See Figures 19.17.1–19.17.8 for symmetric elliptic
integrals with real arguments.
Because the -function is homogeneous, there is no loss of generality in
giving one variable the value or (as in Figure 19.3.2). For
, , and , which are symmetric in , we
may further assume that is the largest of if the variables are
real, then choose , and consider only and
. The cases or correspond to the complete
integrals. The case corresponds to elementary functions.
Figure 19.17.8: ,
, .
Cauchy principal values are shown when .
The function is asymptotic to
as , and to
as .
As it has the limit .
When , it reduces to .
If , then it has the value
when , and
when .
See (19.20.10), (19.20.11), and
(19.20.8) for the cases , , and
, respectively.
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