The following code takes forever to compute in Maple. Basically, I'm trying to find the mean-square-modulus of a complex-valued function on a circle centered at the origin. How should the code be corrected, or is there an online calculator that can achieve this?
evalf(subs(subs([n=3, p=1.2451, z=exp(x*I)/2], subs(s=(p/2)*(1+1/(4*z)^n), subs(t=s+sqrt(s^2-1/(4*z)^n), w=z*t^(2/n)))), Int(abs(w)^2, x=0..2*Pi)/(2*Pi)));
Is this any use? I mean does it give the expected result? By supplying the integrand as an operator (black box) the evalf/Int engine is prevented from poking at it too expensively. This can save time, although there is risk involved as it might miss key discontinuities for some problems.
> U:=subs(subs([n=3, p=1.2451, z=exp(x*I)/2],
> subs(s=(p/2)*(1+1/(4*z)^n),
> subs(t=s+sqrt(s^2-1/(4*z)^n),
> w=z*t^(2/n)))),
> Int(X->eval(abs(w)^2,x=X), 0..2*Pi)/(2*Pi)):
> st:=time():
> evalf(U);
0.3351666815
> time()-st;
0.109
Another (likely less generally useful possibility) might be,
> restart:
> U:=subs(subs([n=3, p=1.2451, z=exp(x*I)/2],
> subs(s=(p/2)*(1+1/(4*z)^n),
> subs(t=s+sqrt(s^2-1/(4*z)^n),
> w=z*t^(2/n)))),
> Int(abs(w)^2, x=0..2*Pi)/(2*Pi)):
> st:=time():
> simplify(U);
-20
0.3351666815 - 0.5131390209 10 I
> time()-st;
3.150