How to plot a curve in complex plane that include e constant

Question

I can understand how a complex numbers such as $10 + 7i$ can be plotted in complex plane with Imaginary and Real axis. But I have no idea how to approach this problem. I have been asked to plot $e^{(0.12+i) \theta\ } $.

I cannot understand what is the role of $ \theta\ $ in this equation. I also do not know how I can raise the power of Euler constant to a complex number. I appreciate any hint, comment, clue, reference, etc to solve this problem.

Answer 1

Hint : $e^{i\theta}=\cos\theta + i \sin\theta$

Parametric form of circle $(r\cos\theta,r\sin\theta)$

Answer 2

Split it: $e^{(a+i)\theta}=e^{a\theta}e^{i\theta}$. Use the Awesome Hint to see that the product contains an exponentially growing and a oscillating part...