The network function F(s) can be written as ratio of two polynomials.

$F\left(s\right)=\dfrac{N\left(s\right)}{D\left(s\right)}=\dfrac{a_0s^m+a_1s^{m-1}+\cdots{}\cdots{}\cdots{}+am-1s^m+am}{b_0s^n+b_1s^{-1}+\cdots{}\cdots{}\cdots{}+a_{n-1}s^n+bn} $

Let N(S)=0

$i.e.\ a_0s^m+a_1s^{m-1}+\cdots{}\cdots{}+am=0,\ this\ has{{}_{\ }^{'}m}^{'}distinct\ roots\ $

Similarly Let D(S) =0 , This has 'n' distinct roots

$p_1,p_2\cdots{}\cdots{}p_n $

$\therefore{}F\left(s\right)=H\dfrac{\left(s-z_1\right)\left(s-z_2\right)\cdots{}\cdots{}\left(s-z_n\right)}{\left(s-p_1\right)\left(s-p_2\right)\cdots{}\cdots{}\left(s-p_n\right)}\ $

H=scale factor

F(s) will be zero if s=Zi where i=1,2…….n

Such complex frequency …