We need to plot the real and imaginary parts of a complex function $k(\omega)$, and cannot find a good way to do this without using "ad hoc tricks."

**Definitions**

- $k$ is a complex-valued function where both the real and the imaginary parts are functions of $\omega$ so that $k(\omega) = \beta(\omega) - i\alpha(\omega)$, where $i$ is the imaginary unit.
- $\omega$ is a real-valued positive number.
- $\alpha(\omega) = -Im(k)$ is a positive real-valued function of the real variable $\omega$.
- $\beta(\omega) = Re(k)$ is a positive real-valued function of the real variable $\omega$.
- $a$ is a real constant number, $0<a<1$
- $A$ and $B$ are real constants.

**The relation between $k$ and $\omega$**

$[k(\omega)]^2 +A[k(\omega)]^{a+1} - B\omega^2 = 0$. $\qquad\quad$ (1)

So, what we really would appreciate help on is to find some water-proof method to plot $\beta = Re(k)$ and $\alpha = -Im(k)$ as functions of $\omega$ given the constants $a,A,B$ and the equation (1) above. Eq. (1) is a special kind of dispersion relation.

Any kind of help is greatly appreciated, we have spent too much time trying to figure this out already!