A question regarding Determinants

Question

Can anyone tell me if the following is true?? Here Det = determinant an I is Sqt[-1]……

LIM ω→0 (1/2I)[ 1/Det(ω-Iδ) -1/Det(ω+Iδ) ] =δ(Det(ω))

Answer 1

No, this $$\lim_{\omega \to 0} \frac 1{2i}\left[\frac 1{\text{det}(\omega -i\delta)} - \frac 1{\text{det}(\omega +i\delta)}\right] = \delta\,\text{det}(\omega)$$ is obviously not true, for a very simple reason.

The right-hand side of the equality depends on the value of $\omega$. But on the left-hand side, $\omega$ is just a dummy variable. The LH expression does not depend on $\omega$ at all.

In fact, the equation commits a mathematical faux pas, in that it uses $\omega$ to mean two different things in the same context. On the left it is a dummy variable, but on the right, it is a free variable with its own set value. These uses are not compatible.

As a dummy variable, any other unused variable could be substituted in place of $\omega$ on the left. For example, your equation could also be written as $$\lim_{\alpha \to 0} \frac 1{2i}\left[\frac 1{\text{det}(\alpha -i\delta)} - \frac 1{\text{det}(\alpha +i\delta)}\right] = \delta\,\text{det}(\omega)$$ Where is is clear that the LH and RH sides cannot be equal.