Mathematical tables abramowitz stegun

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 scientists and engineers.
A greater variety of functions and higher accuracy of tabulation are now required as a result of scientific advances and, especially, of the increasing 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 implement the project, the National Science Foundation requested the National Bureau
of Standards to prepare such a volume and established an Ad Hoc Advisory Committee, 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,
Its success has depended greatly upon the cooperation of
and Irene A. Stegun.
Their efforts together with the cooperation of the Ad HOC
many mathematicians.
Committee
are greatly appreciated.
The particular
contributions
of these and
other individuals are acknowledged at appropriate places in the text. The sponsorship 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.

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Preface

to the Ninth

Printing

The enthusiastic reception accorded the “Handbook of Mathematical
Functions” is little short of unprecedented in the long history of mathematical tables that began when John Napier published his tables of logarithms in 1614. Only four and one-half years after the first copy came
from the press in 1964, Myron Tribus, the Assistant Secretary of Commerce 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 successof the Handbook has not ended our interest in the subject.
On the contrary, we continue our close watch over the growing and changing 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

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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. Abramowitz 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, “Mathematical 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
To provide technical guidance
National Science Foundation
made funds available.
to the Mathematics
Division of the Bureau, which carried out the work, and to provide 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

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,/”


VI

FOREWORD

Active 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 appreciation 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.

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Handbook

of Mathematical

Functions

with

Formulas,
Edited

Graphs,
by Milton

1.

and Mathematical
Abramowitz

and Irene A. Stegun

Introduction

The present Handbook
has been designed to
provide
scientific
investigators
with a comprehensive and self-contained
summary of the mathematical functions that arise in physical and engineering 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 tabulated functions,
particularly
those of computa-

2.

Tables

Accuracy

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. fouror 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 Computing Service Ltd., Great Britain.

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 mathematical properties
stated in the chapter may be
found.
Also listed in the bibliographies
are the
more important
numerical
tables.
Comprehensive lists of tables are given in the Index mentioned 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 Handbook are those commonly
adopted in standard
texts, particularly
Higher Transcendental
Functions, Volumes 1-3, by A. ErdBlyi, W. Magnus,
F. Oberhettinger and F. G. Tricomi (McGrawHill, 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.

of the Tables
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 purpose 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 functions, 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.

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IX /-


.

INTRODUCTION

X

3.

Auxiliary

Functions

One of the objects of this Handbook is to provide 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 procedure clear.
The exponential integral of positive argument
is given by

4.

775
;:;
E

: 89717
89608

4302
8737

d0

g. I

ze*El
. 89823
.89927
90029

(z)
7113
7306
7888

8: 4
ix

:.90227
90129

4695
60”3

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, respectively; p is the given fraction of the argument
interval
p= (x--x0>/(x1-~0>
and jP the required
instance, we have

interpolate.

jo=.89717

ji=.89823

4302

In the present
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 multiplications by l-p and p cumulatively;
a partial
check is then provided by the multiplier
dial

reading unity.
We obtain

[ 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 roportional
out the use of diherences, ‘Brot Edin i: urgh Math. 8 oc. 3.6676

recludes direct interThe logarithmic
singularity
polation near x=0.
The Punctions Ei(x)-In
x
and x-liEi(ln
x-r],
however,
are wellbehaved 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 +Ei(x) is sufficiently well-behaved to admit direct
tabulation,
but for larger values of x, its exponential character predominates.
A smoother and
more readily interpolable
function for large x is
xe-“Ei(x); this has been tabulated for 2 Finally, the range 10 the inverse argument x-l. Twenty-one entries of
xe-“Ei(x), corresponding to x-l = .l(- .005)0, suffice to produce an interpolable
table.

Interpolation

The tables in this Handbook are not provided
with differences or other aids to interpolation,
because 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 conflicted with the requirement

that linear interpolation 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 iterative 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 interpolate to full tabular accuracy.
As an example, consider the following extract
from Table 5.1.
zez El (2)
. 89268 7854
: 89497
89384 9666
6312

and Arguments

parts, with.
(1932).

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

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INTRODUCTION

by the last roundingJ that is, .4403X 10m5, plus
3 X lo-‘, and so certainly cannot exceed .8X lo-‘.
(2) Lagrange’s formula.
In this example, the
relevant formula is the 5-point one, given by

The numbers in the third and fourth columns are
the first and second differences of the values of
xezEl(x) (see below) ; the smallness of the second
difference provides a check on the three interpolations. The required value is now obtained by
linear interpolation
:

f=A-,(p)f_z+A-,(p)f-1+Ao(p>fo+A,(p)fi
+A&)fa
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

7.953

.89774

0379

fn=.3(.89772

.;

&

-2

1
2

3
4
5

7.9
8.1
7.8
8.2
7.7

0999

Yn=ze”G@)
:
:
.
.

89823
89717
89927
89608
90029
89497

7113
4302
7888
8737
7306

9666

Yo.I
89773
:89774

Yo, 1.2. I

Yo. 1, (I

44034
48264
2 90220
4 98773
2 35221

0379)

In cases where the correct order of the Lagrange
polynomial
is not known, one of the prelimina
interpolations
may have to be performed witT
polynomials
of two or more different orders as a
check on their adequacy.
(3) Aitken’s method of iterative linear interpolation.
The scheme for carrying out this process
in the present example is as follows:

10620
.89775

9757)+.7(.89774

= 239773 7192.

10622
7.954

XI

X,-X

Yo.1.a.s.n

.0473
0527
. 1473
-. 1527
. 2473
-. 2527

-.
.89773

71499
2394
1216
2706

. 89773

71938
89773
ii

71930
30

Here
20-x
1 Yo
x,-x
x.--20 Yn
x,-x
1 Yo.1
Yo.1 ,n=x,-x
G--z1 l/O.”

