Computations - RR

Reducing the task by solving a linear system

Solve[{rr12 == r1^2*(rr11 + rr12 + r1 + r2)/z, 
        rr22 == 
          t^2*(rr11 + 2  rr12 + rr22)/z + 2  r2^2*(1 - t)/z}, {rr12, 
        rr22}]

rr11 + 2  rr12 + rr22 /. %[[1]] // Factor

Denominator[%]

Solve[{% == 0, r1 == z/(1 - r1 - t), 
        t == z/(1 - t)}, {r1, t, z}]

% // N

Thus, the smallest singularity of rr is also the smallest singularity of rr11.

Solving the system for rr11

GroebnerBasis[{rr11*(1 - rr11 - 2  r1 - t) - z, 
        r1*(1 - r1 - t) - z, t*(1 - t) - z}, {r1, t, r11, z}, {r1, 
        t}]

F[rr11_, z_] = %[[1]];

Finding a common zero of the polynomial and its derivative

GroebnerBasis[{F[rr11, z], 
        D[F[rr11, z], rr11]}, 
{rr11, 
        z}, {rr11}]

Factor[%[[1]]]

So the smallest singularity must be z0 = (8/81).

z0 = 8/81

Expanding around the singularity

We determine the values of t, r1 and r2 at the singularity:

Solve[t == z0/(1 - t), t]

Since t is monotonically increasing with a value of (1/2) at (1/4),we know that t = (1/9).

t = 1/9;

Solve[r1 == z0/(1 - r1 - t), r1]

r1 is monotonically increasing with a value of (2/5) at (4/25), so we know that r1 = (2/9)(2 - (Sqrt[2])).

r1 = (2/9) (2 - (Sqrt[2]));

Solve[r2 == z0/(1 - t)^2*(r1 + r2), r2]

r2 = (2/63) (2 - (Sqrt[2]));

Next we find the expansion of rr11:

Solve[F[rr11, z0] == 0, rr11]

So the value of r11 at z0 is ((2 (Sqrt[2]))/9).

r0  = ((2 (Sqrt[2]))/9);

D[F[rr11, z], {rr11, 2}] /. {rr11 -> r0, z -> z0} // 
      Simplify

D[F[rr11, z], z] /. {rr11 -> r0, z -> z0} // 
      Simplify

a = Sqrt[2  z0*%/%%] // RootReduce

Then we have rr21~r0 - 
      a (Sqrt[(1 - (z/z0))]).

\(TraditionalForm\`The\ expansion\ of\ rr\ follows\ at\ \(\(once\)\(:\)\)\)

-(-2 (r1^2) (r2^2) + 2 (r1^2) (r2^2) t + 2 (r1^3) z + 
                2 (r1^2) r2 z + 2 (r2^2) z + (r1^2) rr11 z - 
                2 (r2^2) t z + 
                rr11 (z^2))/(((r1^2) - z) (-(t^2) + 
                    z)) /. {rr11 -> r0, z -> z0} // 
      RootReduce

-(-(r1^2) a z - 
                a (z^2))/(((r1^2) - z) (-(t^2) + 
                  z)) /. {rr11 -> r0, z -> z0} // 
      RootReduce

% // N

Now, we readily obtain the constants in our asymptotics:

%%/Gamma[-1/2]*4*Sqrt[Pi] // RootReduce

% // N

1/(4*z0)

% // N