Figure 19.3.6 (See in context.)
as a function of 
and 
for 
, 
.
Cauchy principal values are shown when
.
The function tends to 
as 
,
except in the last case below.
If 
(
), then the function reduces to
with Cauchy principal value
,
which tends to 
as 
.
See (19.6.5) and (19.6.6).
If 
(
), then by (19.7.4)
it reduces to 
, 
,
with Cauchy principal value
,
, by (19.6.5).
Its value tends to as 
by
(19.6.6), and to the negative of the second lemniscate
constant (see (19.20.22)) as
.
: Legendre’s complete elliptic integral of the first kind,
: Legendre’s complete elliptic integral of the third kind,
: cosecant function,
: Legendre’s incomplete elliptic integral of the third kind,
: sine function,
: real or complex argument and
: real or complex modulus
