A function f, analytic in the unit disk, is said to have finite Dirichlet integral if
1![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0008414X00041614/resource/name/S0008414X00041614_eqn1.gif?pub-status=live)
Geometrically, this is equivalent to f mapping the disk onto a Riemann surface of finite area. The class of Dirichlet integrable functions will be denoted by
. The condition above can be restated in terms of Taylor coefficients; if f(z) = Σanzn, then
if and only if Σn|an|2 < ∞. Thus,
is contained in the Hardy class H2.
In particular, every such function has boundary values
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0008414X00041614/resource/name/S0008414X00041614_eqn2.gif?pub-status=live)
almost everywhere and log |f(eiθ)| ∊ L1(dθ).
The zeros zn of a function
must satisfy the Blaschke condition
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0008414X00041614/resource/name/S0008414X00041614_eqn3.gif?pub-status=live)
and f(s) = B(z)F(z), where F(z) has no zeros and
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0008414X00041614/resource/name/S0008414X00041614_eqn4.gif?pub-status=live)
is the Blaschke product with zeros zn; see (5).