# Finding poles of an abstract transfer function

When finding the poles of something like the following transfer function, would I be able to write $$z=\sqrt[L]{\mu}$$ since square roots aren't technically defined on the complex plane?

$$Y(z) = \frac{a_L z^L + a_{L-1}z^{L-1} + ... +a_0}{z^L - \mu}$$

where $$a_j,\mu \in\mathbb{C}$$, $$|\mu|<1$$ and L is positive.

• You're mistaken if you think that roots of complex numbers aren't defined: link Mar 4 '20 at 14:09
• I tried finding the roots of x^5 - 1 = 0 on Wolfram Alpha There is the obvious solution x = 1 and 4 non-obvious solutions x~~-0.30902 ±0.95106 i x~~0.80902 ± 0.58779 i
– Ben
Mar 4 '20 at 14:09
• So to answer your question, some poles will have imaginary values.
– Ben
Mar 4 '20 at 14:10

For $$\mu > 0$$, $$z^L \in \left ( \sqrt[L]{\mu} \right) e^{j 2\pi n/L} \,\forall\, n \in 1 \cdots L$$.
I.e., the roots are evenly spaced on a circle $$\sqrt[L]{\mu}$$ in radius, and there's an $$L$$ of a lot of them.
• Would we have to consider the case for $\mu \leq 0$? ... well I guess $\mu = 0$ just gives $z=0$ but not sure for $\mu <0$. Thanks for your help :) Mar 5 '20 at 10:17
• I'm already concerned that I'm answering a homework problem. For $\mu$ equals any number, $z^L - \mu = 0$ is just an algebra problem -- can you work out what the answer is for $\mu < 0$? Mar 5 '20 at 14:43
• Not a homework problem, I am researching the design of digital filters but I am new to the subject and its been a while since I formally studied any complex analysis. If I was to make a guess, I would assume that calculating the poles for $\mu < 0$ would just give the same roots of unity again but in reverse direction?(each subsequent root would work clockwise around the unit disc) Mar 5 '20 at 16:35