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I am solving a fairly annoying matrix equation order by order in perturbation around a small parameter. I am solving for a list of free parameter that are $\beta 1[1],\beta 0[1],\alpha 0[1], \beta m1[1], \beta m 2[1], Wt[1], \alpha m1[1], Tt[1]$ where all those element depend explicitely on two complex variables $z$ and $\bar{z}$. The equation is

Eq1 = {{(-(1/3))*T[z, barz] + 2*(-1 + \[Mu]2)^2*T[z, barz]*\[Beta]1[1][z, barz] - 
    2*(\[Alpha]m1[1][z, barz] + \[Beta]m1[1][z, barz]) + Derivative[0, 1][\[Alpha]0[1]][z, barz] + 
    (2/3)*Derivative[0, 1][\[Beta]0[1]][z, barz] - Derivative[1, 0][\[Alpha]0[1]][z, barz] - 
    (2/3)*Derivative[1, 0][\[Beta]0[1]][z, barz], -2*Tt[1][z, barz] - (2/3)*(-1 + \[Mu]2)^2*T[z, barz]*
     (3*\[Alpha]0[1][z, barz] + 2*\[Beta]0[1][z, barz]) - 8*(-1 + \[Mu]2)^3*W[z, barz]*\[Beta]1[1][z, barz] + 
    8*\[Beta]m2[1][z, barz] - 2*(Derivative[0, 1][\[Alpha]m1[1]][z, barz] + Derivative[0, 1][\[Beta]m1[1]][z, barz]) - 
    (1/3)*(-1 + \[Mu]2)*(-12*W[z, barz] + 8*(-1 + \[Mu]2)*T[z, barz]*\[Beta]0[1][z, barz] - 
      Derivative[1, 0][T][z, barz]) + 2*(Derivative[1, 0][\[Alpha]m1[1]][z, barz] + 
      Derivative[1, 0][\[Beta]m1[1]][z, barz]), -2*(-1 + \[Mu]2)^2*T[z, barz]^2 + 8*Wt[1][z, barz] + 
    16*W[z, barz]*\[Alpha]0[1][z, barz] - 48*\[Mu]2*W[z, barz]*\[Alpha]0[1][z, barz] + 
    48*\[Mu]2^2*W[z, barz]*\[Alpha]0[1][z, barz] - 16*\[Mu]2^3*W[z, barz]*\[Alpha]0[1][z, barz] + 
    8*(-1 + \[Mu]2)^2*T[z, barz]*\[Beta]m1[1][z, barz] + 8*Derivative[0, 1][\[Beta]m2[1]][z, barz] - 
    8*Derivative[1, 0][\[Beta]m2[1]][z, barz] - (1/3)*Derivative[2, 0][T][z, barz] + 
    (2/3)*\[Mu]2*Derivative[2, 0][T][z, barz] - (1/3)*\[Mu]2^2*Derivative[2, 0][T][z, barz]}, 
  {-\[Alpha]0[1][z, barz] - 2*\[Beta]0[1][z, barz] + Derivative[0, 1][\[Beta]1[1]][z, barz] - 
    Derivative[1, 0][\[Beta]1[1]][z, barz], (1/3)*(2*T[z, barz]*(1 - 6*(-1 + \[Mu]2)^2*\[Beta]1[1][z, barz]) + 
     4*(3*\[Beta]m1[1][z, barz] - Derivative[0, 1][\[Beta]0[1]][z, barz] + Derivative[1, 0][\[Beta]0[1]][z, barz])), 
   -2*Tt[1][z, barz] + (8/3)*(-1 + \[Mu]2)^2*T[z, barz]*\[Beta]0[1][z, barz] - 
    8*(-1 + \[Mu]2)^3*W[z, barz]*\[Beta]1[1][z, barz] - 8*\[Beta]m2[1][z, barz] - 
    2*Derivative[0, 1][\[Alpha]m1[1]][z, barz] + 2*Derivative[0, 1][\[Beta]m1[1]][z, barz] - 
    (1/3)*(-1 + \[Mu]2)*(-12*W[z, barz] + 2*(-1 + \[Mu]2)*T[z, barz]*(3*\[Alpha]0[1][z, barz] - 2*\[Beta]0[1][z, barz]) + 
      Derivative[1, 0][T][z, barz]) + 2*Derivative[1, 0][\[Alpha]m1[1]][z, barz] - 
    2*Derivative[1, 0][\[Beta]m1[1]][z, barz]}, {-(1/(2*(-1 + \[Mu]2)^2)) - 2*\[Beta]1[1][z, barz], 
   -\[Alpha]0[1][z, barz] + 2*\[Beta]0[1][z, barz] - Derivative[0, 1][\[Beta]1[1]][z, barz] + 
    Derivative[1, 0][\[Beta]1[1]][z, barz], (1/3)*(6*\[Alpha]m1[1][z, barz] + 
     T[z, barz]*(-1 + 6*(-1 + \[Mu]2)^2*\[Beta]1[1][z, barz]) - 6*\[Beta]m1[1][z, barz] - 
     3*Derivative[0, 1][\[Alpha]0[1]][z, barz] + 2*Derivative[0, 1][\[Beta]0[1]][z, barz] + 
     3*Derivative[1, 0][\[Alpha]0[1]][z, barz] - 2*Derivative[1, 0][\[Beta]0[1]][z, barz])}}

