I have a question a series expansion of the following equation

$\displaystyle u = \kappa a_1 z^{\lambda + i \varepsilon} - z a_1 \overline{z}^{\lambda - i \varepsilon - 1}(\lambda + i \varepsilon) - c_1 \overline{z}^{\lambda - i \varepsilon} - d_1 \overline{z}^{\lambda + i \varepsilon})$

With the eigenvalues determined from

$\displaystyle 0 = \left(1+\beta\right)e^{i 2\pi \left(\lambda + i \varepsilon\right)} - \beta + 1 $

And are given as:

$\displaystyle \lambda + i \varepsilon = \frac{\left(1 + 2k\right)}{2} - i \frac{1}{2 \pi} \ln\left(\frac{1-\beta}{1 + \beta}\right) \quad \text{where} \quad k\in\mathbb{Z}$

The factors in the expression are:

Real value: $\displaystyle \beta, \varepsilon, \lambda, \kappa, \mu $
Complex value: $\displaystyle a_1, c_1, d_1 $
Complex coordinate: $\displaystyle z = x + i y$

My question is how the eigenfunction series expansion looks like with the above given and what topics do I need to study on to understand the problem? I guess that you can't use orthogonal eigenfunction expansions like the generalized Fourier series because it is a complex expression and because the eigenfunction aren't orthogonal (as far as I understand). This area is new to me and since I am an not a mathematician but a mechanical engineer I would appreciate very much if you will consider that in your replies

Thanks in advance

Regards Brian