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Blaschke product (Definition)
Definition 1   Suppose that $ \{ a_n \}$ is a sequence of complex numbers with $ 0 < \lvert a_n \rvert < 1$ and $ \sum_{n=1}^\infty (1 - \lvert a_n \rvert) < \infty$, then
$\displaystyle B(z) := \prod_{n=1}^\infty \frac{\lvert a_n \rvert}{a_n} \left( \frac{a_n - z}{1-\bar{a}_nz} \right)$    

is called the Blaschke product.

This product converges uniformly on compact subsets of the unit disc, and thus $ B$ is a holomorhic function on the unit disc. Further it is the function on the disc that has zeros exactly at $ \{ a_n \}$. And finally for $ z$ in the unit disc, $ \left\lvert B(z) \right\rvert \leq 1$.

Definition 2   Sometimes $ B_a(z) := \frac{z-a}{1-\bar{a}z}$ is called the Blaschke factor.

With this definition, the Blascke product becomes $ B(z) = \prod_{n=1}^\infty \frac{\lvert a_n \rvert}{a_n} B_{a_n}(z)$.

Bibliography

1
John B. Conway. Functions of One Complex Variable I. Springer-Verlag, New York, New York, 1978.
2
Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.



"Blaschke product" is owned by jirka.
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Also defines:  Blascke factor
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Cross-references: factor, disc, function, unit disc, compact subsets, converges uniformly, product, complex numbers, sequence
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This is version 5 of Blaschke product, born on 2004-04-22, modified 2005-12-07.
Object id is 5795, canonical name is BlaschkeProduct.
Accessed 3556 times total.

Classification:
AMS MSC30C45 (Functions of a complex variable :: Geometric function theory :: Special classes of univalent and multivalent functions )

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