Terms [non]cancellation in series expansions

[Audrius-St]

Hello,

MNWE for a set of simple test functions. Taylor-Maclaurin series expansion in powers of t with multivariate coefficients. My expectation was that substituting t => -t in these series and adding the resulting series to the original series would result in cancellation of the odd power terms in t: t, t^3. However, that's not the case. Insight in how to do this would be appreciated.

# test_term_cancellation.jl

using Symbolics

begin
    @variables θ, p, t
    @variables m, l, g

    # Mulivariate polynomials in powers of t: 1, t, t^2, t^3
    T_θ₊ = θ + (p*t) / (m*(l^2)) + ((-1//2)*g*(t^2)*sin(θ)) / l + (-g*p*(t^3)*cos(θ)) / (6m*(l^3))
    T_p₊ = p + (g*(p^2)*(t^3)*sin(θ)) / (6m*(l^3)) + ((-1//2)*g*p*(t^2)*cos(θ)) / l + (1//6)*m*(g^2)*(t^3)*cos(θ)*sin(θ) - g*l*m*t*sin(θ)

    @show T_θ₊
    @show T_p₊
    println()

    # Reverse the sign of t: t => -t 
    # and thus the signs of t, t^3 terms
    T_θ₋ = expand(substitute(T_θ₊, Dict(t => -t)))
    T_p₋ = expand(substitute(T_p₊, Dict(t => -t)))

    @show T_θ₋
    @show T_p₋
    println()

    # Expectation: upon addition all the t and t^3 terms would cancel. They don't
    T_θₛ = T_θ₊ + T_θ₋
    T_pₛ = T_p₊ + T_p₋

    @show T_θₛ
    @show T_pₛ
    println()

    # Expectation: simplify would cancel all the t and t^3 terms. It doesn't
    T_θₛ_smp = simplify(T_θ₊ + T_θ₋; expand=true, simplify_fractions=false)
    T_pₛ_smp = simplify(T_p₊ + T_p₋; expand=true, simplify_fractions=false)

    @show T_θₛ_smp
    @show T_pₛ_smp
    println()

    # ALternate method? Extract the coefficients and retain only the even ones
    # Test: throws an error
    # Symbolics.coeff(T_θₛ_smp, t)

    # Also throws an error
    # Symbolics.polynomial_coeffs(T_θₛ_smp, t)
end

[hersle]

This cancels the odd coefficients:

julia> expand(simplify(T_θ₊ + T_θ₋))
((2//1)*l*θ - g*(t^2)*sin(θ)) / l