Article 24
Series
for Gudermanian and its Inverse.
Substitute for
in (49), (48) their expansions, Art. 16, and integrate, then
No constants of integration appear, since
vanishes
with ,
and with
. These series are seldom
used in computation, as
is best found and tabulated by means of tables of natural tangents and hyperbolic
sines, from the equation
and a table of the direct function can be used to furnish the numerical values
of the inverse function; or the latter can be obtained from the equation,
|
To obtain a logarithmic expression for
,
let
-
Prob. 63.
- Evaluate ,
.
-
Prob. 64.
- Prove that
is an infinitesimal of the fifth order, when .
-
Prob. 65.
- Prove the relations ,
.