Computations - WZ

Solving a linear system in terms of wz1 and wz2

Solve[{dz1 == 
            2  z1*z2*(dz1 + dz2)/z + z2^2*dz2/z + z*z1' - z1, 
          dz2 == z2^2*(dz1 + dz2)/z + z*z2' - z2, 
          wz1 == 
            dz1 + 2  z1*z2/z*(wz1 + wz2) + z2^2/z*wz2 + 
              2  z1*((dz2 + z*z2')*(z1' + z2') + 
                    z2'*(dz1 + dz2 + z*z1' + z*z2')) + 
              6*z1*z2^2/z*(z1' + z2')*(dz1 + dz2 + z*z1' + z*z2'), 
          wz2 == 
            dz2 + z2^2/z*(wz1 + wz2) + 
              2*z2^3/z*(z1' + z2')*(dz1 + dz2 + z*z1' + 
                    z*z2')}, {dz1, dz2, wz1, wz2}] // Simplify

wz1 + wz2 /. %[[1]] // Simplify

% /. {z1 -> z2^3/z, z1' -> 3  z2^2*z2'/z - z2^3/z^2} // 
      Simplify

% /. {z2' -> (2  z - z2 + z2^2)/(z - 2  z*z2 - 4  z2^3)} // 
      Simplify

Finding the zeroes of the denominator

Solve[{(z^2) - z (z2^2) - 3 (z2^4) == 0, 
          z^2 - z*z2 + z*z2^2 + z2^4 == 0}, {z2, z}] // 
      FullSimplify

Solve[{4 (z2^3) + z (-1 + 2 z2) == 0, 
          z^2 - z*z2 + z*z2^2 + z2^4 == 0}, {z2, z}] // 
      FullSimplify

Expansion of z2

z0 = (1/72) (-35 + 13 (Sqrt[13]));

F[z2_, z_] := z^2 - z*z2 + z*z2^2 + z2^4;

Solve[F[z2, z0] == 0, z2]

s0 = (1/6) (-2 + (Sqrt[13]));

Sqrt[2*z*D[F[z2, z], z]/D[F[z2, z], {z2, 2}]] /. {z2 -> s0, 
          z -> z0} // FullSimplify

Z2 = s0 - %*Sqrt[1 - z/z0] + O[z - z0];

Expansion of wz

Now, we find the expansion of wz around z0:

((z^10) (2 - 4 z2) - 384 (z2^21) - 
                32 z (z2^18) (-24 + 37 z2) + 
                2 (z^4) (z2^11) (72 + 179 z2 - 296 (z2^2)) + 
                (z^9) z2 (-2 + 17 z2 + 14 (z2^2)) + 
                (z^8) (z2^3) (-9 - 17 z2 + 234 (z2^2)) - 
                8 (z^2) (z2^15) (48 - 200 z2 + 253 (z2^2)) + 
                2 (z^7) (z2^5) (9 - 162 z2 + 392 (z2^2)) + 
                2 (z^5) (z2^9) (108 - 511 z2 + 397 (z2^2)) - 
                2 (z^3) (z2^13) (208 - 748 z2 + 971 (z2^2)) + 
                (z^6) (z2^7) (109 - 889 z2 + 
                      1316 (z2^2)))/((z^3) ((((z^2) - z (z2^2) - 3 (z2^4)))^2) (((4 (z2^3) + z (-1 + 2 z2)))^2)) /. {z2 -> 
              Z2, z -> z0 + O[z - z0]} // Simplify;

Series[%, {z, z0, -2}] // RootReduce

((46319 - 12490 (Sqrt[13]))/(808704 ((((1/72) (-35 + 13 (Sqrt[13]))))^2))) // RootReduce

The dominating part is thus given by  ((91 - 5 (Sqrt[13]))/(1248 (1 - z/z0)^2)), which gives the asymptotics we are searching for.

((91 - 5 (Sqrt[13]))/(1248*Gamma[2]))*4*Sqrt[Pi]

1/(4*z0) // RootReduce