S2fl

yo,n=-

Yo.

1.

. .

., m--l.m.n--

1
~n-%n

safz
wa

l/0.1. . . ., n-1.98
Yo.1. . . -, m-1.n

x,-x
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 safeguard 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 arguments, 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),

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
lace of the
written in units of the last decimal
function, as is customary.
The sma Bness of the
high differences provides a check on the function
values
7:9
8.0

Applying,
formula

xe=El(x)
.89717
4302
. 89823 7113

for example,

j~=(l-P)fo+E2(P)~*jo+E4(P)~4jo+

-2
-2

SY
2754
2036

Everett’s

. . .
+Pfl+F2(P)~afl+F4(P)~4fl+

S4f
-34
-39

interpolation
.
. * *

and takin
the numerical values of the interpolation toe flf cients Es(p), E4 ), F,(p) and F,(p)
from Table 25.1, we find t!l at

,,/

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INTRODUCTION

XII
10Qf.6,=

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.

.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)

In the present

+ ~2(P)lk?f0+wJ

f’(z)=(l+Z-‘)f(Z)-1
f”(2)=(1+2-‘)f’(Z)--Z-Qf(2)
f”‘(X)
= (1 -i-z-y’(2)

Inverse

AxmAj/df
dx

;
3

4302

.89717

4302

- .00113
.01074
.00012

7621
0669
1987

-.ooooo
.00056
.ooooo
.a9773

3159
6033
7194
0017

53
9

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 interpo Pate directly for p=.70, .7l and .72 with the aid
of Lagrange’s 5-point formula,
xe=El (x)

X

8. 170

.

89999 -.-_

8.171

.

90000

6

QQ

1 0151
3834

-2
1 0149

8. 172

90001

linear

Hencex=8.17062
An estimate

3983

interpolation

in the new

table

23.
of the maximum

error in this result

is
df~1x10-8_1x10-7
Ajl z
.OlO

example, we have

.9 - .89927

7888
72 2112
7306.89927 7888=101=‘708357’

(x--so) y’k’(x0)/k!

.89717

The desired z is therefore

Inverse
gives

in which Aj 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

‘=.90029

+22-y(2).

Interpolation

With linear interpolation
there is no difference

in principle between direct and inverse interpolation. 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 minimum.
The maximum precision attainable in an
inverse interpolate can be estimated with the aid of
the formula

In the present

p:‘(xo)/k!

i

. . .

5.

-22~Qf’(5)

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.

~(x,=~(xo)+(x-x,,~~+(x-xo,~~~
+(~-x,)q$+

we have

f(x) =xeZEt(x)

Since the maximum value of IEz(p)+Fz(p)I
in the
range Ois fd, the maximum error in a linear
interpolate is approximately

(5) Taylor’s series. In cases where the successive derivatives
of the tabulated function can be
computed fairly easily, Taylor’s expansion

example,

(iii) Aitken’s method.
This is carried out in the
same manner as in direct interpolation.

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INTRODUCTION
n
0

yn=xeeZE1(x)
. 90029
7306

2,

8. 2

4
3

: 90129
89927
. 89823

7888
6033
7113

8. 31
8. 0

8. 17023
17083
8. 17113

5712
1505
8043

%

: 89717
90227

4302

4695

8.
7. 94

8. 17144
16992

0382
9437

Z0.n

QJ.98
8. 1706,l

8. 17062

21 8142
7335

6. Bivariate
Bivariate interpolation
is generally most simply
performed as a sequence of univariate
interpolations.
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

Generation

a.1 2.n

9521
2 5948

The estimate of the maximum
error in this
result is the same as in the subtabulation
method.
An indication of the error is also provided by the

7.

XIII

of Functions

Many of the special mathematical
functions
which depend on a parameter,

called their index,
order or degree, satisfy a linear difference equation (or recurrence relation) with respect to this
parameter.
Examples
are furnished
by the Legendre function P,(z), the Bessel function Jn(z)
and the exponential integral E,(x), for which we
have the respective recurrence relations

zo.l.2.3.74

2244
415
231

8. 17062

2318
265

discrepancy
in the highest
case xo .I ,2.3 A, and ZLI .2.8 .s.

l/n-u
.00029

7306

-. . 00129

00072
-. 00176

6033
2112
2887

-. .00227
00282

4695
5G98

interpolates,

in this

I

Interpolation
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.

from

Recurrence

Relations

(iii) the direction in which the recurrence
applied.
Examples are as follows.
Stability-increasing
Pm(x),

p:(2)

Qnb),

Q:(x)

y&9,

KG)

J-n-&),
&Cd

n

(x
z-t44
(n
Stability-decreasing

Jn+*-~Jn+J.-l=O

P”(X),
nE,+,+xE,,=e-=.

is being

P.,(z)

7t
@
Qnh), Q:(x)

Particularly
for automatic work, recurrence relations provide an important
and powerful
computing 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
overwhelm the wanted function, the process is unstable.
It is important
to realize that st,ability may
depend on (i) the particular
solution of the difference equation being computed;
(ii) the values of
x or other parameters
in the difference equation;

J&4,
Jn+Hcd
Em(z)
F,,(t,

Z.@)

, Zn+&)
(n >r)
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 wellsuited to automatic
work, is described in 19.28,

Example

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1.


INTRODUCTION

XIV

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 formation 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 undiminished 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, completion 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 Yilington 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 Technical Information
in the areas of editorial format,
graphic art layout, printing
detail, preprinting
reproduction
needs, as well as attention to promotional detail and financial support. In addition,
the clerical and typing stafI of the Applied Mathematics 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.

www.pdfgrip.com

M.

ABRAMOWITZ.


1.

