Goursat's Surface
 |  |
A general quartic surface defined by
 |
(1)
|
(Gray 1997, p. 314). The above two images correspond to 
,
and 
, respectively.
Additional cases are illustrated above.
The "rounded cube" case corresponding to 
is a superellipsoid with volume
 |
(2)
|
where 
is the gamma
function.
The volume of the case 
is given by
 |  |  |
(3)
|
 |  | ![int_0^infty-(pi^2)/(16sqrt(2))R{((-1)^(5/8))/(t^(7/4))[(-1)^(1/4)e^(it/8)t^(3/4)×[(-1)^(3/4)J_(1/4)(t/8)-J_(-1/4)(t/8)]^3+(12sqrt(t)Gamma(1/4))/(pi^2)+(2(1+i)Gamma^3(1/4))/(pi^3)]}dt](/National_Library/20160521004321im_/http://mathworld.wolfram.com/images/equations/GoursatsSurface/Inline11.gif) |
(4)
|
where
 |
(5)
|
![R[z]](/National_Library/20160521004321im_/http://mathworld.wolfram.com/images/equations/GoursatsSurface/Inline12.gif)
is the real part
of 
and 
is a Bessel function of the first kind (E. W. Weisstein
and M. Trott, pers. comm., Nov. 9, 2008), which can probably be expressed
in closed form in terms of bivariate hypergeometric functions.
The related surface
 |
(6)
|
for 
an even integer is also considered
by Gray (1997, p. 292) and is a special case of the superellipsoid.
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goursat's surface
