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 H and has non-negative imaginary part. A Nevanlinna function maps the upper half-plane into itself, but is not necessarily injective or surjective. Functions with this property are sometimes also known as Herglotz, Pick or R functions. 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 H and has non-negative imaginary part. A Nevanlinna function maps the upper half-plane into itself, but is not necessarily injective or surjective. Functions with this property are sometimes also known as Herglotz, Pick or R functions. Every Nevanlinna function N admits a representation where C is a real constant, D is a non-negative constant and μ is a Borel measure on R 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 N via and the Borel measure μ can be recovered from N by employing the Stieltjes inversion formula (related to the inversion formula for the Stieltjes transformation): A very similar representation of functions is also called the Poisson representation.