A complex analysis 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 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 N admits a representation
formula_2
where C is a real constant, D is a non-negative constant, formula_3 is the upper half-plane, and μ is a Borel measure on ℝ satisfying the growth condition
formula_4
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
formula_5
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):
formula_6
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.)
These are injective but when p does not equal 1 or −1 they are not surjective and can be rotated to some extent around the origin, such as formula_12.
formula_16
 is a Nevanlinna function if (sufficient but not necessary) formula_17 is a positive real number and formula_18. This is equivalent to the set of such transformations that map the real axis to itself. One may then add any constant in the upper half-plane, and move the pole into the lower half-plane, giving new values for the parameters. Example: formula_19
formula_23
 is a Nevanlinna function.
Importance in operator theory.
Nevanlinna functions appear in the study of Operator monotone functions.
References.
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