.0572 70802 
3.5830 47871 
6.8556 54600 
3.6088 26080 
2.5900 20064 2.6042 90687 2.6183 30499 
2.1411 27368 2.1505 60013 2.1598 30012 
47 
(39)4.7544 50505 
2025 
91125 
41 00625 
1845 28125 
( 9)8.3037 65625 
(11)3.7366 94531 
(13 1.6815 12539 
(14 7.5668 06426 
I (I6 3.4050 62892 
22d9 
1 03823 
48 79681 
2293 45007 
(10 1.0779 21533 
11 5.0662 31205 
ii 
13 2.3811 28666 
15 1.1191 30473 
16 5.2599 13224 
(40)1.3500 46075 
1 10592 
53 08416 
2548 03968 
10)1.2230 59046 
11 5.8706 83423 
13 1 2.8179 28043 
1.3526 05461 
_ _._.. 
57 64801 
6 
7 
9” 
10 
24 40)2.2376 37322 
6.9282 03230 
3.6342 41186 
2.6321 48026 
2.1689 43542 
(40)3.6‘703 36822 
l/2 
l/3 
l/4 
l/5 
6.7082 03932