Elliptic Integral Singular Value k1
The first
Singular Value
, corresponding to
(1)
is given by
(2)
(3)
As shown in
Lemniscate Function
,
(4)
Let
(5)
(6)
(7)
then
(8)
where
is the
Beta Function
and
is the
Gamma Function
. Now use
(9)
and
(10)
so
(11)
Therefore,
(12)
Now consider
(13)
Let
(14)
(15)
(16)
so
(17)
Now note that
(18)
so
(19)
Now let
(20)
(21)
so
(22)
But
(23)
(24)
(25)
so
(26)
(27)
Summarizing (12) and (
27
) gives
© 1996-9
Eric W. Weisstein
1999-05-25