Mathematical
DAVID

Constants

S. LIEPMAN ’

Contents
Page

Mathematical

Table 1.1.

+i,nprime

Constants


20s.

...............

2

..................

2

Some roots of 2, 3, 5, 10, 100, 1000, e, 20s ..........

2

e*n, n=l(l)lO,

....................

2

20s ....................

2

25s

e*tns, n=l(l)lO,
eas, e*‘,

20s

.......................

ln n, log,, n, n=2(1)10,
In 7~,In&,

na, n=1(1)9,
a*“, n=l(l)lO,

........

24D

r(4), l/r($),

3
3

25s ....................

of T, powers and roots involving
26s

3

T,

25s .......

..................

3

.......................

3

15D .....................

Bureau of Standards.

www.pdfgrip.com

3
3

24D. .................

r(2),l/r(z),lnr(2),2~3,a,g,q,~,g,g,~,

2
3

25s .....................

lo, l’, 1” in radians,

1 National

26, 25s

25s ...................

1 radian in degrees,

~,lny,

primes
logI, ?r, log,, e, 25s ...............

n In 10, n=1(1)9,

Fractions

2

3

15D.

........

3


MATHEMATICAL

TABLE

1.1.

1 4142
i7320
2.2360
2..6457
3.3166
3.6055
4.1231
4.3588
4.7958
5.3851
5.5677
6.0827
6.4031
6.5574
6.8556
7.2801
7.6811

7.8102
8.1853
8.4261
8.5440
8.8881
9.1104
9.4339
9.8488

13562
50807
67977
51311
24790
51275
05625
98943
31523
64807
64362
62530
24237
38524
54600
09889
45747
49675
52771
49773
03745

94417
33579
81132
57801

37;:
56887
49978
06459
35539
46398
61766
54067
31271
13450
83002
29821
43284
30200
40104
28051
86860
90665
87244
i7635
31753
31558
14429
05660
79610

50488
72935
96964
05905
98491
92931
05498
35522

1)
lj
2)
2)
3)
3)
3)
4)

2.7182
7.3890
2.0085
5.4598
1.4841
4.0342
1.0966
2.9809
8.1030
2.2026

81828
56098
53692
15003
31591
87934
33158
57987
83927
46579

45:04
93065
31876
31442
02576
92735
42845
04172
57538
48067

52353
02272
67740
39078
60342
12260
85992
82747

40077
16516

1)
2)
4)
5)
6)
8j
9)
LO)
12)
13)

2.3140
5.3549
1.2391
2.8675
6.6356
1.5355
3.5533
8.2226
1.9027
4.4031

69263
16555
64780
13131
23999

29353
21280
31558
73895
50586

enr
27792
24764
79166
36653
34113
95446
84704
55949
29216
06320

69006
73650
97482
29975
42333
69392
43597
95275
12917
29011

26224

72417

14792
99019

64190
79852

47180
12288
94361
37912
59469
10149
41541
24577
85092
95272
49357
13344
38979
94215
95829
87204
17912
72066
00115

In n
55994

66810
11989
43410
22805
05531
67983
33621
99404
79837
46153
05621
16644
92914
98647
48514
64422
70430
69356

0.6931
1.0986
1.3862
1.6094
1.7917
1.9459
2.0794
2.1972
2.3025
2.3978
2.5649

2.8332
2.9444
3.1354
3.3672
3.4339
3.6109
I: %

CONSTANTS

MATHEMATICAL

CONSTANTS
1O’fi

3.1622

77660

16837

93320

1O’fl
10"'

2.1544
1.7782
1.5848

79410
34690
93192

03892
03188
46111

%X
34853

4.6415
2.5118
5.6234
3.9810
1.2599
1.4422
1.1892
1.3160
7.0710
5.7735
4.4721

88833
86431
13251
71705
21049
49570
07115

74012
67811
02691
35954

61277
50958
90349
53497
89487
30740
06275
95249
86547
89625
99957

FKd
08040
25077
31648
83823
20667
24608
52440
76451
93928

4.8104
2.1932

11 2.0787
li 4.5593
1.6487
1) 6.0653
1.3956
1) 7.1653
-I

~-

77380
80050
95763
81277
21270
06597
12425
13105

96535
73801
50761
65996
70012
12633
08608
73789

16555
54566

90855
23677
81468
42360
95286
25043

1)
1)
2)
2j
3)
3)
4)
4)
4)
5)

3.6787
1.3533
4.9787
1.8315
6.7379
2.4787
9.1188
3.3546
1.2340
4.5399

94411e-“71442

52832
36612
06836
78639
63888
87341
46999
08546
52176
66635
19655
54516
26279
02511
98040
86679
92976
24848

32159
69189
42979
80293
70966
84230
20800
83882
54949
51535

- 2)
- 3)
- 5)
- 6)
- 7)
- 9i
-1oj
-11)
-13)
-14j

4.3213
1.8674
8.0699
3.4873
1.5070
6. 5124
2.8142
1.2161
5.2554
2.2711

91826
42731
51757
42356
17275
12136
68457
55670

85176
01068

37722
70798
03045
20899
39006
07990
48555
94093
00644
32409

49774
88144
99239
54918
46107
07282
27211
08397
85552
38387

-

6.5988
5.6145

03584
94835

53125
66885

37077
16982

0102
7712
0205
9897
7815
4509
0308
5424
0000
0413
1139
2304
2787
3617
4623
4913
1. 5682
1.6127
1.6334

99956

12547
99913
00043
12503
80400
99869
25094
00000
92685
43352
48921
53600
27836
97997
61693
01724
83856
68455

101’5
1OOlfl
loo”5
1000"'
lOOO"5

2lB
3’B

:E:
19221

96890
86865
06523
41249
82711
81758
43941
99700
86306
11679
88501
88819
38113
47217

2114

3114

2-m
3-m
5-‘fi

(--

(-i.

