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Jason B.
Jason B.
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That's the basic strategy, but you want a Taylor series in z and zc for w, so let's try this:

Normal[
 Series[gamma1[(z + zc)/2, -(1/2) I (z - zc)] + 
   I gamma2[(z + zc)/2, -(1/2) I (z - zc)], {z, 0, 3}, {zc, 0, 3}]]
(* ((-0.0326007 - 0.0046403 I) e2 - (0.0369689 + 
       0.0632291 I) e2 eps + (0.0197006 - 
       0.0506049 I) e2 eps^2 + (0.0128245 - 
       0.00395608 I) e2 eps^3) z^3 + ((-0.0533697 + 
       0.0831308 I) e2 - (0.122053 - 
       0.0838872 I) e2 eps - (0.0964865 - 
       0.0524172 I) e2 eps^2 - (0.0400624 - 
       0.00400908 I) e2 eps^3) z^2 zc + ((0.0533697 + 
       0.0831308 I) e2 + (0.122053 + 
       0.0838872 I) e2 eps + (0.0964865 + 
       0.0524172 I) e2 eps^2 + (0.0400624 + 
       0.00400908 I) e2 eps^3) z zc^2 + ((0.0326007 - 
       0.0046403 I) e2 + (0.0369689 - 
       0.0632291 I) e2 eps - (0.0197006 + 
       0.0506049 I) e2 eps^2 - (0.0128245 + 
       0.00395608 I) e2 eps^3) zc^3 *)

But this isn't exactly what you want, it has terms like z^2 zc^2, so if you want it to match your expression exactly, you should use the SeriesCoefficient function.

wz = gamma1[(z + zc)/2, -(1/2) I (z - zc)] + 
   I gamma2[(z + zc)/2, -(1/2) I (z - zc)];
Total[SeriesCoefficient[
     wz, {z, 0, #1}, {zc, 0, #2}] z^#1 zc^#2 & @@@ {{3, 0}, {2, 
    1}, {1, 2}, {0, 3}}]
(* (0.0128245 - 
    0.00395608 I) ((-2.21927 - 1.04643 I) e2 - (1.24345 + 
       5.31391 I) e2 eps + (2.51416 - 3.17039 I) e2 eps^2 + (1. + 
       0. I) e2 eps^3) z^3 + ((-0.0533697 + 
       0.0831308 I) e2 - (0.122053 - 
       0.0838872 I) e2 eps - (0.0964865 - 
       0.0524172 I) e2 eps^2 - (0.0400624 - 
       0.00400908 I) e2 eps^3) z^2 zc + ((0.0533697 + 
       0.0831308 I) e2 + (0.122053 + 
       0.0838872 I) e2 eps + (0.0964865 + 
       0.0524172 I) e2 eps^2 + (0.0400624 + 
       0.00400908 I) e2 eps^3) z zc^2 + ((0.0326007 - 
       0.0046403 I) e2 + (0.0369689 - 
       0.0632291 I) e2 eps - (0.0197006 + 
       0.0506049 I) e2 eps^2 - (0.0128245 + 
       0.00395608 I) e2 eps^3) zc^3 *)

That's the basic strategy, but you want a Taylor series in z and zc for w, so let's try this:

Normal[
 Series[gamma1[(z + zc)/2, -(1/2) I (z - zc)] + 
   I gamma2[(z + zc)/2, -(1/2) I (z - zc)], {z, 0, 3}, {zc, 0, 3}]]
(* ((-0.0326007 - 0.0046403 I) e2 - (0.0369689 + 
       0.0632291 I) e2 eps + (0.0197006 - 
       0.0506049 I) e2 eps^2 + (0.0128245 - 
       0.00395608 I) e2 eps^3) z^3 + ((-0.0533697 + 
       0.0831308 I) e2 - (0.122053 - 
       0.0838872 I) e2 eps - (0.0964865 - 
       0.0524172 I) e2 eps^2 - (0.0400624 - 
       0.00400908 I) e2 eps^3) z^2 zc + ((0.0533697 + 
       0.0831308 I) e2 + (0.122053 + 
       0.0838872 I) e2 eps + (0.0964865 + 
       0.0524172 I) e2 eps^2 + (0.0400624 + 
       0.00400908 I) e2 eps^3) z zc^2 + ((0.0326007 - 
       0.0046403 I) e2 + (0.0369689 - 
       0.0632291 I) e2 eps - (0.0197006 + 
       0.0506049 I) e2 eps^2 - (0.0128245 + 
       0.00395608 I) e2 eps^3) zc^3 *)

