Say I had to plot the expression $$\frac{\pi e^{i\theta }}{4\theta}$$ where $\frac{\pi}{4} \le \theta \le \frac{9\pi}{4} $ on an Argand diagram, how would one go about doing so?
If it was just the numerator, this would be easy as it would be an arc of a circle from $\theta$ = $\frac{\pi}{4}$ to $\frac{9\pi}{4} $ with radius of $\pi$ but the inclusion of the denominator confuses me.
Would I have to convert this back into the polar form first?
A useful way to think about the multiplication of complex numbers is that if some complex number $z$ is multiplied by a complex number $w$ then we may consider their product $zw$ to be a rotation and a dilation (stretching) of the original complex number $z$.
In this case if we first have a complex number $z = e^{i \theta}$ then this is simply the equation of a point on the unit circle (a complex number with length 1), and it's graph would in fact be the unit circle. If we let $w = \frac{\pi}{4 \theta}$, then this complex number is in fact a "pure" dilation (i.e. there is dilation when multiplication occurs with another complex number). We can see that as the angle $\theta$ increases then the length of this complex number is inversely proportional to that increase.
Now when we multiply these two what we are doing is taking points on the unit circle and decreasing their length (dilating). The diagram will look like a piece of a spiral on the arc you defined where the length of the vectors decreases as we go around the arc in the positive direction (counter-clockwise). I suggest plugging in the initial and final values on the arc to see where your spiral starts and ends, and then figuring out what comes in the middle by considering the above.