# eccentric anomaly expansion equation

I try to use this simple algorithm (paper) to calculate the Eccentric Anomaly expansion:

a[0]=0;
Do [
t1 = 1-a[k-1]^2/2+a[k-1]^4/24;
t2 = a[k-1]-a[k-1]^3/6;
a[k]=Expand[e*Sin[u]*t1+e*Cos[u]*t2,Trig->True];
Simplify[a[k]];
TeXForm[a[k]] >>"tex.01";
Print [a[k],k], {k,2}]


The result should be this one

but maybe because the algorithm is write with an old version of Mathematica I have as an output this different (in terms of trigonometric function)

How can I re-arrange it to have the one in the paper? And when I have the result of the paper is it possible, then, to regroup the equation in terms of power series of sin(u), something like that:

Thanks a lot !

Clear[a]

a[0] = 0;
a[k_] := a[k] =
e*Sin[u]*(1 - a[k - 1]^2/2 + a[k - 1]^4/24) +
e*Cos[u]*(a[k - 1] - a[k - 1]^3/6) //
TrigReduce // Expand;



TraditionalForm /@ (a[2] // Collect[#, Table[Sin[n*u], {n, 5}]] &)


There seem to be several issues here.

1. The Mathematica code seems strange, arguably miswritten, though it appears you transcribed it correctly.

• Simplify[a[k]]; doesn't appear to do anything; it's output is never assigned
• Placing TeXForm[a[k]] >>"tex.01"; within the loop writes that file, then overwrites it in the next iteration. Perhaps >>> (PutAppend) was intended?
2. TeXForm reorders terms as a result of using TraditionalForm, and it seems you do not want that reordering

3. Even when Simplify is actually applied the results (using v10) differ from what you show. Expand seems to help.

Perhaps this gets you close enough:

a[0] = 0;

Do[t1 = 1 - a[k - 1]^2/2 + a[k - 1]^4/24;
t2 = a[k - 1] - a[k - 1]^3/6;
a[k] = Expand[e*Sin[u]*t1 + e*Cos[u]*t2, Trig -> True];
a[k] = Simplify @ a[k],
{k, 2}
]

TeXForm[HoldForm @@ {Expand @ a[2]}]


$e \sin (u)-\frac{1}{4} e^3 \sin (u)+\frac{1}{64} e^5 \sin (u)+e^2 \cos (u) \sin (u)-\frac{1}{24} e^4 \cos (u) \sin (u)+\frac{1}{4} e^3 \cos (2 u) \sin (u)-\frac{1}{48} e^5 \cos (2 u) \sin (u)+\frac{1}{24} e^4 \cos (3 u) \sin (u)+\frac{1}{192} e^5 \cos (4 u) \sin (u)$