((-

60287

30427
92853
11026
11156
83872
63720
43592
09997
95790

-

1)
1)
lj

~--~

--~-

-----

*
*

*

-----

55238

39995
34242
71802
36048
45167
31361
13891
76367
59152

e--nr

2)
1)

log10

53094
96913
xz:::
50008
33051
59282
93827
56840
05440
67360
60802
04600
96908

40271
62459
44443
78038
24234

172321
952452
344642
007593
124774
053527
516964
904905
179915
619436
534874
495346
090274
067528
832720
291643
680957
667634
728425
*See page xx.

www.pdfgrip.com

12

63981
l!id62
27962
36018
83643
14256
91943
39324
00000
15822
30683
37827
95282
01759
89895
83427
06699
71973
57958

19521
43729
39042
80478
63250
83071
58564
87459
00000

50407
67692
39285
89615
%Ei
26796
49968
54945
65264

37389
50279
74778
62611
87668
22163
12167
~:~::
50200
::%i
X%f
32847
66704
08451
09412
05088


MATHEMATICAL

TABLE

1.1.

CONSTANTS

MATHEMATICAL

3.8501
3. 9702
4.0775
4.1108
4. 2046
4.2626
4. 2904
4.3694
4.4188
4.4886
4. 5747

47601
91913
37443
73864
92619
79877
59441
47852
40607
36369

10978

In n
71005
55212
90.571
17331
39096
04131
14839
46702
79659
73213
50338

85868
18341
94506
12487
60596
54213
11290
14941
79234
98383
28221

209507
444691
160.504

513891
700720
294545
921089
729455
754722
178155
167216

(-1)

1. 1447
9. 1893

29885
85332

84940
04672

01741
74178

43427
03296

(
(
(
(

(

1)
1)
1)
1)
1)

2.3025
4.6051
6.9077
9.2103
1. 1512
1. 3815
1. 6118
1. 8420
2. 0723

85092
70185
55278
40371
92546
51055
09565
68074
26583

nln 10
99404

98809
98213
97618
49702
79642
09583
39523
69464

56840
13680
70520
27360
28420
74104
19788
65472
11156

17991
35983
53974
71966
08996
10795
12594
14393
16192

1)

1)
2)
2j
3)
3)
4)
4)

3.1415
9.8696
3. 1006
9. 7409
3.0601
9.6138
3.0202
9.4885
2.9809
9.3648

92653
04401
27668
09103
96847
91935
93227
31016
09933
04747

58979
08935
02998
40024
85281
75304
77679
07057
34462
60830

32384
86188
20175
37236
45326
43703
20675
40071
11666
20973

62643
34491
47632
44033
27413
02194
14206
28576

50940
71669

1. 5707
1. 0471
7.8539
1. 7724
1.4645
1.3313
2. 1450
2.3597
5. 5683
2. 2459
2. 5066
1. 2533
2. 2214

96326
97551
81633
53850
91887
35363
29397
30492
27996
15771
28274
14137
41469

79489
19659
97448
90551
56152
80038
11102
41469
83170
83610
63100
31550
07918

66192
77461
30961
60272
32630
97127
56000
68875
78452
45473
05024
02512
31235

31322

54214
56608
98167
20143
97535
77444
78474
84818
42715
15765
07883
07940

57. 2957
0. 0174

79513
53292

08232
51994

08767
32957

98155’
69237r

CONSTANTS-Continued
log10

loglog

(-1)
(-1)

logl0e

(
(
(
(
(
(

(
(
C
(
(
(
(
(

(-1)

t:

I-, -,
r (714)

In
In
In
ln

r(i/3)
r(2/3)
r(lj4)
r(3/4)

1)

97857
75869
52011
29835
74802
58348
22860
27091
78092
90006
71734

93571
60078
64214
01076
70082

71907
12045
29044
37607
64491
f ?624

74644
90456
41902
70338
64341
52860
59010
14279
39038
27847
48517

14219
32992
60656
85749
49132
92829
74387
94821
32760
23543
84362

4.9714
4.3429

98726
44819

94133
03251

85435
82765

12683
11289

92653
85307
77960
37061
96326
55592
14857
74122
33388

5s”9”19
17958
76937
43591

79489
15387
51285
87183
23081

32384
64769
97153
72953
66192
59430
52669
45907
39146

62643
25287
87930
85057
31322
77586
23850
70115
16379

3. 1415
6. 2831
!a. 4247
1) 1. 2566

1) 1.5707
1) 1. 8849
1) 2. 1991
1)2.5132
1) 2.8274

n

7P

0. 5772

15664

90153

1. 7724
2. 6789
1. 3541
3. 6256
1. 2254
0.8929
0. 9027
0. 9064
0. 9190
0. 9854
0.3031
1.2880
0.2032

53850
38534
17939
09908
16702
79511
45292
02477
62526
20646
50275
22524
80951

905516
707748
426400
221908
465178
569249
950934
055477
848883
927767
147523
698077
431296

28606

06512

;
3
4
x
7
i
10
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)

3. 1830

1.. 0132
3.2251
1. 0265
3.2677
1. 0401
3. 3109
1. 0539
3.3546
1. 0678

98861
11836
53443
98225
63643
61473
36801
03916
80357
27922

83790
42337
31994
46843
05338
29585
77566
53493
20886

68615

67153
77144
89184
35189
54726
22960
76432
66633
91287
33662

77675
38795
42205
15278
28250
89838
59528
17287
39854
04078

4.7123
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

88980
90204
82938
95835
40632
55444
40770
72081
71221
26726
22804
45608
81580

38468
78639
15836
47756
55295
64942
35411
23757
25166
69229
01432
02865
78533

98576
09846
62470
28694
68146
48285
61438
59679
56168
06151
67793
35587
03477

93965
16858

15881
80795
70208
87030
19885
10077
90820
35273
99461
98921
75996

0. 0002
0. 0000

90888
04848

20866
13681

57215
10953

96154r
59936r

48223

37662

-2j
-2)
-3)
-31

-4j
-4)
-5)
-5)

