Article 39
Explanation of
Tables.
In Table I the numerical values of the hyperbolic functions
are tabulated
for values of
increasing from 0 to 4 at intervals of .02. When
exceeds 4, Table IV may be used.
Table II gives hyperbolic functions of complex arguments, in which
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and the values of are
tabulated for values of
and of
ranging separately from 0 to 1.5 at intervals of .1. When interpolation
is necessary it may be performed in three stages. For example, to find
: First
find , by
keeping
at 1.3 and interpolating between the entries under
and
; next
find , by
keeping
at 1.4 and interpolating between the entries under
and
, as before; then by
interpolation between
and find
, in
which
is kept at .82. The table is available for all values of
,
however great, by means of the formulas on page :
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It does not apply when
is greater than 1.5, but this case seldom occurs in practice. This table can also be
used as a complex table of circular functions, for
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and, moreover, the exponential function is given by
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in which the signs of
and are to be taken
the same as the sign of ,
and the sign of
on the right is to be the product of the signs of
and
of on
the left.
Table III gives the values of ,
and of the gudermanian angle ,
as
changes from 0 to 1 at intervals of .02, from 1 to 2 at intervals of .05, and from 2 to 4
at intervals of .1.
In Table IV are given, the values of
,
,
, as
increases from 4 to 6 at intervals of .1, from 6 to 7 at intervals of .2, and from 7 to 9
at intervals of .5.
In the rare cases in which more extensive tables are necessary,
reference may be made to the tables32 of Gudermann, Glaisher, and
Geipel and Kilgour. In the first the Gudermanian angle (written
) is taken as the
independent variable, and increases from 0 to 100 grades at intervals of .01, the corresponding
value of
(written )
being tabulated. In the usual case, in which the table is entered with the value of
,
it gives by interpolation the value of the gudermanian angle, whose
circular functions would then give the hyperbolic functions of
.
When
is large, this angle is so nearly right that interpolation is not reliable.
To remedy this inconvenience Gudermann’s second table gives directly
,
,
, to nine figures,
for values of
varying by .001 from 2 to 5, and by .01 from 5 to 12.
Glaisher has tabulated the values of
and
, to nine significant
figures, as varies
by .001 from 0 to .1, by .01 from 0 to 2, by .1 from 0 to 10, and by 1 from 0 to 500. From these
the values of ,
are
easily obtained.
Geipel and Kilgour’s handbook gives the values of
,
, to seven
figures, as
varies by .01 from 0 to 4.
There are also extensive tables by Forti, Gronau, Vassal, Callet, and Hoüel;
and there are four-place tables in Byerly’s Fourier Series, and in Wheeler’s
Trigonometry.
In the following tables a dash over a final digit indicates that the number has
been increased.
TABLE I.—HYPERBOLIC FUNCTIONS.
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TABLE II.—VALUES OF
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0 | | | | | | | | |
.1 | | ” | ” | | | | | |
.2 | | ” | ” | | | | | |
.3 | | ” | ” | | | | | |
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.4 | | ” | ” | | | | | |
.5 | | ” | ” | | | | | |
.6 | | ” | ” | | | | | |
.7 | | ” | ” | | | | | |
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.8 | | ” | ” | | | | | |
.9 | | ” | ” | | | | | |
1.0 | | ” | ” | | | | | |
1.1 | | ” | ” | | | | | |
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1.2 | | ” | ” | | | | | |
1.3 | | ” | ” | | | | | |
1.4 | | ” | ” | | | | | |
1.5 | | ” | ” | | | | | |
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0 | | | | | | | | |
.1 | | | | | | | | |
.2 | | | | | | | | |
.3 | | | | | | | | |
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.4 | | | | | | | | |
.5 | | | | | | | | |
.6 | | | | | | | | |
.7 | | | | | | | | |
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.8 | | | | | | | | |
.9 | | | | | | | | |
1.0 | | | | | | | | |
1.1 | | | | | | | | |
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1.2 | | | | | | | | |
1.3 | | | | | | | | |
1.4 | | | | | | | | |
1.5 | | | | | | | | |
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TABLE II.—VALUES OF
AND .
