lacunary - Mathnotes

Zeros

:::define "Zero" A point z0 is called a zero of an order m of a complex function f if f is analytic at z0 and

0=f(z0)=f′(z0)=⋯=f(m−1)(z0),f(m)(z0)≠0. :::

:::define "Simple Zero" Zeroes of order 1 are often called simple zeros. :::

Theorem \@{theorem-1}

If a function f(z) can be expressed in the form

f(z)=(z−z0)mg(z)

valid in some circle |z−z0|<R, where g(z) is analytic at z0 and g(z0)≠0, then f(z) has a zero of order m at z0.

Intuition \@{theorem-1-intuition}

We can see this from the definition of the taylor series. Assume we're at a zero, that is, that f(z0)=0. Then

f(z)=f(z0)+f′(z0)(z−z0)+f″(z0)2!(z−z0)2+⋯,

and since f(z0) is 0, we're left with all terms that have (z−z0) in them, and so we can factor them out and get

f(z)=(z−z0)(f′(z0)+f″(z0)2!(z−z0)+f″(z0)3!(z−z0)2+⋯.

Now, if f′(z0) does not have a zero at z0, then f′(z0) is not zero and f(z) has a first order zero at z0. If f′(z0) has a zero at z0, we can repeat and move onto the next term until, going through m terms, until we find a term that doesn't have a zero at z0, in which case we will have factored out (z−z0)m and we'll be left with an analytic g(z) with g(z0)≠0.