Taylor Expansion of and Exponential function but multivariable

Question

I have tried find the Taylor series expansion for my multivariable function, f[x_, y_, z_] := Exp[I*(x^2 + y^2 + z^2)^(1/2)] of order 3.

This is what I tried, Expandedfunction = Normal[Series[f[(x - x0)*t + x0, (y - y0)*t + y0, (z - z0)*t + z0], {t, 0, 3}]] /. t -> 1

I followed this answer/question: Multivariable Taylor expansion does not work as expected

However, my output does not get rid of the x0, y0, and z0. This is my Mathematica output,

E^(I Sqrt[x0^2 + y0^2 + z0^2]) - (
 I E^(I Sqrt[
   x0^2 + y0^2 + z0^2]) (-x x0 + x0^2 - y y0 + y0^2 - z z0 + 
    z0^2))/Sqrt[x0^2 + y0^2 + z0^2] + 
 1/2 E^(I Sqrt[
   x0^2 + y0^2 + 
    z0^2]) (-((-x x0 + x0^2 - y y0 + y0^2 - z z0 + z0^2)^2/(
     x0^2 + y0^2 + z0^2)) + (
    I (x0^2 y^2 - 2 x x0 y y0 + x^2 y0^2 + x0^2 z^2 + y0^2 z^2 - 
       2 x x0 z z0 - 2 y y0 z z0 + x^2 z0^2 + y^2 z0^2))/(x0^2 + 
      y0^2 + z0^2)^(3/2)) + 
 1/3 E^(I Sqrt[
   x0^2 + y0^2 + z0^2]) ((
    3 I (-x x0 + x0^2 - y y0 + y0^2 - z z0 + z0^2) (x0^2 y^2 - 
       2 x x0 y y0 + x^2 y0^2 + x0^2 z^2 + y0^2 z^2 - 2 x x0 z z0 - 
       2 y y0 z z0 + x^2 z0^2 + y^2 z0^2))/(
    2 (x0^2 + y0^2 + z0^2)^(
     5/2)) + ((-x x0 + x0^2 - y y0 + y0^2 - z z0 + z0^2) (x0^2 y^2 - 
       2 x x0 y y0 + x^2 y0^2 + x0^2 z^2 + y0^2 z^2 - 2 x x0 z z0 - 
       2 y y0 z z0 + x^2 z0^2 + y^2 z0^2))/(x0^2 + y0^2 + z0^2)^2 - (
    I (-x x0 + x0^2 - y y0 + y0^2 - z z0 + 
       z0^2) (-((-x x0 + x0^2 - y y0 + y0^2 - z z0 + z0^2)^2/(
        x0^2 + y0^2 + z0^2)) + (
       I (x0^2 y^2 - 2 x x0 y y0 + x^2 y0^2 + x0^2 z^2 + y0^2 z^2 - 
          2 x x0 z z0 - 2 y y0 z z0 + x^2 z0^2 + y^2 z0^2))/(x0^2 + 
         y0^2 + z0^2)^(3/2)))/(2 Sqrt[x0^2 + y0^2 + z0^2]))

I thought that my expanded function would not have x0, y0, and z0 terms. My end goal is to evaluate my function f[x, y, z] at some point {x, y, z}, but using the Taylor approximated function of order 3 instead. Later, I want to compare my answers such that when I increase the order, the approximated Taylor Polynomial @ {x, y, z} approaches the "exact" function f[x, y, z].

For example for the point {1, 2, 3}, I want to see the following to happen if I increase the order of the Taylor Polynomial.

Expandedfunction/.{x->1, y->2, z->3} ~ f[1, 2, 3]

What am I doing wrong or misunderstanding?

Answer 1

The {x0,y0,z0} is the point about which the series is expanded. I think you want it to be {0,0,0}, but you also can choose an other point. Try this

f[x_, y_, z_] = Exp[I*(x^2 + y^2 + z^2)^(1/2)];

ef[x_, y_, z_, x0_, y0_, z0_, n_Integer] := 
   Normal[Series[
     f[(x - x0)*t + x0, (y - y0)*t + y0, (z - z0)*t + z0], {t, 0, 
       n}]] /. t -> 1

ef[x, y, z, 0, 0, 0, 3]

(*   1 + 1/2 (-x^2 - y^2 - z^2) + I Sqrt[x^2 + y^2 + z^2] - 
     1/6 I (x^2 + y^2 + z^2)^(3/2)   *)

ef[1, 1, 1, 0, 0, 0, 3]

(*   -(1/2) + (I Sqrt[3])/2   *)