Series expansion of large expression

Question

I have the following equation:

eq=E0[x,y,z]+E1[x,y,z]*Cos[phi]+E2[x,y,z]*Cos[phi]^2+E3[x,y,z]*Sin[phi]

Now E0,E1,E2 and E3 are already series expansions of expressions in x,y,z. Therefore I would like to series expand the result of

Solve[D[eq,phi]==0,phi]

However, this is some giant arctan expression and Mathematica seems to take forever.

Question: Is there a more clever approach to this (not calling Series at the end only), or is there some way I can tune the Series command?

Here are the real expressions I am dealing with, if someone wants to try something out:

Ecp = (s1*s3*Abs[J]/DE^2/2 - 1/DE*s2*s3*KroneckerDelta[1, s1])*(t1^2 -
      t2^2) /. KroneckerDelta[1, s1] -> (1 + s1)/2 /. 
 t1 -> tt1*DE /. t2 -> tt2*DE /. xi -> xii*DE /. J -> Jj*DE
Ec2p = s2/DE^2*KroneckerDelta[1, s1] (t1^2 - t2^2)^2/Abs[J] /. 
  KroneckerDelta[1, s1] -> (1 + s1)/2 /. t1 -> tt1*DE /. 
t2 -> tt2*DE /. xi -> xii*DE /. J -> Jj*DE
Esp = s2*s3/DE^2*t1*t2*xi*
    Abs[J]/(Abs[xi] KroneckerDelta[-1, s1] + 
       Abs[J] KroneckerDelta[1, s1]) /. 
   KroneckerDelta[1, s1] -> (1 + s1)/2 /. 
  KroneckerDelta[-1, s1] -> (1 - s1)/2 /. t1 -> tt1*DE /. 
t2 -> tt2*DE /. xi -> xii*DE /. J -> Jj*DE
Solve[D[E00 + Cos[phi]*Ecp + Cos[phi]^2*Ec2p + Sin[phi]*Esp, phi] == 0, phi]

The small parameters are tt1,tt2,Jj,xii. Usually I then introduce a "smallness-paramter" as suggested here on SE to perform the multi series.

Answer 1

To solve this problem under the assumption that the stated quantities are small, you can insert a perturbation expansion for the unknown phi and solve for its expansion coefficients in the smallness parameter:

First I'm copying the definitions from the question to be self-contained:

Ecp = (s1*s3*Abs[J]/DE^2/2 - 
         1/DE*s2*s3*KroneckerDelta[1, s1])*(t1^2 - t2^2) /. 
      KroneckerDelta[1, s1] -> (1 + s1)/2 /. t1 -> tt1*DE /. 
    t2 -> tt2*DE /. xi -> xii*DE /. J -> Jj*DE
Ec2p = s2/DE^2*KroneckerDelta[1, s1] (t1^2 - t2^2)^2/Abs[J] /. 
      KroneckerDelta[1, s1] -> (1 + s1)/2 /. t1 -> tt1*DE /. 
    t2 -> tt2*DE /. xi -> xii*DE /. J -> Jj*DE
Esp = s2*s3/DE^2*t1*t2*xi*
        Abs[J]/(Abs[xi] KroneckerDelta[-1, s1] + 
           Abs[J] KroneckerDelta[1, s1]) /. 
       KroneckerDelta[1, s1] -> (1 + s1)/2 /. 
      KroneckerDelta[-1, s1] -> (1 - s1)/2 /. t1 -> tt1*DE /. 
    t2 -> tt2*DE /. xi -> xii*DE /. J -> Jj*DE

Now I define the derivative that you're trying to set to zero:

deriv = 
 D[E00 + Cos[phi]*Ecp + Cos[phi]^2*Ec2p + Sin[phi]*Esp, phi]

(*
==> (DE s2 s3 tt1 tt2 xii Abs[DE Jj] Cos[phi])/(
 1/2 (1 + s1) Abs[DE Jj] + 
  1/2 (1 - s1) Abs[DE xii]) - (DE^2 tt1^2 - 
    DE^2 tt2^2) (-(((1 + s1) s2 s3)/(2 DE)) + (s1 s3 Abs[DE Jj])/(
    2 DE^2)) Sin[phi] - ((1 + s1) s2 (DE^2 tt1^2 - DE^2 tt2^2)^2 Cos[
   phi] Sin[phi])/(DE^2 Abs[DE Jj])
*)

Now I make the assumption that tt1, tt2, Jj, xii are all proportional to the small parameter ϵ and that I can then also write phi = ϕ0 + ϵ ϕ1 to first order in that small parameter. The expansion of deriv in powers of ϵ has two nonvanishing terms if we go to third order. Setting these two non-vanishing terms to zero allows us to solve for the two unknowns ϕ0 and ϕ1. This solves the problem of finding phi to first order in the small parameter.

Solve[
 Thread[CoefficientList[
    Normal@Series[
      Simplify[
       deriv /. 
         Thread[{tt1, tt2, Jj, 
            xii} -> ϵ {tt1, tt2, Jj, xii}] /. 
        phi -> (ϕ0 + ϵ ϕ1), 
       DE Jj ϵ > 0 && DE xii ϵ > 0], {ϵ, 0,
        3}], ϵ] == 0], {ϕ0, ϕ1}]

(*
==> {{ϕ0 -> 
   0, ϕ1 -> -((
    4 Jj tt1 tt2 xii)/((1 + s1) (tt1^2 - tt2^2) (Jj + Jj s1 + xii - 
       s1 xii)))}, {ϕ0 -> -Pi, ϕ1 -> -((
    4 Jj tt1 tt2 xii)/((1 + s1) (tt1^2 - tt2^2) (Jj + Jj s1 + xii - 
       s1 xii)))}, {ϕ0 -> Pi, ϕ1 -> -((
    4 Jj tt1 tt2 xii)/((1 + s1) (tt1^2 - tt2^2) (Jj + Jj s1 + xii - 
       s1 xii)))}}
*)

In order to get useful results, I had to add some assumptions about certain products of your parameters being positive. You may have to check whether those assumptions are appropriate for your data.