Zero by zero division and complex variable.

Question

To my knowledge, zero divided by zero is neither zero, nor infinity, it can't be understood. Let a complex function of a real variable $w$ be $$ Z(w) = iw = 0 + iw $$ Now, generally the argument of this variable is $ \arctan (\frac{w}{0}) = 90^\circ $. Now, if $Z(w)$ is a transfer function and I'm measuring the frequency response phase angle at $w = 0$, How is $\angle Z = 90^\circ$, given that at $w=0$ the argument becomes $\arctan(\frac{0}{0})$ ?

Answer 1

You don't need a division to define an argument, as it would be generally better to use the double argument arctangent function.

This way: $\angle Z =\arctan2(w,0) = 90^\circ$ if $w>0$.

If $w=0$ as well, then the argument of the complex number is undefined, but you might just use a convention to say it is zero or other convenient value for some special application, since it also has no radius, it should hardly matter,