>>>>> "Aleksas" == Aleksas Domarkas <aleksasd873 at gmail.com> writes:
Aleksas> On Mon, 04 Nov 2013? Raymond Toy wrote: aught maxima what
Aleksas> the value of elliptic_ec(-1) is: ?> elliptic_ec(-1); ?>
Aleksas> sqrt(2)*elliptic_ec(1/2) ?> However, maxima also knows
Aleksas> that elliptic_ec(1/2) is ? >?
Aleksas> gamma(3/4)^2/(2*sqrt(%pi))+%pi^(3/2)/(4*gamma(3/4)^2) ?
Aleksas> >???Should elliptic_ec(-1) evaluate to ??>?
Aleksas> ?gamma(3/4)^2/(sqrt(2)*sqrt(%pi))+%pi^(3/2)/(2^(3/2)*gamma(3/4)^2)
Aleksas> ? >?? instead of sqrt(2)*elliptic_ec(1/2)? ??>??? Ray
Aleksas> ?
Aleksas> ?
Aleksas> ?
Aleksas> Example. Compute elliptic_ec(-1).
Aleksas> ?By definition:? elliptic_ec (m) = integrate(sqrt(1 -
Aleksas> m*sin(x)^2), x, 0,
Aleksas> %pi/2).
Aleksas> (%i1) S:integrate(sqrt(1+sin(x)^2), x, 0, %pi/2); (%o1)
Aleksas> integrate(sqrt(sin(x)^2+1),x,0,%pi/2)
Aleksas> ?This integral is equal to
Aleksas> integrate(sqrt(cos(y)^2+1),y,0,%pi/2): (%i2)
Aleksas> S1:changevar(S, y=%pi/2-x, y, x),expand; (%o2)
Aleksas> integrate(sqrt(cos(y)^2+1),y,0,%pi/2)
Aleksas> (%i3) S1:subst(cos(y)^2=1-sin(y)^2,S1); (%o3)
Aleksas> integrate(sqrt(2-sin(y)^2),y,0,%pi/2) (%i4)
Aleksas> S2:subst(2-sin(y)^2=2*(1-sin(y)^2/2),S1); (%o4)
Aleksas> sqrt(2)*integrate(sqrt(1-sin(y)^2/2),y,0,%pi/2)
Aleksas> ?By definition? of elliptic_ec this integral is equal to
Aleksas> (%i5) sqrt(2)*elliptic_ec(1/2); (%o5)
Aleksas> sqrt(2)*elliptic_ec(1/2)
This is a nice derivation that elliptic_ec(-1) is
sqrt(2)*elliptic_ec(1/2), but I didn't follow what you're trying to
say about how maxima should simplify elliptic_ec(-1).
Ray