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I have two real polynomials that depend in variables (xi1, xi2) and parameters (eps, e2). I apply the following complexification:

w(xi1, xi2, eps, e2) = gamma1(xi1, xi2, eps, e2) + I gamma2(xi1, xi2, eps, e2)

Now I wish to define the variable z = xi1 + I xi2, and find a neat expression w in z and it's conjugate, zc, as follows:

w(z, zc) = c30 z^3 + c21 z^2 zc+ c12 z zc^2 + c03 zc^3 + o(3)

In order to find the coefficients.

How may I ask this to Mathematica?

---------------------------- actual code -----------------------

gamma1[xi1,xi2,eps,e2]=(0. - 1.72945 I) e2 (-0.3567 xi1 + 0.229 xi2 + 0.7398 (-0.0423 eps xi1 + 0.4227 eps xi2))^3 - (0. + 0.5 I) (eps (0.5929 xi1 - 0.3806 xi2 - 1.2296 (-0.0423 eps xi1 + 0.4227 eps xi2)) + 38.4563 e2 (-0.3567 xi1 + 0.229 xi2 + 0.7398 (-0.0423 eps xi1 + 0.4227 eps xi2))^3);
gamma2[xi1,xi2,eps,e2]=(-1.72945 + 0. I) e2 (-0.3567 xi1 + 0.229 xi2 + 0.7398 (-0.0423 eps xi1 + 0.4227 eps xi2))^3 + (0.5 + 0. I) (eps (0.5929 xi1 - 0.3806 xi2 - 1.2296 (-0.0423 eps xi1 + 0.4227 eps xi2)) + 38.4563 e2 (-0.3567 xi1 + 0.229 xi2 + 0.7398 (-0.0423 eps xi1 + 0.4227 eps xi2))^3);

w[xi1, xi2, eps, e2] = gamma1[xi1, xi2, eps, e2] + I gamma2 [xi1, xi2, eps, e2];

I have two real polynomials that depend in variables (xi1, xi2) and parameters (eps, e2). I apply the following complexification:

w(xi1, xi2, eps, e2) = gamma1(xi1, xi2, eps, e2) + I gamma2(xi1, xi2, eps, e2)

Now I wish to define the variable z = xi1 + I xi2, and find a neat expression for w in terms of z and it's conjugate, zc, as follows:

w(z, zc) = c30 z^3 + c21 z^2 zc+ c12 z zc^2 + c03 zc^3 + o(3)

In order to find the coefficients.

How may I ask this to Mathematica?

---------------------------- actual code -----------------------

gamma1[xi1,xi2,eps,e2]=(0. - 1.72945 I) e2 (-0.3567 xi1 + 0.229 xi2 + 0.7398 (-0.0423 eps xi1 + 0.4227 eps xi2))^3 - (0. + 0.5 I) (eps (0.5929 xi1 - 0.3806 xi2 - 1.2296 (-0.0423 eps xi1 + 0.4227 eps xi2)) + 38.4563 e2 (-0.3567 xi1 + 0.229 xi2 + 0.7398 (-0.0423 eps xi1 + 0.4227 eps xi2))^3);
gamma2[xi1,xi2,eps,e2]=(-1.72945 + 0. I) e2 (-0.3567 xi1 + 0.229 xi2 + 0.7398 (-0.0423 eps xi1 + 0.4227 eps xi2))^3 + (0.5 + 0. I) (eps (0.5929 xi1 - 0.3806 xi2 - 1.2296 (-0.0423 eps xi1 + 0.4227 eps xi2)) + 38.4563 e2 (-0.3567 xi1 + 0.229 xi2 + 0.7398 (-0.0423 eps xi1 + 0.4227 eps xi2))^3);

w[xi1, xi2, eps, e2] = gamma1[xi1, xi2, eps, e2] + I gamma2 [xi1, xi2, eps, e2];

(xi1, xi2, eps, e2) = gamma1(xi1, xi2, eps, e2) + I gamma2(xi1, xi2, eps, e2)

a1 = 0;
a2 = 0;
a3 = 0;
a4 = 0;
a5 = -0.0423;
a6 = 0.4227;
b1 = 0;
b2 = 0;
b3 = 0;
b4 = 0;
b5 = -1.5293;
b6 = -0.8906;
h1[a1, a2, a3, a4, a5, a6, xi1, xi2, eps] = a1 xi1^2 + a2 xi2^2 + a3 eps^2 + a4 xi1 xi2 + a5 xi1 eps + a6 xi2 eps; 
h2[b1, b2, b3, b4, b5, b6, xi1, xi2, eps] = b1 xi1^2 + b2 xi2^2 + b3 eps^2 + b4 xi1 xi2 + b5 xi1 eps + b6 xi2 eps;
f11 = -0.0322 eps - 0.0028 eps^2;
f12 = -0.3761 + 0.207 eps + 0.0283 eps^2;
f21 = 0.3761 - 0.0502 eps + 0.0044 eps^2;
f22 = 0.0322 eps + 0.0440 eps^2;
f = {{f11, f12}, {f21, f22}};
eig = Eigenvalues[f];
v[eps_] = Eigenvectors[f];
p = v[0];
ip = Inverse[p];
d21 = 0.5929;
d22 = -0.3806;
d23 = -1.2296;
cub[xi1, xi2, y3_] = (-0.3567 xi1 + 0.229 xi2 + 0.7398 y3 )^3;
g1[xi1, xi2, y3_, eps, e2] = -3.4589 e2 cub[xi1, xi2, y3];
g2[xi1, xi2, y3_, eps, e2] = eps (d21 xi1 + d22 xi2 + d23 y3) + 38.4563 e2 cub[xi1, xi2, y3];
gamma1[xi1, xi2, eps, e2] = ip[[1, 1]] g1[xi1, xi2, h1[a1, a2, a3, a4, a5, a6, xi1, xi2, eps], eps, e2] + ip[[1, 2]] g2[xi1, xi2, h1[a1, a2, a3, a4, a5, a6, xi1, xi2, eps], eps, e2];
gamma2[xi1, xi2, eps, e2] = ip[[2, 1]] g1[xi1, xi2, h1[a1, a2, a3, a4, a5, a6, xi1, xi2, eps], eps, e2] + ip[[2, 2]] g2[xi1, xi2, h1[a1, a2, a3, a4, a5, a6, xi1, xi2, eps], eps, e2];
effe = ip.f;
ef = effe.f;
xi = {xi1, xi2};
w[xi1, xi2, eps, e2] = gamma1[xi1, xi2, eps, e2] + I gamma2 [xi1, xi2, eps, e2];

