Nevanlinna function

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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 H and has non-negative imaginary part. A Nevanlinna function maps the upper half-plane to itself or to 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.

Integral representation[edit]

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.[2]

Examples[edit]

  • Some elementary examples of Nevanlinna functions follow (with appropriately chosen branch cuts in the first three). ( can be replaced by for some real number )
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
A sheet of such as the one with
(an example that is surjective but not injective)
is a Nevanlinna function if (but not only if) is a positive real number and 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:
  • and are examples which are entire functions. The second is neither injective nor surjective.
  • If S is a self-adjoint operator in a Hilbert space and f is an arbitrary vector, then the function
is a Nevanlinna function.
  • If M(z) and N(z) are Nevanlinna functions, then the composition M(N(z)) is a Nevanlinna function as well.

References[edit]

  1. ^ A real number is not considered to be in the upper half-plane.
  2. ^ See for example Section 4, "Poisson representation", of Louis de Branges. Hilbert spaces of entire functions. Prentice-Hall. . De Branges gives a form for functions whose real part is non-negative in the upper half-plane.