Computations - WS

Solving a linear system in terms of ws1 and ws2

Solve[{ds1 == ds2*s2 + z*s1' - s1, 
          ds2 == (ds1 + ds2)*s2^2/z + z*s2' - s2, 
          ws1 == ds1 + ws2*s2 + 2*s1*s2*s2'*(ds2 + z*s2'), 
          ws2 == 
            ds2 + (ws1 + ws2)*s2^2/z + 
              2  s2^3/z*(s1' + s2')*(ds1 + ds2 + z*s1' + 
                    z*s2')}, {ds1, ds2, ws1, ws2}] // Simplify

ws1 + ws2 /. %[[1]] // Simplify

% /. {s1 -> z/(1 - s2), s1' -> 1/(1 - s2) + s2*s2'} // 
      Simplify

% /. {s2' -> 1/(3  s2^2 - 4  s2 + 1)} // Simplify

Finding the zeroes of the denominator

Solve[{1 - 3  s2 == 0, s2 == z/(1 - s2)^2}, {s2, 
        z}]

Solve[{1 - s2 == 0, s2 == z/(1 - s2)^2}, {s2, 
        z}]

Solve[{(s2^2) + (s2^3) - z == 0, 
        s2 == z/(1 - s2)^2}, {s2, z}]

Expansion of s2

z0 = 4/27;

F[s2_, z_] := s2*(1 - s2)^2 - z;

Solve[F[s2, z0] == 0, s2]

s0 = 1/3;

Sqrt[2*z*D[F[s2, z], z]/D[F[s2, z], {s2, 2}]] /. {s2 -> s0, 
        z -> z0}

S2 = s0 - %*Sqrt[1 - z/z0] + O[z - z0];

Expansion of ws

Now, we find the expansion of ws around 4/27:

-(z (4 (s2^8) + 8 (s2^9) - 4 (s2^10) + 
                    (s2^6) (4 - 38 z) + (s2^2) (7 - 10 z) z + 
                    (z^2) + 24 (s2^3) (z^2) + (s2^7) (-16 + 15 z) + 
                    (s2^5) (8 + 42 z) + 
                    s2 z (-1 - 2 z + 4 (z^2)) - 
                    (s2^4) (4 + 25 z + 
                          15 (z^2))))/((((1 - 3 s2))^2) (((-1 + s2))^3) ((((s2^2) + (s2^3) - z))^2)) /. {s2 -> S2, 
          z -> z0 + O[z - z0]} // Simplify

Series[%, {z, z0, -1}]

The dominating part is given by (80/(59049 (((z - (4/27)))^2))) =(5/(81 (((1 - ((27 z)/4)))^2))), which gives the asymptotics we are searching for.

5/(81*Gamma[2])*4*Sqrt[Pi]

1/(4*z0)