$A a e^{-aT} = \frac{2i}{3\Re(T)}\left(\omega_0 + A e^{-aT}\right)$ solving for $T$ I am having some troubles in understanding how to solve (if possible) this complex equation:
$$A a e^{-aT} = \frac{2i}{3\Re(T)}\left(\omega_0 + A e^{-aT}\right)$$
where


*

*$A, a, \omega_0$ are real constants;

*$i$ is the imaginary unit;

*$T$ is a complex number, and it can be willingly written as $T = \sigma + i\zeta$;

*$\Re(T)$ is the real part of $T$.
I tried to write $\Re(T)$ as $\sigma$, or also as $\frac{T + T^*}{2}$ but this did not bring me to a solution.
Is there some way to solve this equation for $T$? 
I also tried to use a Taylor series for the exponential but I actually don't know what the constants are (they may be small quantities or big ones or some and some), hence it is not a good way to proceed.
Thanks in advance!
 A: An idea is to begin seperating the real and imaginary parts (this was too long to put in a comment). Use:
$$\exp\left[-\text{a}\cdot\left(\sigma+\zeta i\right)\right]=\exp\left[-\text{a}\sigma\right]\cdot\exp\left[-\text{a}\zeta i\right]=\frac{\cos\left(\text{a}\zeta\right)-\sin\left(\text{a}\zeta\right)i}{\exp\left[\text{a}\sigma\right]}\tag1$$
So, we get for the LHS:
1.$$\Re\left(\text{A}\cdot\text{a}\cdot e^{-\text{a}\cdot\text{T}}\right)=\Re\left\{\text{A}\cdot\text{a}\cdot\frac{\cos\left(\text{a}\zeta\right)-\sin\left(\text{a}\zeta\right)i}{\exp\left[\text{a}\sigma\right]}\right\}=\text{A}\cdot\text{a}\cdot\frac{\cos\left(\text{a}\zeta\right)}{\exp\left[\text{a}\sigma\right]}\tag2$$
 2.$$\Im\left(\text{A}\cdot\text{a}\cdot e^{-\text{a}\cdot\text{T}}\right)=\Im\left\{\text{A}\cdot\text{a}\cdot\frac{\cos\left(\text{a}\zeta\right)-\sin\left(\text{a}\zeta\right)i}{\exp\left[\text{a}\sigma\right]}\right\}=-\text{A}\cdot\text{a}\cdot\frac{\sin\left(\text{a}\zeta\right)}{\exp\left[\text{a}\sigma\right]}\tag3$$
Now, for the RHS:
$$\frac{2i}{3\cdot\Re\left(\text{T}\right)}\cdot\left(\omega_0+\text{A}\cdot e^{-\text{a}\cdot\text{T}}\right)=\frac{2i}{3\cdot\sigma}\cdot\left(\omega_0+\text{A}\cdot\frac{\cos\left(\text{a}\zeta\right)-\sin\left(\text{a}\zeta\right)i}{\exp\left[\text{a}\sigma\right]}\right)=$$
$$\frac{2}{3\cdot\sigma}\cdot\left(\omega_0i+\text{A}\cdot\frac{\sin\left(\text{a}\zeta\right)+\cos\left(\text{a}\zeta\right)i}{\exp\left[\text{a}\sigma\right]}\right)\tag4$$
So, we can write:
1.$$\Re\left\{\frac{2i}{3\cdot\Re\left(\text{T}\right)}\cdot\left(\omega_0+\text{A}\cdot e^{-\text{a}\cdot\text{T}}\right)\right\}=\frac{2}{3\cdot\sigma}\cdot\left(\text{A}\cdot\frac{\sin\left(\text{a}\zeta\right)}{\exp\left[\text{a}\sigma\right]}\right)\tag5$$
 2.$$\Im\left\{\frac{2i}{3\cdot\Re\left(\text{T}\right)}\cdot\left(\omega_0+\text{A}\cdot e^{-\text{a}\cdot\text{T}}\right)\right\}=\frac{2}{3\cdot\sigma}\cdot\left(\omega_0+\text{A}\cdot\frac{\cos\left(\text{a}\zeta\right)}{\exp\left[\text{a}\sigma\right]}\right)\tag6$$
So, we can set up a system of equations:
$$
\begin{cases}
\text{A}\cdot\text{a}\cdot\frac{\cos\left(\text{a}\zeta\right)}{\exp\left[\text{a}\sigma\right]}=\frac{2}{3\cdot\sigma}\cdot\left(\text{A}\cdot\frac{\sin\left(\text{a}\zeta\right)}{\exp\left[\text{a}\sigma\right]}\right)\\
\\
-\text{A}\cdot\text{a}\cdot\frac{\sin\left(\text{a}\zeta\right)}{\exp\left[\text{a}\sigma\right]}=\frac{2}{3\cdot\sigma}\cdot\left(\omega_0+\text{A}\cdot\frac{\cos\left(\text{a}\zeta\right)}{\exp\left[\text{a}\sigma\right]}\right)
\end{cases}\tag7
$$
This leads towards, this simplified system of equations:
$$
\begin{cases}
\text{a}\cdot\frac{\cos\left(\text{a}\zeta\right)}{\exp\left[\text{a}\sigma\right]}=\frac{2}{3\cdot\sigma}\cdot\frac{\sin\left(\text{a}\zeta\right)}{\exp\left[\text{a}\sigma\right]}\\
\\
\frac{\sin\left(\text{a}\zeta\right)}{\exp\left[\text{a}\sigma\right]}=-\frac{1}{\text{A}\cdot\text{a}}\cdot\frac{2}{3\cdot\sigma}\cdot\left(\omega_0+\text{A}\cdot\frac{\cos\left(\text{a}\zeta\right)}{\exp\left[\text{a}\sigma\right]}\right)
\end{cases}\tag8
$$
So:
$$\text{a}^2\cdot\frac{\cos\left(\text{a}\zeta\right)}{\exp\left[\text{a}\sigma\right]}=-\frac{4}{9\cdot\sigma^2}\cdot\left(\frac{\omega_0}{\text{A}}+\frac{\cos\left(\text{a}\zeta\right)}{\exp\left[\text{a}\sigma\right]}\right)\tag9$$
