Artwork Plaque

In a comprehensive analysis by Blaschke $B(z)$ ${\ displaystyle B (z)}$ $B (z)$ An analytic function in a unit circle is called that has zeros (their finite or countable number) at predetermined points $\{z_{n}\}_{1}^{k}$ ${\ displaystyle \ {z_ {n} \} _ {1} ^ {k}}$ ${\ displaystyle \ {z_ {n} \} _ {1} ^ {k}}$ where $k$ ${\ displaystyle k}$ $k$ - a finite positive number or infinity (it is called a Blaschke sequence ). If the sequence of zeros is infinite, then an additional condition is imposed on it - the convergence of the series $\sum _{n}(1-|z_{n}|).$ ${\ displaystyle \ sum _ {n} (1- | z_ {n} |).}$ ${\ displaystyle \ sum _ {n} (1- | z_ {n} |).}$

The product of Blaschke is built from the so-called Blaschke factors $B(z)=\prod B(z_{n},z)$ ${\ displaystyle B (z) = \ prod B (z_ {n}, z)}$ ${\ displaystyle B (z) = \ prod B (z_ {n}, z)}$ of the following form:

$B(z_{n},z)={\frac {|z_{n}|}{z_{n}}}{\frac {z-z_{n}}{1-{\overline {z_{n}}}z}}.$ ${\ displaystyle B (z_ {n}, z) = {\ frac {| z_ {n} |} {z_ {n}}} {\ frac {z-z_ {n}} {1 - {\ overline {z_ {n}}} z}}.}$ ${\ displaystyle B (z_ {n}, z) = {\ frac {| z_ {n} |} {z_ {n}}} {\ frac {z-z_ {n}} {1 - {\ overline {z_ {n}}} z}}.}$

If $z_{n}=0$ ${\ displaystyle z_ {n} = 0}$ ${\ displaystyle z_ {n} = 0}$ is considered $B(0,z)=z$ ${\ displaystyle B (0, z) = z}$ ${\ displaystyle B (0, z) = z}$ .

Artwork Plaque

More articles: