If $f(z)$ is either zero or has a singularity at a point $c$, and if the function $g(z)=(z-c)^nf(z)$ is defined and nonzero at $z=c$, then $f$ has a pole or zero of order $n$ at $c$. To me, this is an interesting classification, but as someone without any education in complex analysis (but looking to learn more), I don’t intuitively understand why this classification is important. The only place I’ve ever seen the order of a pole or zero mentioned is in the Weierstrass Factorisation Theorem, but since I’ve neither seen the proof nor really understood the theorem I don’t see the purpose of the $z^m$ term, where $m$ is the order of the zero at zero – a most confusing statement by Wikipedia.
What does the order of these points tell us about the behaviour of the function such that the classification became necessary? Why did mathematicians bother to distinguish between simple poles and zeros versus non-simple ones?
Clearly it’s not a trivial definition since it shows up in some theorems, and guessing from its place in a factorisation theorem I might guess they’re sometimes relevant in the same way that multiplicity is sometimes relevant – but I don’t know and can’t seem to find out.
Thanks.
Best Answer
Perhaps an analogy with real numbers may help. Define two positive real numbers are commeasurable if they have a greatest common divisor. This is an equivalence relation on positive real numbers. Another way to define this relation is the two real numbers are equivalent if their quotient is a positive rational number. Every positive rational number can be uniquely factored into positive or negative powers of prime numbers.
In the analogy with functions, the prime factors of a rational number in the numerator correspond to function zeros and the factors in the denominator correspond to function poles. Rational functions are relatively simple to analyze in analogy with rational numbers. Given an analytic function, the zeros and poles with multiplicity combine to form a rational function. Divide the analytic function by this rational function and the quotient has no zeros or poles although it may still have essential singularities. This simplifies the analysis of the given function.