1’
1”
In Y

-0.

l/~U/‘4
l/Ul/3)
l/r(2/3)
mm)
mw
m4/3)
mw)
mw
uw4
In r(4/3)
In r(5/3)
In r(5/4)
in r(7/4)

.
*See page II.

www.pdfgrip.com

n

1.6720
1.7242
1.7708
1.7853
1.8260
1.. 8512
1. 8633
I.. 8976
1. 9190
1. 9493
1. 9867

0.
0.
0.
0.
0.
:l.
l.
l.
:l.
-0.

-0.
-0.
-0.

5495

39312

98164

5641
3732
7384
2758
8160
1198
1077
1032
0880
1131
1023
0982
0844

89583
82173
88111
15662
48939
46521

32167
62651
65252
91641
14832
71836
01121

547756
907395
621648
830209
098263
722186
432472
320837
131017
740343
960640
421813
020486

*

i


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2. Physical

Constants

and Conversion

A. G. MCNISH

Factors

1

Contents
Table
Table
Table
Table
Table
Table

2.1.
2.2.
2.3.
2.4.
2.5.
2.6.

Common Units and Conversion Factors . . . . . . . . .
Names and Conversion Factors for Electric and Magnetic
Units . . . . . . . . . . . . . . . . . . . . . . .

Adjusted Values of Constants . . . . . . . . . . . . .
Miscellaneous Conversion Factors. . . . . . . . . . . .
Conversion Factors for Customary U.S. Units to Metric
Units . . . . . . . . . . . . . . . . . . . . . . .
Geodetic Constants . . . . . . . . . , . . . . . . . .

* National Bureau of Standards.

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Page
6
6
7
8
8
8


2. Physical

Constants

and

Table

Quantity

and

Conversion

Common

Factors

for

Electric

and

=

SI

emu
name

name

I

Magnetic

=

force

Magnetomotive
force
Magnetic flu*
Magnetic flux density
Electric displacement

ampere
coulomb
volt
ohm
henry
farad
amp. turns/
meter
amp. turns
weber
tesla

Conversion

I tbampere
1tbcoulomb

abvolt
abohm
centimeter

statampere
statcoulomb
statvolt

statohm

10-l
LO-’
108
100
100
10-g
4*x

centimeter
oersted
gil bert
maxwell
gauss
--._-_-______

Units

=

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*

IO-3*

4rX lo-I*
108
10’

__---___----__-______-_____

-3/10**
-(1/3)X
10-z
-(1/3)X
10-B
-3x 105*

10-J*

I_-..____..____

-

-

Example: If the value assigned to a current
*Divide this number by 4?r if unrationalized
6

and

SI unit/
emu unit

esu
name

-

Current
Charge
Potential
Resistance
Inductance
Capacitance
Magnetizing

Units
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 infinitesimal
cross-section.

If F is in newtons,
and rrn has the numerical
value 477 X lo-‘,
then I1 and Zr are in amperes.
The customary equations
define the other electric and
magnetic
units of SI such as the volt, ohm,
farad, henry, etc. The force between electric charges in a vacuum in this system is
given by Q, Qn/4nrerg= F, re having the numerical
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 defines the CGS unrationalized
electromagnetic
system (emu), re then taking the numerical
value of 1/c2. Setting
re equal to unity defines the CGS unrationalized
electrostatic

system (esu), r,,, then taking
the numerical
value of l/cz.

Mass-the
kilogram
-fixed by the international 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 cesiurn 133.
Temperature-the
degree-fixed
on a thermodynamic
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 proportionThe
ality constant in each defining equation.
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
=

2.1.

often used in physics

~

(1 meter - 1650763.73h).

2.2. Names

Factors

rise to the CGS system,
and chemistry.

The tables in this chapter supply some of
the more commonly
needed physical

constants and conversion
factors.
All scientific measurements
in the fields of
mechanics
and heat are based upon four international
arbitrarily
adopted
units,
the
magnitudes
of which are fixed by four agreed
on standards:
Lengththe meter -fixed
by the vacuum
wavelength
of radiation
corresponding
to the
transition
2Plu-5Da
of krypton 86

Table

Conversion

-

is 100 amperes its value in abamperes is 100X10-‘=lO.

system is involved; other numbers are unchanged.

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3. Elementary

Analytical

MILTON

Methods

ABRAMOWITZ l

Contents
Elementary
3.1.

3.2.
3.3.
3.4.
3.5.
3.6.
3.7.
3.3.
3.9.
3.10.

Analytical

Methods

Page
10

.................

Binomial Theorem and Binomial Coefficients; Arithmetic
and
Geometric Progressions; Arithmetic,
Geometric, Harmonic
and Generalized Means
...............
Inequalities ......................
Rules for Differentiation
and Integration
.........
Limits, Maxima and Minima
..............
Absolute and Relative Errors ..............
Infinite Series .....................
Complex Numbers and Functions ............
Algebraic Equations ..................
Successive Approximation
Methods
...........
Theorems on Continued Fractions ............

Numerical

3.11.

Methods

Table

3.1.

19
19
23

............................

Powers and Roots

13
14
14
16
17
18
19
19

.......................

Use and Extension of the Tables
............
3.12. Computing Techniques.

................

References

10
10
11

. . . . . . . . . . . . . . . . . .

