> restart; #Uniformizatopn of the twice punctured disc: trace -> punctures. > assume(r,real);assume(t,real); > N:=20: #Discretization parameter > M:=20: #Number of Schwarz iterations > tr:=6.0: #Input > lambda:=tr/2+sqrt(tr^2/4-1): #Finding r > Y:=solve({-a+b-c+d=0,-lambda*a+b+lambda*c-d=0,(a+b)*(lambda*c+d)+(c+d)*(lambda*a+b)=0,a*d-b*c=1},[a,b,c,d]): > aa:=rhs(Y[1,1]): > bb:=rhs(Y[1,2]): > cc:=rhs(Y[1,3]): > dd:=rhs(Y[1,4]): > r:=((aa+bb)/(cc+dd)+1)/2; 0.4142135624 > t:=j->Pi*(j+1/2)/(N+1): > s:=j->Pi*cot(t(j)/2)/(1-r): > ds:=j->Pi/(2*(1-r)*(sin(t(j)/2))^2): > u:=i->Pi*r^2*sin(t(i))/((1-r)*(2-2*r+2*r*cos(t(i))+r^2-r^2*cos(t(i)))): > v:=i->2*Pi*(1-r+r*cos(t(i)))/(2-2*r+2*r*cos(t(i))+r^2-r^2*cos(t(i))): > with(LinearAlgebra):with(plots): > A:=(i,j)->evalf((sin(v(i))/(cosh(s(j)-u(i))-cos(v(i)))+sin(v(i))/(cosh(s(j)+u(i))-cos(v(i))))*ds(j)/(2*(N+1))): > AA:=Matrix(N,A): > X[1]:=Vector(N,1): > for j from 1 to M do X[j+1]:=AA.(-X[j]) end do: > Y:=Matrix(N,M+1,(i,j)->X[j][i]): > Z:=Y.Vector(M+1,1): > b:=evalf(Pi*(1-2*r)/(1-r)): > p:=(i,j)->evalf((sin(b)/(cosh(s(j))-cos(b)))*ds(j)/(N+1)): > #B:=Matrix(N,p): > C:=Matrix(1,N,p): > ANS:=C.Z; ANS:=Vector[column](%id = 301684424) > #This is the solution of the Dirichlet problem at the center. Now final computation: > q:=evalf(cos(Pi*ANS[1])); 0.05054956414 > a:=(1/2)*(q+1/q); 9.916556902 > s:=-a+sqrt(a^2-1); -0.050549564 > mu:=(-1+sqrt(1-s^2))/s; 0.02529094811