BeginPackage["Schofield`"]

$Assumptions = {θc > 0, θc > 1, gC[_] ∈ Reals, B > 0, γ > 0, ξ0 > 0}

β[D_:2] := Piecewise[
  {
    {1/8, D == 2},
    {0.326419, D == 3},
    {1/2, D == 4},
    {β, True}
  }
]

δ[D_:2] := Piecewise[
  {
    {15, D == 2},
    {4.78984, D == 3},
    {3, D == 4},
    {δ, True}
  }
]

α[D_:2] := Piecewise[
  {
    {0, D == 2},
    {0.11008, D == 3},
    {0, D == 4},
    {α, True}
  }
]

Δ[D_:2] := β[D] δ[D]
t[θ_] := ((θ)^2 - 1)
(*
hBasis = Orthogonalize[x^# & /@ Range[0, 20], Function[{f, g}, Integrate[f g (1 - x^2)^2, {x, -1, 1}]]]
h[n_][θ_] := (1 - (θ/θc)^2) Sum[gC[i] hBasis[[2 * i + 2]], {i, 0, n}] /. x -> θ / θc
*)
h[n_][θ_] := (1 - (θ/θc)^2) Sum[gC[i]θ^(2 * i + 1), {i, 0, n}]

RFLow[B_, θc_][θ_] := (2 E^(1/(
   B \[Theta]c)) \[Theta]c ExpIntegralEi[-(1/(B \[Theta]c))] + 
  E^(1/(B (-\[Theta] + \[Theta]c))) (\[Theta] - \[Theta]c) \
ExpIntegralEi[1/(B \[Theta] - B \[Theta]c)] - 
  E^(1/(B \[Theta] + 
    B \[Theta]c)) (\[Theta] + \[Theta]c) ExpIntegralEi[-(1/(
     B \[Theta] + B \[Theta]c))])
RFHigh[ξ0_][ξ_] := (1+(ξ/ξ0)^2)^(5/6) - 1

RF[n_][θ_] := AL RFLow[B, θc][θ] + AH RFHigh[θ0][θ] + Sum[A[i] θ^(2(i+1)), {i, 0, n}]

ruleB[g_] := B - π / (2 1.3578383417066) (- g'[θc] / t[θc]^Δ[2])
ruleθ0[g_] := Simplify[g[I θ0]/(-t[I θ0])^Δ[2]/I] - 0.18930

eqLowRHS[F_][m_] := SeriesCoefficient[F[θ], {θ, θc, m}, Assumptions -> Join[{θ < θc, θ > 1}, $Assumptions]]

eqLowLHS[h_][m_] :=D[
  t[θ]^2 (Gl[h[θ] t[θ]^-Δ[2]] + Log[t[θ]^2]/(8 π)),
  {θ, m} ] / m! /. θ -> θc

eqLow[F_, h_][m_] := eqLowRHS[F][m] - eqLowLHS[h][m]

eqHighRHS[F_][m_] := D[F[θ], {θ, m} ] /. θ -> 0

eqHighLHS[h_][m_] := D[(-t[θ])^2 (Gh[h[θ] (-t[θ])^-Δ[2]] + Log[(-t[θ])^2]/(8 π)), {θ, m} ] /. θ -> 0

eqHigh[F_, h_][m_] := eqHighRHS[F][m] - eqHighLHS[h][m]

eqMid[F_, h_][m_] := D[
  F[θ] - t[θ]^2 Log[h[θ]^2]/(8 Δ[2]π) - h[θ]^((2-α[2])/Δ[2]) Φ[η]
    /. η -> t[θ] / h[θ]^(1 / Δ[2]),
  {θ, m} ] /. θ -> 1

Φs = {
  -1.197733383797993,
  -0.318810124891,
  0.110886196683,
  0.01642689465,
  -2.639978 10^-4,
  -5.140526 10^-4,
  2.08856 10^-4,
  -4.4819 10^-5
}

Gls = {
  0,
  -1.3578383417066,
  -0.048953289720,
  0.038863932,
  -0.068362119,
  0.18388370,
  -0.6591714,
  2.937665,
  -15.61,
  96.76,
  -679,
  5340
}

Ghs = {
  0,
  0,
  -1.8452280782328,
  0,
  8.333711750,
  0,
  -95.16896,
  0,
  1457.62,
  0,
  -25891,
  0,
  502000
}

dRule[sym_][f_, i_] := Derivative[i[[1]] - 1][sym][0] -> f (i[[1]] - 1)!

ΦRules = MapIndexed[dRule[Φ], Φs];
GlRules = MapIndexed[dRule[Gl], Gls];
GhRules = MapIndexed[dRule[Gh], Ghs];

rules[g_] := Join[ΦRules, GlRules, GhRules]
(*ξ0 := 0.18930*)
gC[0] := 1
tC[0] := 1

eq[F_, g_][m_] := Flatten[Join[{ruleB[g], ruleθ0[g]},{eqLow[F, g][#](*, eqMid[F, g][#]*)} & /@ Range[0, m], eqHigh[F, g] /@ Range[2, m, 2]]] //. rules[g]

newSol[eqs_, oldSol_, newVars_, δ_:0, opts___] := FindRoot[
  eqs,
  Join[
    oldSol /. Rule -> List,
    Thread[{newVars, δ * RandomVariate[NormalDistribution[], Length[newVars]]}]
  ],
  MaxIterations -> 500,
  opts
]

EndPackage[]