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Consider the function $$f(t)= \frac{\pi c^2}{l^2} \sum_{n=1}^\infty \frac{ n }{\omega_n}\cos(\omega_nt)$$ where $\omega_n= \sqrt{(\frac{n \pi c}{l})^2-(\frac{r_0}{2})^2}.$ If we neglect the term $(\frac{r_0}{2})^2$ from $\omega_n$, this series will be $$f(t)=\frac{c}{l} \sum_{n=1}^\infty \cos(\frac{n \pi c}{l}t)$$ which is the same as Dirac comb $$\Delta_T(t)= \frac{1}{T}\sum_{k=-\infty}^\infty e^{i2 \pi kt/T}$$ with $T=2l/c$. Now I have two questions:

1) What happen if we don't neglect the term $(\frac{r_0}{2})^2$ from $\omega_n$? I mean how we can represent the function $f(t)$ (with term $(\frac{r_0}{2})^2$) as a series which is the same as Dirac comb (or an other series like Dirac comb)?

2) What happen if we put $(-1)^n$ in the series of $f(t)$? I mean how we can represent the function $$g(t)=\frac{\pi c^2}{l^2} \sum_{n=1}^\infty \frac{ (-1)^nn }{\omega_n}\cos(\omega_nt)$$ (with term $(\frac{r_0}{2})^2$) as a series which is the same as Dirac comb (or an other series like Dirac comb)? It would be appreciated if someone can help me to find the solutions.

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