Power series expansion in terms of a function

Question

I have a two variable function z[x,y] = f[x,y] + g[x,y], such that I know the functional form of f[x,y] but not of g[x,y]. I have to do some symbolic calculations with the function z[x,y], but I would like to keep only the first order in g[x,y] (treating g as small). So, for example, I would like Mathematica to approximate (z[x,y])^3 = (f[x,y] + g[x,y])^3 = f[x,y]^3 + 3*f[x,y]^2*g[x,y], or (D[z[x,y], x])^2 = (D[f[x,y], x])^2 + 2*D[f[x,y], x]*D[g[x,y], x]. Is there a way to do it? I have tried the most naive way, namely to use Series['exp'[z],{g, 0, 1}], treating g[x,y] as a parameter rather than a function, but (as expected) it doesn't work. Do you know a way to do it?

Thank you very much in advance for anyone who will reply!

Answer 1

Here a simple approach which assumes g and its derivatives(!) to be small:

z[x,y]:= f[x,y]+eps g[x,y]

Normal[Series[(f[x, y] + eps g[x, y])^3, {eps, 0, 1}]] /. eps -> 1
(*f[x, y]^3 + 3 f[x, y]^2 g[x, y]*)

Normal[Series[D[z[x, y] ^2, x], {eps, 0, 1}]] /. eps -> 1
(*2 f[x, y] Derivative[1, 0][f][x, y] + 2 g[x, y] Derivative[1, 0][f][x, y] + 2 f[x, y] Derivative[1, 0][g][x, y]*)

Answer 2

You can use O to drop higher-order terms:

Clear[f, g, z]
z[x_, y_] := f[x, y] + g[x, y]

z[x, y]^3 + O[g[x, y]]^2

If you want an expression without the O, use Normal:

z[x, y]^3 + O[g[x, y]]^2 // Normal

(* Out: f[x, y]^3 + 3 f[x, y]^2 g[x, y] *)

This works for the derivative as well:

D[z[x, y], x]^2 + O[g[x, y]]^2 // Normal // Expand

(* Out:
   Derivative[1, 0][f][x, y]^2 + 
     2*Derivative[1, 0][f][x, y] * Derivative[1, 0][g][x, y] + 
       Derivative[1, 0][g][x, y]^2
*)