Impedance calculation from voltage and current (exponential) waveforms
My solution is in accordance with the expected result:

From FOURIER TRANSFORMS:

$$ \mathcal{F} \{ e^{-at}\sin(\omega_0t)u(t)\} = \frac{\omega_0}{(a+j\omega)^2+\omega_0^2} $$
$$ \mathcal{F} \{ e^{-at}\cos(\omega_0t)u(t)\} = \frac{a+j\omega}{(a+j\omega)^2+\omega_0^2} $$

and the inductor current as:

$$ i(t) = I_0e^{\alpha t}\sin(\omega_0t)u(t) $$

In frequency domain:

$$ \boxed{I(\omega)= \frac{I_o\omega_0}{(-\alpha+j\omega)^2+\omega_0^2}} $$

Then the inductor voltage is:

$$ v(t) = L\frac{\mathrm{d}i(t) }{\mathrm{d}t } $$

or 

$$ v(t)=LI_0 \left[\alpha e^{\alpha t}\sin(\omega_0t)+ \omega_0e^{\alpha t} \cos(\omega_0t) \right ]u(t) $$

Similarly, in frequency domain:

$$ V(\omega)=LI_0 \left [  \frac{\alpha\omega_0}{(-\alpha+j\omega)^2+\omega_0^2} + \frac{\omega_0(-\alpha+j\omega)}{(-\alpha+j\omega)^2+\omega_0^2} \right ] $$

or 

$$ \boxed{V(\omega)=LI_0\omega_0 \left [ \frac{j\omega}{(-\alpha+j\omega)^2+\omega_0^2} \right ] }$$

The impedance is:

$$ Z(\omega)=\frac{V(w)}{I(\omega)} $$

or 

$$ \boxed{Z(\omega)= j\omega L} $$

From LAPLACE TRANSFORMS, the same result can be achivied:

$$ I(s) = \frac{I_0 \omega_0}{(-\alpha + s)^2+ \omega_0^2} $$

and

$$ V(s)= LI_0\omega_0 \frac{s}{(-\alpha + s)^2+ \omega_0^2} $$

The impedance is:

$$ Z(s)=\frac{V(s)}{I(s)} = sL $$

doing \$ s = j\omega\$

$$ Z(j\omega)= j\omega L $$

EDIT:

The main goal of my answer was to point out an error in the calculation of a particular Fourier transform present in question. Although the OP does not explicitly make clear the fact that the INDUCTANCE IS UNKNOWN - it seems to me that more important is the determination of the INDUCTANCE rather than the IMPEDANCE of inductor from the measurements in time.

Note that, in this case, the OP will be faced with a linear system identification procedure (since in practice it will be a series RL circuit due to the inductor's DCR). There are many techniques already used by LCR meters. In other hand seems the OP intention is to do some inovative work using growing exponentials. One problem I realize that,  in the question, would be difficult to generate a exponentally growing sine CURRENT. Maybe a VOLTAGE with a similar shape could be applied on RL circuit. Measurement in time suggest a easier  form of curve fitting (expected versus measured) to determine the parameters L and R.