On Mon, 04 Nov 2013 Raymond Toy wrote:
aught maxima what the value of elliptic_ec(-1) is:
> elliptic_ec(-1);
> sqrt(2)*elliptic_ec(1/2)
> However, maxima also knows that elliptic_ec(1/2) is
> gamma(3/4)^2/(2*sqrt(%pi))+%pi^(3/2)/(4*gamma(3/4)^2)
> Should elliptic_ec(-1) evaluate to
> gamma(3/4)^2/(sqrt(2)*sqrt(%pi))+%pi^(3/2)/(2^(3/2)*gamma(3/4)^2)
> instead of sqrt(2)*elliptic_ec(1/2)?
> Ray
Example. Compute elliptic_ec(-1).
By definition: elliptic_ec (m) = integrate(sqrt(1 - m*sin(x)^2), x, 0,
%pi/2).
(%i1) S:integrate(sqrt(1+sin(x)^2), x, 0, %pi/2);
(%o1) integrate(sqrt(sin(x)^2+1),x,0,%pi/2)
This integral is equal to integrate(sqrt(cos(y)^2+1),y,0,%pi/2):
(%i2) S1:changevar(S, y=%pi/2-x, y, x),expand;
(%o2) integrate(sqrt(cos(y)^2+1),y,0,%pi/2)
(%i3) S1:subst(cos(y)^2=1-sin(y)^2,S1);
(%o3) integrate(sqrt(2-sin(y)^2),y,0,%pi/2)
(%i4) S2:subst(2-sin(y)^2=2*(1-sin(y)^2/2),S1);
(%o4) sqrt(2)*integrate(sqrt(1-sin(y)^2/2),y,0,%pi/2)
By definition of elliptic_ec this integral is equal to
(%i5) sqrt(2)*elliptic_ec(1/2);
(%o5) sqrt(2)*elliptic_ec(1/2)
(%i6) sol1:makegamma(%);
(%o6) sqrt(2)*(gamma(1/4)^2/(8*sqrt(%pi))+%pi^(3/2)/gamma(1/4)^2)
(%i7) beta(a,b)=makegamma(beta(a,b));
(%o7) beta(a,b)=(gamma(a)*gamma(b))/gamma(b+a)
(%i8) subst([a=1/4,b=3/4],%);
(%o8) sqrt(2)*%pi=gamma(1/4)*gamma(3/4)
(%i9) solve(%,gamma(1/4));
(%o9) [gamma(1/4)=(sqrt(2)*%pi)/gamma(3/4)]
(%i10) sol2:subst(%,sol1);
(%o10) sqrt(2)*(gamma(3/4)^2/(2*sqrt(%pi))+%pi^(3/2)/(4*gamma(3/4)^2))
Answer: sol1 or sol 2.
(%i11) [sol1,sol2];
(%o11) [sqrt(2)*(gamma(1/4)^2/(8*sqrt(%pi))+%pi^(3/2)/gamma(1/4)^2),
sqrt(2)*(gamma(3/4)^2/(2*sqrt(%pi))+%pi^(3/2)/(4*gamma(3/4)^2))]
(%i12) float(%), numer;
(%o12) [1.910098894513856,1.910098894513856]
best
Aleksas