w(xi1, xi2, eps, e2) = gamma1(xi1, xi2, eps, e2) + I gamma2(xi1, xi2, eps, e2)

gamma1[xi1,xi2,eps,e2]=(0. - 1.72945 I) e2 (-0.3567 xi1 + 0.229 xi2 + 0.7398 (-0.0423 eps xi1 + 0.4227 eps xi2))^3 - (0. + 0.5 I) (eps (0.5929 xi1 - 0.3806 xi2 - 1.2296 (-0.0423 eps xi1 + 0.4227 eps xi2)) + 38.4563 e2 (-0.3567 xi1 + 0.229 xi2 + 0.7398 (-0.0423 eps xi1 + 0.4227 eps xi2))^3);
gamma2[xi1,xi2,eps,e2]=(-1.72945 + 0. I) e2 (-0.3567 xi1 + 0.229 xi2 + 0.7398 (-0.0423 eps xi1 + 0.4227 eps xi2))^3 + (0.5 + 0. I) (eps (0.5929 xi1 - 0.3806 xi2 - 1.2296 (-0.0423 eps xi1 + 0.4227 eps xi2)) + 38.4563 e2 (-0.3567 xi1 + 0.229 xi2 + 0.7398 (-0.0423 eps xi1 + 0.4227 eps xi2))^3); 

w[xi1, xi2, eps, e2] = gamma1[xi1, xi2, eps, e2] + I gamma2 [xi1, xi2, eps, e2];

---------------------------- actual code -----------------------

a1 = 0;
a2 = 0;
a3 = 0;
a4 = 0;
a5 = -0.0423;
a6 = 0.4227;
b1 = 0;
b2 = 0;
b3 = 0;
b4 = 0;
b5 = -1.5293;
b6 = -0.8906;
h1[a1, a2, a3, a4, a5, a6, xi1, xi2, eps] = a1 xi1^2 + a2 xi2^2 + a3 eps^2 + a4 xi1 xi2 + a5 xi1 eps + a6 xi2 eps; 
h2[b1, b2, b3, b4, b5, b6, xi1, xi2, eps] = b1 xi1^2 + b2 xi2^2 + b3 eps^2 + b4 xi1 xi2 + b5 xi1 eps + b6 xi2 eps;
f11 = -0.0322 eps - 0.0028 eps^2;
f12 = -0.3761 + 0.207 eps + 0.0283 eps^2;
f21 = 0.3761 - 0.0502 eps + 0.0044 eps^2;
f22 = 0.0322 eps + 0.0440 eps^2;
f = {{f11, f12}, {f21, f22}};
eig = Eigenvalues[f];
v[eps_] = Eigenvectors[f];
p = v[0];
ip = Inverse[p];
d21 = 0.5929;
d22 = -0.3806;
d23 = -1.2296;
cub[xi1, xi2, y3_] = (-0.3567 xi1 + 0.229 xi2 + 0.7398 y3 )^3;
g1[xi1, xi2, y3_, eps, e2] = -3.4589 e2 cub[xi1, xi2, y3];
g2[xi1, xi2, y3_, eps, e2] = eps (d21 xi1 + d22 xi2 + d23 y3) + 38.4563 e2 cub[xi1, xi2, y3];
gamma1[xi1, xi2, eps, e2] = ip[[1, 1]] g1[xi1, xi2, h1[a1, a2, a3, a4, a5, a6, xi1, xi2, eps], eps, e2] + ip[[1, 2]] g2[xi1, xi2, h1[a1, a2, a3, a4, a5, a6, xi1, xi2, eps], eps, e2];
gamma2[xi1, xi2, eps, e2] = ip[[2, 1]] g1[xi1, xi2, h1[a1, a2, a3, a4, a5, a6, xi1, xi2, eps], eps, e2] + ip[[2, 2]] g2[xi1, xi2, h1[a1, a2, a3, a4, a5, a6, xi1, xi2, eps], eps, e2];
effe = ip.f;
ef = effe.f;
xi = {xi1, xi2};
w[xi1, xi2, eps, e2] = gamma1[xi1, xi2, eps, e2] + I gamma2 [xi1, xi2, eps, e2];