close all
clear all
clc

a = 1;
q = 1.2;
F = @(z) exp(z.^(-1));
G = @(z) (F(z)-1).*z;
m = @(z) (a*(F(z).^2-1)+q*G(z).^2)/2;
d = @(z) m(z).^2 + a*q*G(z).^2;
P1 = @(z) m(z) + sqrt(d(z));
P2 = @(z) m(z) - sqrt(d(z));

N = 1000;
phi = (0:N-1)/N;
z = exp(-2i*pi*phi);
vP1 = P1(z);
vP2 = P2(z);
vi = abs(vP1)-abs(vP2);
vP1b = vP1.*(vi>=0)+vP2.*(vi<0);
vP2b = vP1.*(vi<0)+vP2.*(vi>=0);

figure
subplot(2,2,1)
plot(phi,real(F(z)),phi,imag(F(z)),phi,abs(F(z)))
subplot(2,2,2)
plot(phi,real(G(z)),phi,imag(G(z)),phi,abs(G(z)))
plot(...
    ...%phi,abs((F(z).^2).*(a*a)./(vP1b+a)),...
    ...%phi,abs((F(z).^2).*(a*vP1b)./(vP1b+a)),...
    ...%phi,abs(-2*m(z)-a),...
    phi,abs(a*F(z).^2+q*G(z).^2),...
    phi,abs(vP1b))
subplot(2,2,3)
plot(phi,real(vP1b),phi,imag(vP1b),phi,abs(vP1b))
subplot(2,2,4)
plot(phi,real(vP2b),phi,imag(vP2b),phi,abs(vP2b))