Consider the function $f(u)=|u|^{2k}$ where $u$ is a complex number, and $k$ is an integer. I need to 'linearize' this equation* around the value $\phi$. In other words I need a series expansion of $f(\phi + r)$ where $\phi$ and $r$ are complex, and $r$ is small. I would appreciate either solutions or suggestions on the correct approach to this problem or literature references.
[*] I ask this question because I am a physicist learning about linear stability of stationary solutions of nonlinear differential equations; and I am trying to follow example 2 on this Wikipedia page, and I am not confident about how series expansions work for functions that involve absolute values of complex numbers.
Maybe it's easier if you forget about the fact that you're working with complex numbers and instead pretend you're just in $\mathbb R^2$. Then the linearization of $f$ at $\vec x_0$ is simply $f(\vec x_0+\vec h)\approx f(\vec x_0)+\nabla f(\vec x_0)\cdot \vec h$. Rename $x_0$ and $h$ to your liking.