But this isn't exactly what you want, it has terms like z^2 zc^2, so if you want it to match your expression exactly, you should use the SeriesCoefficient function.

wz = gamma1[(z + zc)/2, -(1/2) I (z - zc)] + 
   I gamma2[(z + zc)/2, -(1/2) I (z - zc)];
Total[SeriesCoefficient[
     wz, {z, 0, #1}, {zc, 0, #2}] z^#1 zc^#2 & @@@ {{3, 0}, {2, 
    1}, {1, 2}, {0, 3}}]
(* (0.0128245 - 
    0.00395608 I) ((-2.21927 - 1.04643 I) e2 - (1.24345 + 
       5.31391 I) e2 eps + (2.51416 - 3.17039 I) e2 eps^2 + (1. + 
       0. I) e2 eps^3) z^3 + ((-0.0533697 + 
       0.0831308 I) e2 - (0.122053 - 
       0.0838872 I) e2 eps - (0.0964865 - 
       0.0524172 I) e2 eps^2 - (0.0400624 - 
       0.00400908 I) e2 eps^3) z^2 zc + ((0.0533697 + 
       0.0831308 I) e2 + (0.122053 + 
       0.0838872 I) e2 eps + (0.0964865 + 
       0.0524172 I) e2 eps^2 + (0.0400624 + 
       0.00400908 I) e2 eps^3) z zc^2 + ((0.0326007 - 
       0.0046403 I) e2 + (0.0369689 - 
       0.0632291 I) e2 eps - (0.0197006 + 
       0.0506049 I) e2 eps^2 - (0.0128245 + 
       0.00395608 I) e2 eps^3) zc^3 *)
Source Link
Jason B.
Jason B.
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  • 298

You have,

gamma1[xi1_, xi2_] = (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_] = (-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);

Just rewrite your variables in terms of z and zc,

Solve[{z == xi1 + I xi2, zc == xi1 - I xi2}, {xi1, xi2}]
(* {{xi1 -> (z + zc)/2, xi2 -> -(1/2) I (z - zc)}} *)

Then substitute them into the original expression:

gamma1[(z + zc)/2, -(1/2) I (z - zc)] // FullSimplify
gamma2[(z + zc)/2, -(1/2) I (z - zc)] // FullSimplify
(* (0.00641225 - 0.00197804 I) e2 ((0.838054 - 1.0568 I) z + 
    1. eps z - (0.612024 + 1.20191 I) zc - (0.98017 + 
       0.198158 I) eps zc)^3 - (0. + 
    0.5 I) (eps (((0.29645 + 
            0.1903 I) + (0.026006 + 0.259876 I) eps) z + ((0.29645 - 
            0.1903 I) + (0.026006 - 
             0.259876 I) eps) zc) + (0.043984 + 
       0.142584 I) e2 ((0.838054 - 1.0568 I) z + 
       1. eps z - (0.612024 + 1.20191 I) zc - (0.98017 + 
          0.198158 I) eps zc)^3) *)
(* (-0.00197804 - 0.00641225 I) e2 ((0.838054 - 1.0568 I) z + 
    1. eps z - (0.612024 + 1.20191 I) zc - (0.98017 + 
       0.198158 I) eps zc)^3 + (0.5 + 
    0. I) (eps (((0.29645 + 
            0.1903 I) + (0.026006 + 0.259876 I) eps) z + ((0.29645 - 
            0.1903 I) + (0.026006 - 
             0.259876 I) eps) zc) + (0.043984 + 
       0.142584 I) e2 ((0.838054 - 1.0568 I) z + 
       1. eps z - (0.612024 + 1.20191 I) zc - (0.98017 + 
          0.198158 I) eps zc)^3) *)