taylor expansion of scalar fields

Question

Starting of with electrodynamics I have to compute the taylor expansion around $\vec{r} = 0$ of

$\psi (\vec{r}) = |\vec{r} - \vec{r_0}|^{\frac{3}{2}}$ where $\vec{r_0}$ is a constant vector up to second order and

$\psi(\vec{r}) = e^{i\vec{k}\cdot \vec{r}}$ where $\vec{k}$ is a constant vector up to arbitrary order.

I don't have problems with multidimensional taylor expansions as long as there are no vectors involved and functions look like $f(x,y,z) = y \cdot \sin(xz) + xz^2$ for example. Therefore, in the cases above I feel lost.

Can someone explain how to solve the exercise?

Answer 1

Since you are already OK with multidimensional Taylor expansions, note that if we call $\vec r = \begin{pmatrix}x \\ y \\ z\end{pmatrix}$, $\vec r_0 = \begin{pmatrix}x_0 \\ y_0 \\ z_0\end{pmatrix}$, and $\vec k = \begin{pmatrix}k_x \\ k_y \\ k_z\end{pmatrix}$, then

$$|\vec r - \vec r_0|^{3 \over 2} = |\vec r - \vec r_0|^{2 \cdot \frac{3}{4}} = \left[(x-x_0)^2+(y-y_0)^2+(z-z_0)^2\right]^\frac{3}{4}$$

and

$$e^{i\vec k \cdot \vec r} = e^{i(k_x \cdot x+k_y \cdot y+k_z \cdot z)}$$

Can you proceed from here?

Answer 2

I think Taylor series expansion of $\phi(\vec r+\vec l)$ ,where $\vec r$ is variable and $\vec l$ is a constant vector, can be written as $\phi(\vec r+\vec l)=\phi(\vec r)+\vec l.\vec \nabla\phi(\vec r)+....$ in analogy with the general form of taylor series expansion $f(x-a)$=$f(a)+xf'(a)+\frac {x^2}{2!}f''(a)+....$; but not sure about it. I have seen this formula used in solution of a problem in a book, but I'm not sure about it.