How to Implement a Two-Parameter Asymptotic Expansion in Mathematica?

[squenshl]

Homework Statement

How do I implement the series k(tilde) = k₀ + $\sum_{i,j}$ k_i,j $\epsilon$₁ⁱ $\epsilon$₂^j into Mathematica.
This is a 2 parameter asymptotic expansion and I am plugging this into my equation I am trying to solve.
Also I don't know how to implement the O[$\epsilon$1} and O[$\epsilon$2] terms in the file as well.
Note I am using the LogicalExpand[...] then Solve[..] to get my constants k1, k2, ...

Homework Equations

k₀ = -π/sqrt(3)

The Attempt at a Solution

I tried k-tilde = k0+Sum[k[i,j]*$\epsilon$1^i*$\epsilon$2^j,{i,1,5},{j,1,5}] in Mathematica but this isn't correct.
Someone please help.

 

[mighty2000]

I would suggest breaking down the problem into smaller steps and using Mathematica's built-in functions to solve it. Here's one possible approach:

1. Define the parameters and constants:
k0 = -π/Sqrt[3]; (*define k0*)
k[i_,j_] := Subscript[k, i,j]; (*define k[i,j] as a function*)
ε1 = Subscript[ε, 1]; (*define ε1*)
ε2 = Subscript[ε, 2]; (*define ε2*)

2. Use Table to generate the terms in the series:
ktilde = k0 + Sum[k[i,j]*ε1^i*ε2^j,{i,1,5},{j,1,5}] (*generate the series*)

3. Use LogicalExpand to simplify the series:
ktilde = LogicalExpand[ktilde] (*simplify the series*)

4. Solve for the constants:
Solve[ktilde == your_equation, {k[1,1], k[1,2], k[2,1], k[2,2], k[3,1], k[3,2], k[4,1], k[4,2], k[5,1], k[5,2]}] (*solve for the constants*)

5. Substitute the solutions into the original series:
ktilde = ktilde /. sol (*substitute the solutions into the series*)

6. Use Simplify to further simplify the series:
ktilde = Simplify[ktilde] (*simplify the series*)

7. Finally, add the O[ε1] and O[ε2] terms to the series:
ktilde = ktilde + O[ε1] + O[ε2] (*add the O[ε1] and O[ε2] terms*)

This should give you the desired result. Keep in mind that there may be other approaches to solving this problem, so feel free to explore and find what works best for you.