Short-circuited quarter wavelength wave resonator

This post tells about short-circuited quarter wavelength line resonator. Quarter wavelength line resonator is depicted below, it is short circuited at one end.

Input impedance for this resonator with length $l$ is $Z_{in}=Z_0\tan h(\alpha +j\beta )l=Z_0\frac{1–j\tan h\alpha lcot\beta l}{\tan h\alpha l–jcot\beta l}$.

If $l=\frac{\lambda }{4}, \omega ={\omega }_0, and \omega ={\omega }_0+∆\omega$.

Here we can conclude that $\beta l=\frac{\pi }{2}+\frac{\pi ∆\omega }{2{\omega }_0}$, then $cot\beta l\approx –\frac{\pi ∆\omega }{2{\omega }_0}$. And $\tan h\alpha l\approx \alpha l$, then $Z_{in}\approx \frac{Z_0}{\alpha l+{\displaystyle \frac{\mathrm{j\pi }∆\mathrm{\omega }}{2{\omega }_0}}}$