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
A Nevanlinna function maps the upper half-plane to itself or a real constant,[1] but is not necessarily injective or surjective.
is the upper half-plane, 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.
[2] Some elementary examples of Nevanlinna functions follow (with appropriately chosen branch cuts in the first three).