# Multivariables series expansion up to some power of all the variables [duplicate]

I have a function f[x, y, z] that I would like to expand up to a given power of xyz.

For now, I am using Series[f, {x,0,6}, {y,0,6}, {z,0,6}], but that makes an expansion up to power 6 in x, in y and in z, which contains extra terms compared to the expansion I would like.

How can I do it?

• Feb 3 '16 at 18:20

taylor = (vars - point).# &;
init := D[f[vars], {vars, j}] /. Thread[vars -> point];
taylorPoly[m_] := Sum[1/j! Nest[taylor, init, j], {j, 0, m}]


Example

vars = {x, y, z};
point = {0, 0, 0};
f[vars_] = Sin[y - x ] + Exp[x - y + 2 z];

taylorPoly[2] // FullSimplify
1/2 (2 + 4 z + (x - y + 2 z)^2)

taylorPoly[3] // FullSimplify
1/6 (6 + (3 + 2 x - 2 y) (x - y)^2 + 12 z + 6 (x - y) (2 + x - y) z +
12 (1 + x - y) z^2 + 8 z^3)

taylorPoly[6] // FullSimplify
1/720 ((x - y)^2 (360 + 240 x + 30 x^2 + x^4 -
4 (60 + 15 x + x^3) y + 6 (5 + x^2) y^2 - 4 x y^3 + y^4) +
12 (x - y) (x (60 + x (20 + x (5 + x))) - 60 (-2 + y) -
x (40 + x (15 + 4 x)) y + (20 + 3 x (5 + 2 x)) y^2 - (5 +
4 x) y^3 + y^4) z +
60 (24 + 24 x + 12 x^2 + 4 x^3 + x^4 -
4 (6 + x (6 + x (3 + x))) y + 6 (2 + x (2 + x)) y^2 -
4 (1 + x) y^3 + y^4) z^2 +
160 (6 + x (6 + x (3 + x)) - 6 y - 3 x (2 + x) y + 3 (1 + x) y^2 -
y^3) z^3 + 240 (2 + 2 x + x^2 - 2 (1 + x) y + y^2) z^4 +
192 (1 + x - y) z^5 + 64 z^6 + 720 (1 + 2 z))