I have to fit experimental data to an analytical expression involving complex variables. The expression is set such that it cannot be decomposed into real and imaginary parts (in an algebraic form), e.g., $$Z(\omega) = \frac{coth\left(H \sqrt{i \omega/ \alpha_z} \right)}{\pi R^2 k_z \sqrt{i \omega/ \alpha_z}}$$
My data points are $\hat{Z}(\omega)$ (hat to distinguish experimental data from analytical variable).
Intuitively, I would expect an equivalent least-squares solution for the complex domain which minimizes the error (* means complex conjugate): $$ \varepsilon = \sqrt{\frac{1}{N} \sum_i^N \left(Z(\omega_i) - \hat{Z}(\omega_i) \right) \cdot \left(Z^*(\omega_i) - \hat{Z}^*(\omega_i) \right)}$$
How do I go about it? Are there other solutions to this problem? How would a computer implementation look like, e.g., in MATLAB?
Note: $\omega$ is independent variable, $Z$, $\hat{Z}$ dependent variable, $H$, $R$ constants and $k_z$, $\alpha_z$ are the parameters to be back-computed via curve-fitting.
Appropriate transformation of the initial equation leads to a linear relationship involving an inverse function which can be numerically computed :
NOTE : The function $F(X)$ involved in the numerical calculus is a multivaluated function that is, for an input value $X_k$ the function is subject to returns several values $F_k$ in some cases ($X$ close to $0$.).
From a few tests with simulated data, it appears that this causes some difficulties in the numerical process to determine the convenient $F_k$ among the possible ones. This depends a lot of the range of the data. No major difficulty in cases of $0.001205<X<1$ . That is why realistic numerical examples of data would have been better than simulated data in order to investigate this problem.