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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 $\arctan(\frac{0}{0})$ ?

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  • $\begingroup$ $\arctan (\frac{w}{0}) = 90^\circ$ is wrong because $\frac w 0$ is undefined. And the complex number $z=0 + i0$ does not have an argument. $\endgroup$
    – Martin R
    Commented Nov 22, 2021 at 12:32

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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,

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  • $\begingroup$ O yes! I see, I was thinking about the frequency response of a pole at origin, frequency response doesn't care about w=0, so as a convention we take it as 90 degrees, it doesn't matter like you said. $\endgroup$
    – User
    Commented Nov 22, 2021 at 12:52

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