(continued)
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0 | | | | | | | | |
.1 | | | | | | | | |
.2 | | | | | | | | |
.3 | | | | | | | | |
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.4 | | | | | | | | |
.5 | | | | | | | | |
.6 | | | | | | | | |
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.8 | | | | | | | | |
.9 | | | | | | | | |
1.0 | | | | | | | | |
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1.3 | | | | | | | | |
1.4 | | | | | | | | |
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0 | | | | | | | | |
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.2 | | | | | | | | |
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.5 | | | | | | | | |
.6 | | | | | | | | |
.7 | | | | | | | | |
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.8 | | | | | | | | |
.9 | | | | | | | | |
1.0 | | | | | | | | |
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1.3 | | | | | | | | |
1.4 | | | | | | | | |
1.5 | | | | | | | | |
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TABLE II.—VALUES OF
AND .
(continued)
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0 | | | | | | | | |
.1 | | | | | | | | |
.2 | | | | | | | | |
.3 | | | | | | | | |
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.4 | | | | | | | | |
.5 | | | | | | | | |
.6 | | | | | | | | |
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.8 | | | | | | | | |
.9 | | | | | | | | |
1.0 | | | | | | | | |
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1.2 | | | | | | | | |
1.3 | | | | | | | | |
1.4 | | | | | | | | |
1.5 | | | | | | | | |
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.6 | | | | | | | | |
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.9 | | | | | | | | |
1.0 | | | | | | | | |
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1.3 | | | | | | | | |
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TABLE II.—VALUES OF
AND .
(continued)
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.5 | | | | | | | | |
.6 | | | | | | | | |
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.9 | | | | | | | | |
1.0 | | | | | | | | |
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1.2 | | | | | | | | |
1.3 | | | | | | | | |
1.4 | | | | | | | | |
1.5 | | | | | | | | |
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0 | | | | | | | | |
.1 | | | | | | | | |
.2 | | | | | | | | |
.3 | | | | | | | | |
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.4 | | | | | | | | |
.5 | | | | | | | | |
.6 | | | | | | | | |
.7 | | | | | | | | |
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.8 | | | | | | | | |
.9 | | | | | | | | |
1.0 | | | | | | | | |
1.1 | | | | | | | | |
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1.2 | | | | | | | | |
1.3 | | | | | | | | |
1.4 | | | | | | | | |
1.5 | | | | | | | | |
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TABLE III.
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00 | | | .60 | | | 1.50 | | |
.02 | | | .62 | | | 1.55 | | |
.04 | | | .64 | | | 1.60 | | |
.06 | | | .66 | | | 1.65 | | |
.08 | | | .68 | | | 1.70 | | |
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.10 | | | .70 | | | 1.75 | | |
.12 | | | .72 | | | 1.80 | | |
.14 | | | .74 | | | 1.85 | | |
.16 | | | .76 | | | 1.90 | | |
.18 | | | .78 | | | 1.95 | | |
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.20 | | | .80 | | | 2.00 | | |
.22 | | | .82 | | | 2.10 | | |
.24 | | | .84 | | | 2.20 | | |
.26 | | | .86 | | | 2.30 | | |
.28 | | | .88 | | | 2.40 | | |
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.30 | | | .90 | | | 2.50 | | |
.32 | | | .92 | | | 2.60 | | |
.34 | | | .94 | | | 2.70 | | |
.36 | | | .96 | | | 2.80 | | |
.38 | | | .98 | | | 2.90 | | |
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.40 | | | 1.00 | | | 3.00 | | |
.42 | | | 1.05 | | | 3.10 | | |
.44 | | | 1.10 | | | 3.20 | | |
.46 | | | 1.15 | | | 3.30 | | |
.48 | | | 1.20 | | | 3.40 | | |
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.50 | | | 1.25 | | | 3.50 | | |
.52 | | | 1.30 | | | 3.60 | | |
.54 | | | 1.35 | | | 3.70 | | |
.56 | | | 1.40 | | | 3.80 | | |
.58 | | | 1.45 | | | 3.90 | | |
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TABLE IV.
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4.0 | | | | 5.5 | | | |
4.1 | | | | 5.6 | | | |
4.2 | | | | 5.7 | | | |
4.3 | | | | 5.8 | | | |
4.4 | | | | 5.9 | | | |
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4.5 | | | | 6.0 | | | |
4.6 | | | | 6.2 | | | |
4.7 | | | | 6.4 | | | |
4.8 | | | | 6.6 | | | |
4.9 | | | | 6.8 | | | |
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5.0 | | | | 7.0 | | | |
5.1 | | | | 7.5 | | | |
5.2 | | | | 8.0 | | | |
5.3 | | | | 8.5 | | | |
5.4 | | | | 9.0 | | | |
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