24

n’“, k=l(l)lO,
24, l/2, l/3, l/4, l/5
n=2(1)999, Exact or 10s

The author acknowledges the assistance of Peter J. U’Hara
the preparation
and checking of the table of powers and roots.

1 National

Bureau of Standards.

(Deceased.)

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and Kermit

C. Nelson in


3. Elementary

Analytical

3.1. Binomial
Theorem
and Binomial
Coefficients; Arithmetic
and Geometric
Progressions;
Arithmetic,
Geometric,
Harmonic
and
Generalized
Means
Binomial

Methods

3.1.9
Sum

of Arithmetic

a-t-b+d)+b+24+

a”-lb+@
a”-w+

. . . +b”

Coefficients

of Geometric

(see chapter

integer)

s,=a+ar+a?+

n =nc7t=n(n-l)

24)

. .k., (n--k+l)=

Arithmetic

Mean

Geometric

3.1.12

A

. - . +a.
Mean

of

Mean

n Quantities

G

(at>O,k=1,2,.

G= (a,&. . . a,,)l’”
Harmonic

. .,n)

H

of n Quantities

3.1.13

3.1.5

.+$)
3.1.6

1+c)+@+.

(ah>O,k=1,2,.

1-G)+@-

3.1.7

of Binomial

Coefficients

0

M(t)=O(t
lim M(t) =max.
t+m
3.1.17 t&M(t)=min.
3.1.16

1
1
1
1
1

7---8---9---lo_--_
ll---12----

2
3
4
5
6

1 7
1 8
1 9
1 10
1 11
1 12

;
1
6
4’
1
10 10
5
15 20 15

1
6

a2,

(a,,%,.

* * *, a,) =mtLx. a

. .,a,)=min.a

M(l)=A

3.1.20

M(-l)=H
3.2. Inequalities

1
8
36

1
9
45

Relation
1
10
55

For a more extensive table see chapter 24.
page

3.1.19

1

21 35 35 21
7
28 56 70 56 28
36 84126126
84
45120210252210120
55165330462462330165
66220495792924792495220

@I,

liiM(t)=G

3.1.18
3---4---5..--6-e-m

some ak zero)

z

3.1.8

2---I

Mean

M(t) ==(i g a:yf

3.1.14

. . . +(--I)“@=0

3.1.15
Table

. .,n)

. . +c)=2”
General&d

*See

(--l
n

(k-;-l)

3.1.4

a(l-P)
l--P

n Quantities

of

&h+az+

3.1.11

@=(n:k)=(-l)k

n Terms

to

lim s.=a/(l-r)
n-t-

n!
(n-k)!k!

0k

3.1.3

Progreamion

. . . +a+‘=---

3.1.2
*

(a+z))

3.1.10

(n a positive
Binomial

7+-l)&;

last term in series=Z=a+(n-1)d

an-%2
Sum

+C)

n Terms

to

. . . +(a++-114

=na+;

Theorem

3.1.1
(a+b)“=a”+c)

Progression

Between
and

Arithmetic,
Generalized

Geometric,
Means

Harmonic

3.2.1
A> G>H,

3.2.2

II.

10

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equality

if and only if al=az=

min. a
a

. . . =a,,


ELEMENTARY

3.2.3

min. aequality

3.2.5

Minkowski’s

If p>l

Inequalities

Inequality

for

Sums

and &, bk>O for all x:,

3.2.12

if tor sTriangle

11

METHODS

a

holds if all ak are equal, or tand an an is zero

M(t)<&!(s)

3.2.4

ANALYTICAL

(&

(ak+bk)p~‘p<(&

a;)“‘+($

holds if and only

t?qUdity

i-f

bi$“j
(c=con-

bk=C&

stant>O).

Iall-la21_
Minkowski’s

3.2,6

Inequality

5or Integrals

If P>l,
Chehyshev’s

Inequality

If alla2>a,z

. . . >a,

b,>b,>b,>

. . . >bn

3.2.7

n 5

akbk>

-(kc,

k=l

Hiilder’s

If ;++,p>1,

2

3.2.13

(Jb
a

‘> (&

Inequality

for

I?(~)+9(z)I~~s)llp~(~b

a

a

lJ~(z)l%iz)l’p
+(Jb

“)

a

Iscd lqp

equality holds if and only if g(z) =cf(x)
stant>O).

Sums

(c=con-

fj>l
3.3. Rules

for

Differentiation

and

Integration

Derivatives

equality holds if and only if jbkl=cIuEIP-’
stant>O).
If p=q=2
we get
Cauchy’s

(c=con-

$; (cu) =c $9 c constant

3.3.1

3.3.2

Inequality

-g (u+v)++g

3.2.9
[&

akb,]2<&

a; &

b: (equality

for &=Cbk,

& (uv) =u E+v

3.3.3

2

c constant).
Hiilder’s
4

Inequality

for

3.3.4

Integrals

& (u/v)=

vdu/dx-udv/dx
v2 -

,

1r;++=1,p>1,

q>l

.g u(v) =g

3.3.5

gi

3.2.10

& (u”) =uD e %+1.n U 2)

3.3.6
equality holds if and only
(c=constant>O).
If p=p=2 we get
Schwa&s

3.2.11

Inequality

if

jgCs)I=clflr)Ip-’

Leibniz’s

Theorem

for

Differentiation

of an Integral

3.3.7
b(c)

d
&

s

a(c)

f

(2,

wx
b(c)

3

a(cj

s

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b
,J(x,c)dx+f@,

4

$--f

(a,

4

2


12

ELEMENTARY

Leibniz’s

Theorem

for

Differentiation

ANALYTICAL

The following

of a Product

3.3.8
s

METHODS

formulas

S

P(x)dx

n>l

is an integer.