The outcome looks like

Solve[Eq1 == 0, {\[Beta]1[1][z, barz], \[Beta]0[1][z, barz], \[Alpha]0[1][z, barz], \[Beta]m1[1][z, barz], \[Beta]m2[1][z, barz], 
   Wt[1][z, barz], \[Alpha]m1[1][z, barz], Tt[1][z, barz]}]

{{\[Beta]1[1][z, barz] -> -(1/(4*(-1 + \[Mu]2)^2)), \[Beta]0[1][z, barz] -> 
    (1/2)*(Derivative[0, 1][\[Beta]1[1]][z, barz] - Derivative[1, 0][\[Beta]1[1]][z, barz]), \[Alpha]0[1][z, barz] -> 0, 
   \[Beta]m1[1][z, barz] -> (1/12)*(-5*T[z, barz] + 4*Derivative[0, 1][\[Beta]0[1]][z, barz] - 
      4*Derivative[1, 0][\[Beta]0[1]][z, barz]), \[Beta]m2[1][z, barz] -> 
    (1/2)*((-(1/2))*T[z, barz] + \[Mu]2*T[z, barz] - (1/2)*\[Mu]2^2*T[z, barz])*
      (-Derivative[0, 1][\[Beta]1[1]][z, barz] + Derivative[1, 0][\[Beta]1[1]][z, barz]) + 
     (1/24)*(6*Derivative[0, 1][\[Beta]m1[1]][z, barz] + Derivative[1, 0][T][z, barz] - 
       \[Mu]2*Derivative[1, 0][T][z, barz] - 6*Derivative[1, 0][\[Beta]m1[1]][z, barz]), 
   Wt[1][z, barz] -> (T[z, barz] - 2*\[Mu]2*T[z, barz] + \[Mu]2^2*T[z, barz])^2/(4*(-1 + \[Mu]2)^2) + 
     (1/6)*(T[z, barz] - 2*\[Mu]2*T[z, barz] + \[Mu]2^2*T[z, barz])*
      (T[z, barz] - 2*Derivative[0, 1][\[Beta]0[1]][z, barz] + 2*Derivative[1, 0][\[Beta]0[1]][z, barz]) + 
     (1/24)*(6*T[z, barz]^2 - 12*\[Mu]2*T[z, barz]^2 + 6*\[Mu]2^2*T[z, barz]^2 - 
       24*Derivative[0, 1][\[Beta]m2[1]][z, barz] + 24*Derivative[1, 0][\[Beta]m2[1]][z, barz] + 
       Derivative[2, 0][T][z, barz] - 2*\[Mu]2*Derivative[2, 0][T][z, barz] + 
       \[Mu]2^2*Derivative[2, 0][T][z, barz]), \[Alpha]m1[1][z, barz] -> 
    (1/2)*(Derivative[0, 1][\[Alpha]0[1]][z, barz] - Derivative[1, 0][\[Alpha]0[1]][z, barz]), 
   Tt[1][z, barz] -> -3*W[z, barz] + 3*\[Mu]2*W[z, barz] - Derivative[0, 1][\[Alpha]m1[1]][z, barz] + 
     Derivative[1, 0][\[Alpha]m1[1]][z, barz]}}

¨ from this, you see that $\beta 1[1] = -\frac{1}{4(1-\mu^2)^2}$ which does not depend on $z$ or $\bar{z}$ and so the next one that is $\beta 0[1] = \partial_z \beta 1[1] + \partial_\bar{z} \beta 1[1] $ should evaluate to zero. Is there a way to tell Mathematica to do this and not to have to watch each component by hand ?

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  • $\begingroup$ The new in M12 function AsymptoticSolve is extremely useful for these sorts of things. If you provide your equations, I might be able to help you figure out how to use it. $\endgroup$ – Carl Woll Jun 27 at 18:29
  • $\begingroup$ @CarlWoll Woll I added the equation. Thank you $\endgroup$ – Ezareth Jun 27 at 18:47

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