I have a question a series expansion of the following equation

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

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

And are given as:

 \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: \beta, \varepsilon, \lambda, \kappa, \mu
Complex value:  a_1, c_1, d_1
Complex coordinate: 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