where P(x)

(ux”+ fJx+cy

(t&g

u+(y)

g

g

+(g

ds

g

are useful for evaluating
is a polynomial

and

3.3.16
+...+~)d~rg+...+cg

dx
2
(ax2+br+c)=(4c&c-bz)~

S
dX
-&=1g

3.3.9

(b2--4uc
a

2az+b-

3.3.17

I 2azfbf

#x
-d2y
dy -3
dy2=dr2
0 zc

3.3.10

$=

-B

g-3

Integration

3.3.12

(g--j

($)-"

-2

S

by Parts

S

3.3.20

jidu=w+dti

(a+ bx$c+dxj=k

jkudx=(jidx)
Integrals

v-s(judx)

of Rational

(Integration

Algebraic

2 dx

dx
1
=- arctan E!
u2+b2;C2 ub
U

Functions

3.3.22

const,ants are omitted)

S(ux+f,)"dx=(ux+6)"+1
4n+1>
S
S

S
(n#-1)

3.3.24

3.3.15

$&In

)ax+b)

S

t(u+bx)

of Irrational

dx
=h2
(c+dx)11’2

3.3.27

=+

3.3.28

=h2

3.3.29

3.3.30

(x2;1-“a2)2=&

arctan

~+20~(xf+U2)

3.3.25

Integrals

3.3.26

c+dx
bc In I-a+bx I

dx
(a+bx)“P(c+dx)=[d(bc~ud)]1~2

S

=[d(ad&]1/2

Algebraic

arctan
arcsin

(ad # bc)

S___
lnb2+
b2x21
S
S

3.3.21

3.3.23

3.3.14

(P-4ac=O)

3.3.19

S
3.8.13

(b2-4m>O)

=2az+b

3.3.18
3.3.11

(b2--4uc)t
(P-4acy

Functions

-d(a+

1 W
bx) 1’2

C b(c+dx)
2bdx+ad+
bc-ad

bc
>

ln J[bd(n+ bx)]1/2+ b(c+dx)1/21
arctan ~~~~-J”
d(u+bx)1’2-[d(ad-bc)]1’2
In d(a+bx)1’2+[d(ud-bc)]“2

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(b>O, d(bd>O)

(d(u&-bc)
I

@b-J--4>O)

ELEMENTARY

If zn=un+ivn,
3.7.23

then ~~+l:=u,+,+iv~+~

u,+~=xu~-~v,;

ANALYTICAL

where

3.8. Algebraic
Solution

v,+,=xv,+yu,

9?z” and 92” are called harmonic

17

METHODS

polynomials.

3.7.24

Equations

of Quadratic

3.8.1

Given az2+ bz+c=O,

21,2=-

0-

2”, j-k

gf, p= b2-4ac,
.zi+,za= -b/u,

3.7.25

If a>O, two real roots,
p=O, two equal roots,
a
Roots

3.7.26

z*=&=rte+rs=r+

cos @+iri

sin $0

Solution

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
sign is taken tc

2uv=y and where the ambiguous
be the same as the sign of y.
3 . 7 . 28

I

l&l-I221

I

Let
sl=[r-+(q3+r2)q+,

Catwhy-Riemann

Equations

au
-=-,

av

au
-=--

av

a~

by

by

ax

;,

; f$=-$

a

If z=Tefff,
3.7.31

sz=[T’-((p3+?3*]*

then

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+$
which
3.7.30

Equations

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).

_<1z1~~2111211+I~21

Functions,

roots.

$+P>O,

root if - ?r<0 5 7r)
(k=l,2,3,
a . ., n-1)

Inequalities

Complex

If

of Cubic

Z~Z~=C/U

Given Z3+a2z2+ulz+a0==0, let

3.8.2

Zl/n,Tl/nefe/n , (principal

Other roots are Pet(B+2rn’ln

3.7.29

Equations

If zl, z2, z3 are the roots of the cubic equation
g=;

