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 formula_1 and has non-negative imaginary part. A Nevanlinna function maps the upper half-plane to itself or to 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 representation where is a real constant, is a non-negative constant, formula_3 is the upper half-plane, and is a Borel measure  on 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 (related to the inversion formula for the Stieltjes transformation): A very similar representation of functions is also called the Poisson representation. Examples. Some elementary examples of Nevanlinna functions follow (with appropriately chosen branch cuts in the first three). (formula_7 can be replaced by formula_8 for any real number formula_9.)