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Hi, there! Just wondering if there is a way to compute numerical integration of some complex valued functions (meromorphic ones, most of the time). I intend to explore Weierstrass representation for minimal surfaces in Sage, and the only examples I can get are the ones were it can compute a primitive for the integrand, like in the Ennerper case:

var('rho,a,w') f=w g=1/w theta=0

Phi_1=(f/2(1/g-g)).integrate(w) Phi_2=(I/2f(1/g+g)).integrate(w) Phi_3=(f).integrate(w) def FF1(rho,a): return real_part(exp(Itheta)Phi_1(w=rhoexp(Ia))) def FF2(rho,a): return real_part(exp(Itheta)Phi_2(w=rhoexp(Ia))) def FF3(rho,a): return real_part(exp(Itheta)Phi_3(w=rhoexp(I*a)))

parametric_plot3d([FF1,FF2,FF3],(rho,0,2),(a,0,2*pi),plot_points=[10,100], mesh=true)

Maybe this numerical integration could be done in a meshgrid or something like that. If so, how one can do it?

Best, Alexandre

Hi, there! Just wondering if there is a way to compute numerical integration of some complex valued functions (meromorphic ones, most of the time). I intend to explore Weierstrass representation for minimal surfaces in Sage, and the only examples I can get are the ones were it can compute a primitive for the integrand, like in the Ennerper case:

var('rho,a,w')
f=w
g=1/w
theta=0
theta=0
  
Phi_1=(f/2(1/g-g)).integrate(w)
Phi_2=(I/2f(1/g+g)).integrate(w)
Phi_1=(f/2*(1/g-g)).integrate(w)
Phi_2=(I/2*f*(1/g+g)).integrate(w)
Phi_3=(f).integrate(w)
def FF1(rho,a):
  return real_part(exp(Itheta)Phi_1(w=rhoexp(Ia)))
real_part(exp(I*theta)*Phi_1(w=rho*exp(I*a)))
def FF2(rho,a):
  return real_part(exp(Itheta)Phi_2(w=rhoexp(Ia)))
real_part(exp(I*theta)*Phi_2(w=rho*exp(I*a)))
def FF3(rho,a):
  return real_part(exp(Itheta)Phi_3(w=rhoexp(I*a)))real_part(exp(I*theta)*Phi_3(w=rho*exp(I*a)))
  
parametric_plot3d([FF1,FF2,FF3],(rho,0,2),(a,0,2*pi),plot_points=[10,100], mesh=true)mesh=true)