Table Of ContentP U B L I S H E R ' S N O T I C E
T O R E V I E W E R S A N D R E A D E R S
C O N C E R N I N G T H E Q U A L I T Y O F P R O D U C T I O N
A N D P U B L I S H E D P R I C E O F T H I S W O R K
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I. R. M A X W E L L
Publisher at Pergamon Press
U.S.S.R. Academy of Sciences
Institute of Precision Mechanics and
Computational Technology
MATHEMATICAL TABLES
TABLES OF
In r [z]
FOR COMPLEX ARGUMENT
A. A. ABRAMOV
Translated from the Russian by
D. G. F R Y , B.A. Hons.
P E R G A M O N PRESS
OXFORD LONDON NEW YORK. • PARIS
1960
PERGAMON PRESS LTD.,
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c
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19C0
Pergamon Press Ltd.
Library of Congress Card Number 60-14845
Printed in Great Britain by
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IKEEDDUCTION
The Institute of Precision Mechanics and Computational Technology
of the U.S.S.R. Academy of Sciences compiled these six-figure tables
of lnTfz), z~x + iy for the use of a number of scientific research
institutes. The tables have been compiled for the range l < z < 2 ,
0 < y < 4 . Fonmlao for the calculation of lnT (z) outside the given
rectangle are listed. An auxiliary nomograph is given on a separate
sheet of paper to facilitate the interpolation procedure. The
tables have been calculated for steps in x and y of 0.01 and with
an accuracy of one unit in the last (the sixth) decimal place.
The tables were compiled by a group headed by A.A.Aferamov in the
Department of Approximate Calculations. The necessary evaluations
were made by Z.S. Romanova. The calculations were performed on
keyboard machines in the Department's calculating section. The
work was supervised by A.A. Bunatyan. Some of the calculations
(including the rounding off of the results to the sixth place and
the calculation of second differences) were carried out on
analytical computers in the Institute's Computational Research
Laboratory under the supervision of M. G-* Bappoport. Verification
was supervised by A. I. Vsorova. The tables were prepared for the
press by L. V. Bochek, and G. S. Khovanskii compiled the auxiliary
nomograph.
The Use of the Tables
The numerical material in the tables is arranged in the following
manner: each two facing pages contairn values of the function In r (z)
for the rectangle x ' < x < ^ , y' < y <y" which correspond to these
pages. The lefthand page contains the values for the real part and
the righthand page the values for the imaginary part of the corres-
ponding values of lnr(z).
The arrangement of the material by pages in the table is as
follows: the first 40 pages contain the values of In r (z) for
0 < * / < 0.50, the next 40 for 0.50 < y < 1.00 and so on: there are
in all eight such sections corresponding to intervals in y of length
0.50. Within each group the material is arranged in order of
increasing x
%
To find the pair of pages which will contain the required value
of In r (x + iy) one has first to find the group of pages with the
required value of y (the interval of variation of y is printed at
the top of the page at the outside edge) and then to find the page
1
2 TABLES FOR A COMPLEX ARGUMENT
with the required value of x within the group (the values of x are
given in the first column of each page).
Example. To find In T(l,07 +i3,21).
Here the group of pages for 3*00 O < - 3 * 5 0 is pp. 252-291
on p. 254 w we find Re In F (z) = -3.457847,
on p. 255 « find ImlnF(z) = 1.391384
lnr(1.07 + 13.21) = — 3.457847 + i 1.391384.
Since lnr(z) =flnf (z)]* lnF(z) can be found directly from the
table for l < R e z < 2 , — 4 < I m z<0; thus In F (1.07 —3.21 i) =
= — 3.457847 — 1,391384 i.
Interpolation is needed for the calculation of lnr(z) for inter-
mediate values. Let z~x + iy lie in the square z, z, z, z, where
0 f 2 3
z, z z , z are the tabular values of the argument of z; a , U , U , W
0 v 2 3 0 { 2 3
*re the corresponding values of the raal part, v v <v , v are
w v 2 2
the corresponding values of the imaginary part of lnT (z) = u + iv.
