hence, by integration,
the integration-constant being zero, since
This series is convergent, and can be used in computation, only when
. Another series,
convergent when ,
is obtained by writing the above derivative in the form
is the integration-constant, which will be shown in Art. 19 to be equal to
A development of similar form is obtained for
in which is
again equal to
[Art. 19, Prob. 46]. In order that the function
may be real,
must not be less
than unity; but when
exceeds unity, this series is convergent, hence it is always available for computation.
From (32), (33), (34) are derived:
- Show that the series for ,
are always available for computation.
- Show that one or other of the two developments of the inverse hyperbolic
cosecant is available.