I am learning complex analysis and need help understanding and relating the concept of poles and zeros of a complex function.
I understand that if a complex function $f(z)$ has an isolated singularity at $z = z_{0}$ then it can be expanded into a Laurent's series in the annulus region $0 < \lvert z – z_{0} \rvert < r$. Then, the point $z = z_{0}$ is said to be a pole of order $m$ of $f(z)$ if the principle part of the Laurent's series has finite number of elements.
I understand it this far.
Now, if $f(z)$ is analytic in a domain $D$ and $z_{0}$ is a point withing $D$ then $f(z)$ can be expanded into a Taylor series about $z = z_{0}$ in the form
$$
f(z) = \sum_{n = 0}^{\infty} a_{n}(z – z_{0})^{n}
$$
if $a_{0} = a_{1} = \dots = a_{m – 1} = 0$ and $a_{m} \ne 0$ then we say that $f(z)$ has a zero of order $m$ at $z = z_{0}$.
I have doubts with the following points:
1) How are poles and zeroes of $f(z)$ related or differ from one another? I am not able to see them together.
2) For a given function, say $f(z) = \sin(\frac{1}{z})$, how do I find a pole zero (without going into the theory)?
3) Also, the book says that nature of singularity at $z = \infty$ will be same as that of function $f(\frac{1}{z})$ at $z = 0$. How is this so? Is it related to the same concept?
Best Answer
1) The poles of $f$ are the zeroes of $1/f$ and vice versa.
2) $f(z)=\sin(1/z)$ is defined and holomorphic everywhere except $z=0$, where it has an essential singularity. Zeroes when $1/z=\pi n i \implies z=-i/\pi n$
3) $f$ takes all complex values near each of $\{0, \infty \}$, and near both has infinitely many zeroes. Not sure exactly what they mean by "nature" though.