(These values are arranged in the following manner in the tables
Z/ftl/l V
— on the lefthand page
„ — on the righthand page).
u u
3 2
The interpolation is carried out separately for the real and
imaginary parts of lnT(z) respectively by the formulas:
u = u + c (K — u )
0 A 0
U = lig + $ (tt — tt )
2 3
lmhiF(z) =
= V + r (V — V) — (A)lm
{ 2
5 = Vo + M^i — V )
0
v = v + S(v — v )
t t 9
The values of (A ) and (A )i (smoothed second differences in y of
2 Re 2 m
Relnr(z) and Imlnr(z) ) are given in the tables between the four
corresponding supporting values at t7he points The inter-
polation error does not exceed 5*10~ , i.e. six places are retained
in interpolation. If four correct places are sufficient in inter-
polation, the bilinear part only o5f the f omnia need be used, the
error of which is less than 2 .10~ .
Examples:
1 . To find InT (1.52108 + £2.17835).
INTRODUCTION 3
u = —1.671735 + 0.108 (—1.663026 + 1.671735) = —1.670794;
u = —1.683000 + 0.108 (—1.674254 + 1.683000) = —1.682055;
Re In T (z) = —1.670794 + 0.835 (—1.682055 + 1.670794) —
v = 0.897539 + 0.108 (0.908776 — 0.897539) = 0.898753;
v - 0.906258 + 0.108 (0.917513 — 0.906258) = 0.907474;
Im In T (z) = 0.898753 + 0.835 (0.907474 —0.898753) —
In T (1.52108 +12.17836) =.—. 1,680197 + #.906034
2 . To find lnT (1.32795 —i3.9?861) t© four places, fa use. the formula
lnr(z)=[lnr(z)*j*
5 = 0,795; ?j = 0,86f;
o = —4.182878 + 0.795 (— 4.168902+ 4.182878).= — 4;171767;
u = —4.196541 +0.795 (—4.182541 + 4.196541) = — 4.185411;
Re In T (z) = — 4.171767 + 0.861 ( - 4.18541* + 4,171767) =• — 4.183514;
v = 2,717758 + 0.795 (2.731406 — 2.717758) = 2.728608;
v = 2.731743 + 0.795 (2.745397 — 2,731743) = 2.742598;
Im In T (z) — 2.728608 + 0.861 (2.742598 — 2.728608) = 2.740653
In T(l .32795 — t'3.97861) = —4.1835 +1'2.7407
The calcalation of lnr(s) outside the quadrangle 1 < ^ < 2 ; 0 < y < 4
For the range Re(z)>0; lm(z)>0 outside the given quadrangle
calculations are made by the formulae:
l) the asymptotic expansion fornula:
i.e.
Imlnr(z) =
4 TABLES FOR A COMPLEX ARGUMENT
1
B are Bernoulli s numbers
n
ln27r = 0.91893854.
In the diagrams of recommended formulae given opposite this
series is cut short at n - 0, 1, 2 or 3.
2) the multiplication formula:
lnF (z) +\nT
2ir,
When n 2 the formula known as the duplication formula is obtained
s
from the previous formila:
2 in 2 = 1.38629436, In 4TT = 1.26551213.
3) 1a formula that yields a displacement parallel to the real
axis.
In T (z + k) = In (z + k — 1) + In (z + k — 2) + . .. + Inz + In T (z).
In the range Re(*)<0, lm(z)<0 the calculations are made by the
fornula:
In T (z) = In x— In sin r.z — In T (1 — z).
If we rewrite this formula in the form
In r (z) = In K - In T (1 — z) — In
m2 x
and disregard the tern — e , we obtain
Inr (z) « In* — In T (1 — z) — im + ln2 + -j- ,
i.e.
