I'm dealing with the following type of function where $\omega_a$, $\gamma$ and $a$ are parameters: $$ f(\omega)\propto \sqrt{\omega_\textrm{a}-\omega-\textrm{i}\gamma}\; \arctan\left(\frac{a}{\sqrt{\omega_\textrm{a}-\omega-\textrm{i}\gamma}}\right) $$
Im currently thinking of ways to approximate this function by a rational function (in $\omega$). The issue I'm having is that the quantities under the square root as well as in the arctan are complex. I've really no idea how to do it - I didn't hear any functional analysis but I feel like it has to do something with analytical continuation. I would be happy about any tips how to approach the problem!
What I did first was trying to ignore that the arguments are complex-valued, so I did some Taylor expansion, Padé-approximation and such, but then the real and imaginary part of the approximant were completely different from the original function.
I've stumbled upon this representation of the arcus tanges for complex arguments (taken from wikipedia): $$ \arctan(a+b\,\mathrm i) = \left\{ \begin{array}{ll} \displaystyle \frac12\,\left(\arctan \frac{a^2+b^2-1}{2a} + \frac\pi2\,\textrm{sgn}(a) \right) & \; a\neq0 \\ 0 & \; a=0,\, |b|\leq1 \\ \displaystyle \frac\pi2\,\textrm{sgn}(b) & \; a=0,\, |b|>1 \\ \end{array} \right\} \\ + \mathrm i \cdot \frac12\,\operatorname{artanh} \frac{2b}{a^2+b^2+1} $$ But since the argument in my function $f(\omega)$ involves a square root I do have a lot of terms to the power of $\frac{1}{2}$, $\frac{3}{2}$ and so on.
Regarding the range of values of $\omega \in [\omega_L, \omega_U]$ I can say that $\omega_L<\omega_a<\omega_U$ but it is not really restricted. Typical values are: $10^{15} <\omega < 10^{16},\;\omega_a = 3\cdot 10^{15}$ and $a=5\cdot10^{7}$. With these values we have
$$ \max_\omega \Re\left(\cfrac{a}{\sqrt{\omega_a+\omega+\textrm{i}\gamma}} \right) \approx 1.385 \qquad\qquad \min_\omega \Re\left(\cfrac{a}{\sqrt{\omega_a+\omega+\textrm{i}\gamma}}\right) \approx 0.625 $$
$$ \max_\omega \Im\left(\cfrac{a}{\sqrt{\omega_a+\omega+\textrm{i}\gamma}}\right) \approx 0.0152 \qquad\qquad \min_\omega \Im\left(\cfrac{a}{\sqrt{\omega_a+\omega+\textrm{i}\gamma}}\right) \approx 0.0150 $$