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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, [1] 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, 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). ( can be replaced by for any real number .)
Nevanlinna functions appear in the study of Operator monotone functions.
![]() | This article includes a list of general
references, but it lacks sufficient corresponding
inline citations. (March 2021) |
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, [1] 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, 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). ( can be replaced by for any real number .)
Nevanlinna functions appear in the study of Operator monotone functions.