Z~+Z~+Z~=-CIC~

Laplace’s

~~~2+~$,+&~,:=~~

Equation

The functions U(X, y) and v(x, y) are callec
harmonic functions and satisfy Laplace’s equation
Cartesian

Polar

3.7.33

r ;

(r g)+$=r

of Quartic

Equations

Given 24+~323+a3~~+~1l~+ag=O,
real root u1 of the cubic equation

3.8.3

Coordinates

r

i!?+$E~2+g2=o

3.7.32

Solution

U3

-

a2U2

i-

(~23

-

4ao)U

-

(a; j- Uoa$ - kW3)

find the
=

and determine the four roots of the quartic
solutions of the two quadratic equations

Coordinates

;

(r g+g=o

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0

as


18

ELEMENTARY

ANALYTICAL

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

METHODS
Method

of Iteration

then

-a3,

4142==0.

Convergence

of

an

fcxk)

xk

2, . . .).

Approximution

Process

(n=l,

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), xoNewton’s

3.9.6
x=N’l”

Method

Applied

to Real

nth

Roots

Given x”=N,
if zk is an approximation
then the sequence

3.9.2 Let zl, z2, x3, . . . be an infinite sequence
of approximations
to a number f. Then, if
1%n+~-~I<&n-tlk,

Rule

f’

Then, if f’(z)>0
and the constants cn are
negative and bounded, the sequence x,, converges
monotonically
to the root [.
If c,,=c=constantand f’(z)>O,
then the
process converges but not necessarily monotonically.
of

Approximations

Methods

3.9.1 Let z=zl be an approximation
to x=[
where f(t) =0 and both x1 and [ are in the interval
a$rWe define

Degree

of Successive

xk+l=

(n=l,

b.

If z=zk is an approximation
to the solution
z= I of f(z) =0 then the sequence

Comments

GI+1=G+C&n)

for aLzSb,

Newton’s

Approximation

General

Method

will

3.9.5

-&,

z1z2z3z4=ao.

Czj2,=u2,

3.9. Successive

Newton’s

z2 j2,2t=

IF’(s)J
(2) a-

If zl, z2, z3, z4 are the roots,
z2 j=

Substitution)

3.9.4 The iteration
scheme Q+~=F(z~)
converge to a zero of z=F(z) if
(1)

pl+p2=a2,plp2+pl+q2=a2,p~q2+p2q~=al,

(Successive

xk+l=-

;

[$i+(n-l)xk]

2, . . .)
will converge quadratically

to a.

where A and k are independent 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 convergence is quadratic.
Regula

Falsi

(False

Position)
Aitken’e

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,

-Hi

f,=

f 1~o-JoX,

jlVfO

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-

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 interpolation,

G-Process

(5k--k+1)*=;tk~k+2-2:+1.

A*&

is an improved

estimate

OGtk) then Z=s+O(P),

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A*Xk

’

of x.
Ix\<~.

In fact, if zk”x+*


ELEMENTARY

ANALYTICAL

METHODS

*
3.10.

Theorems

on Continued

Fractions

A,B,_l-A,-lB,=(-l)n-’

(4)

(5)

Definitions

kiI al;

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,

bz
=-- 1
l- b,+l-

=b,,+&e&.

..

. . . +;=-&

d+$+
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

x+A ...l

1
--a0

_- b3
b,+l-

--&

I

. . . $yu

1

2

t(-1,n----5

aoGa2

aof

1

=-

aox

n1

n

_ . . . a,
_- a12

___
al-x+

uo+

b,

* ’ ‘--b,+l

I12-xf

%-1X

* . . +un-2

is called the nth convergent of j.
A
(2) If lim -A exists, the infinite continued fracIt-+- 88
If uf= 1 and the
tion j is said to be convergent.
bt are integers there is always convergence.
Theorems
(1)

If

and br are positive

at

then j2n
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.

0

.2

.4

.6

.8

FIGURE 3.1
1 i

y:=xn*
2, 5.

*n=0,,5t 29 1,

Numerical
3.11.

Example

using Table

Use and

Extension

1. Computti
3.1.

of the Tables

xl9 and x4’ for x=29

Methods
Linear

I

interpolation

(919.826)“4-5.507144.
By Newton’s method
N=919.826,

3p=x9. x10

Table

=6.10326 1248. 102’

Repetition

for fourth

yields the same result.

roots with

3845

Thus,

~“~=5.50714 3845/10$=1.74151. 1796,
~-~“=zt/x=.18983

= (1.25184 9008. 1036)2/29

gives

3.1

1
4 ~7~3+3(5.507144)-]=5.50714
[ .

= (1.45071 4598. 1013)(4.207072333. 1014)
x4’= (x*4)2/x

in

05683.

=5.40388 2547. lO6*
3.12.
Example

2.

(9.19826)“‘=

Compute

x-3’4 for x=9.19826.

(919.826/100)1’4= (919.826)1’4/10t

Computing

Techniques

Example
3. Solve the quadratic
equation
x2- 18.2x+.056 given the coeflicients as 18.2 f .l,
*see

page

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II.


..

20

ELEMENTARY

.056f

.OOI.

ANALYTICAL

<

METHODS

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

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

4” -43.2
f(x)

The smaller root may be obtained more accurately
from
*
.05fi/18.1969= .0031& .OOOl.
Example

5
6
7

Compute (-3 + .0076i)i.

4.

From 3.7.26, (-3+.0976i)~=u+iv
Y
r!y
u=2G? I,-= (

.3
72.6
181.5

We obtain by linear inverse interpolation:

where

O-(-.3)
72.6-(-.3)=5’oo4’

x,=5+

*, j”= (t”+y’)t
>

Thus

Using Newton’s method, f’(x) =3x2-

r=[(-3)2+(.0076)2]~=(9.00005776)~=3.00000

1

Ij= 3.00000 9627- (-3)
2

f=

9627

21 =zo-f&J/f’

.73205 2196

We note that the principal square root has been
computed.
Example

Solve the quartic equation

6.

~‘-2.37752

Into

Quadratic

(22 + p12 + qd w

by Inverse

00526
'

'

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,

Factors

+ p2x + 92)

Interpolation

Starting with the trial value pI = 1 we compute
successively

QI

42

p1= a’--am
42-pll

pz=an-p1

9.
4. 053
526

--2. 1. 093
543

2. 2

4. 115

-3.

- 1.. 284
165

106

QI

Inverse interpolation

-2.

550

172

between qI=2.2
:md thus,

- 2: E

Qz

P2

PI

4. 51706
4.,51684
4. 51661

7640
2260
6903

-2.
-2.
-2.

55259
55282
55306

257
851
447

17506
. 17530
. 17553

Y h)

765
358

955

.00078
. 00001
-. 00075

552
655
263

gives q,=2.00420 2152, and we get finally,

_.
2. 00420

4. 520

Inverse interpolation
2.003 gives ql=2.0041,

5. 383

. 729

2. 0041
2. 0042
2. 0043

P2

Ykll)

. 011

Y(Qd=ql-t92+p*P2
- a2

_____

::

171

~~----

2.003

q2=;

C--.07215 9936jz5
57.020048

I

4922x3+6.07350 5741.x’
-11.17938 023s+9.05265 5259=0.

Resolution

18.1 we get

(d

=5.004-

.0076
u=&=2(1.73205
21g6)=.00219 392926

PI

-

.

2152

Qz
4. 51683

PI
7410

-2.

55283

Y (Ql)

P2

358

-

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17530

8659

-.

00000

0011

and pl=