Re In r (z) « — Re In T (1 ~ z) + In 2* + icy;
Imlnf —Im InT
The error to which this formula gives rise does not exceed
1—
This estimation can be used to compile a table for the conditions
which y should satisfy (independently of x) for the required accur-
acy to be observed*
A
When calculating by this formula one can, for example, use the
"Tables of the logarithms of complex numbers and transformation
from cartesian to polar coordinates" published by the U.S.S.R.
Academy of Sciences. 1952.
INTRODUCTldN 5
2 3
^
KS/A range of application of
the asymptotic formula;
fiTim range of application of
the displacement formula;
range of application of
the duplication formula;
range for which the tables
have been compiled.
_ \\WW//^///A
Z 3
The numbers of n and k in the asymptotic and displacement
formulas have been included on the diagrams. They are
not to scale.
6 TABLES FOR A COMPLEX ARGUMENT
Required
Given the conditions for The values of lnr(z) in
accuracy
ranges symmetrical to the
7 ranges Re (z) > 0, Im (2) > 0
5-10- 6 — 2 / > 2.31 and Re (2) < 0, Im (z) < 0 are
5.10-5 - 2 / > 1.95 obtained directly from the
5.10--4 — 1 / > 1 . 5 8 f omnia:
5.10- — y > 1 . 2 2
lnT(2) =[lnT ( 2 ) * ] *
The diagrams given above show the division into parts of the
range Rez>l, Im z > 0 and indicate in each case the formula recom-
mended in relation to the required accuracy of 8.
When 0 < R e < 1 it is sufficient, in addition, to use the formula
lnr(z) = lnr(2 + l) —lnz.
The f ornulae here recommended should not be thought to be oblig-
atory. In a number of cases it may be more expedient to use one of
the other fornulae. Thus* for example, when x is close to 0 and
is sufficiently large it may be possible to m k e direct use of the
asymptotic fornula, for example, with n = 0 or n = 1, which is more
convenient than the calculation of the value of lnT(2 + l)by the
formulae here given and subsequent use of the fornula
l n r ( 2 ) = l n r (2 + 1) —lnz.
Or, for example, the asymptotic fornula may be preferred to the
displacement formula if there are no suitable tables for the calcul-
ation of In 2.
In addition, the plane is divided into rectangles (or even strips)
for the sake of simplicity and for part of the range this increases
the number of terms in the asymptotic fornula which are indicated
as being essential.
The Use of the Nomograph
The nomograph has baen constructed for the calculation of the tern
in quadratic interpolation.
A is the second difference expressed in units of the sixth place,
2
r
* = —, i = — are the divided increments of the coordinates of
z = x+iy. To find a , the edge of a ruler is applied to the S scale
at the given reading and to the point where the lines for r and A
t 2
with the given readings intersect. The point where the edge of the
ruler intersects scale a will be the answer in units of the sixth
place. If the solving line does not intersect scale a, the roles
of 5 and 7 ] must be reversed and the sign in the result changed. In
the nomograph A is positive. If A is negative the answer should
2 2
be sought for a positive A and the sign changed.
2
Example. Given 5 = 0.115. t) = 0,28. A = 85.
2
INTRODUCTION 7
We apply the edge of the ruler to the reading 0.115 on the scale
of 5 and to the point at which the reading 85 on the A line and
2
the reading 0.28 on the TJ line intersect. We read off Che answer
on the a scale and find it to he a 4.3.
s
Compilation of the Tables
The tables were calculated in the following way.
0,5 0.5*
OS 03*
0.7 0.3
0.8 O.Z^i
* At 0J~A
t 0 —J
Values of lnTfz) were first calculated by the asymptotic fornula
in a wide lattice with step k s 0.16. Points were selected more
frequently near the lefthand boundary of the lattice and their
values obtained by the formula In V (z) = lnT (z + 1) — In z from the
values calculated to the right of LB hy interpolations.
