Nevanlinna function
In mathematics, in the field of complex analysis, a Nevanlinna function is a complex function which is an analytic function on the open upper half-plane and has a non-negative imaginary part. A Nevanlinna function maps the upper half-plane to itself or a real constant, but is not necessarily injective or surjective. Functions with this property are sometimes also known as Herglotz, Pick or R functions.
Integral representation
Every Nevanlinna function admits a representationwhere is a real constant, is a non-negative constant, is the upper half-plane, and is a Borel measure on [real numbers|] satisfying the growth condition
Conversely, every function of this form turns out to be a Nevanlinna function.
The constants in this representation are related to the function via
and the Borel measure can be recovered from by employing the Stieltjes inversion formula :
A very similar representation of functions is also called the Poisson representation.
Examples
Some elementary examples of Nevanlinna functions follow.- A sheet of such as the one with.
- .
- A Möbius transformation
- and are examples which are entire functions. The second is neither injective nor surjective.
- If is a self-adjoint operator in a Hilbert space and is an arbitrary vector, then the function
- If and are both Nevanlinna functions, then the composition is a Nevanlinna function as well.