Solving complex equations

I am trying to solve two complex equations, but as I've never really dealt with complex numbers and equations before, I've run into trouble.

The equations are as follows:

$T_{L} = i \left[ \frac{2\pi}{n} \cdot \frac{r}{d} \cdot e^{i\phi} \right]$

$J_{A} = k_{y}^{2} \cdot \frac{C_{L_{\alpha}}}{C_{M_{\alpha}}} \left[ iP\xi_{0} - \xi_{0}^{'}\right]$

For the first one:

$e^{i\phi} = - \sin(\phi) + i\cos(\phi)$

;thus:

$T_{L} = - \frac{2\pi r \cdot \sin(\phi)}{nd} + \frac{2\pi r \cdot i\cos(\phi)}{nd}$

;which becomes:

$T_{L} = \left[ \frac{2\pi}{n} \cdot \frac{r}{d} \right]^{2}$

;and through a wild goose chase the second one simply becomes:

$J_{A} = k_{y}^{2} \cdot \frac{C_{L_{\alpha}}}{C_{M_{\alpha}}} \left[ - (P\xi_{0})^{2} - \xi_{0}^{'}\right]$

I honestly do not know what I am doing, a friend of mine gave me some "hints" but he doesn't really know either, and I wouldn't be surprised if it's all wrong. The solution to the first one makes sense (when you know what it's supposed to do) but I still have a feeling that I'm just making humbug here.

Any help would be very much appreciated!

Well, what, exactly, are you trying to do with them? Are $T_L$ and $J_A$ constants and you want to solve for r and $\phi$ ?

$T_{L}$ and $J_{A}$ are ballistic constants, lateral throw-off and aerodynamic jump.

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$T_{L} = i \left[ \frac{2\pi}{n} \cdot \frac{r}{d} \cdot e^{i\phi} \right]$

Lateral throw-off is the dispersion (deviation from the mean path; measured in radians) due to static imbalance in the projectile. Lateral throw-off is always at a right angle to the orientation at which the centre of mass emerges from the muzzle, e.g. if the CM emerges at 12 o'clock, the lateral throw-off is directed towards 3 o'clock for a right-hand spin.

As I understand it, the complex part is to properly model the angle of lateral throw-off AND the angle of orientation, e.g. x rad towards -π/2 rad (3 o'clock). I want to find the absolute value, the angle of dispersion without the second component that tells in what direction (I already know it's at right angles).

In the book it also says: "TL = Lateral throw-off (the tangent of the deflection angle)."

n = twist rate, calibres/turn
r = radius of gyration, m
d = projectile diameter, m
φ = roll orientation angle at emergence from the muzzle, probably deg

The actual yaw angle due to static imbalance is:

$\phi = \frac{\omega r}{v}$

;so I wasn't surprised to see the familiar form in the "solution" for $T_{L}$ .

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$J_{A} = k_{y}^{2} \cdot \frac{C_{L_{\alpha}}}{C_{M_{\alpha}}} \left[ iP\xi_{0} - \xi_{0}^{'}\right]$

$k_{y}^{2} = \frac{I_{y}}{md^{2}}$ (although probably not needed for a solution)

$P = \frac{I_{x}}{I_{y}} \cdot \frac{\omega d}{v}$ (although probably not needed for a solution)

Ix = angular mass, kg·m^2
Iy = angular mass, kg·m^2
m = projectile mass, kg
d = projectile diameter, m
CLα = lift force constant, k/1
CMα = overturning moment constant, k/1
ξ0 = yaw angle at the muzzle, rad
ξ0' = yaw rate at the muzzle, rad/s
ω = angular velocity (spin) of projectile, rad/s
v = velocity relative to wind, m/s

Aerodynamic jump is the dispersion due to initial yaw angle and rate (of tip-off rate) as it emerges from the muzzle. This would also be an angle (rad) that describes the deviation from the mean path.

This is all from the book "Modern Exterior Ballistics: The Launch And Flight Of Symmetric Projectiles" by Robert L. McCoy. Because it's a proper thesis on this particular subject, most of the equations are full models that describe the physics in full mathematical form, even though it could be described in simpler forms.

It is those simpler forms that I am looking for, e.g. the drag force is explained in full vector form with a few other things thrown in, but it is equal to F=kpA if you use a scalar form and make a few things implicit, etc.