Nevanlinna function

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Complex analysis function

title: "Nevanlinna function" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["complex-analysis"] description: "Complex analysis function" topic_path: "general/complex-analysis" source: "https://en.wikipedia.org/wiki/Nevanlinna_function" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary 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 , \mathcal{H} , 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

: N(z) = C + D z + \int_{\mathbb{R}} \bigg(\frac{1}{\lambda - z} - \frac{\lambda}{1 + \lambda^2} \bigg) \operatorname{d} \mu(\lambda), \quad z \in \mathcal{H},

where C is a real constant, D is a non-negative constant, \mathcal{H} is the upper half-plane, and μ is a Borel measure on ℝ satisfying the growth condition

: \int_{\mathbb{R}} \frac{\operatorname{d} \mu(\lambda)}{1 + \lambda^2}

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

: C = \Re \big( N(i) \big) \qquad \text{ and } \qquad D = \lim_{y \rightarrow \infty} \frac{N(i y)}{i y}

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):

: \mu \big( (\lambda_1, \lambda_2 ] \big) = \lim_{\delta\rightarrow 0} \lim_{\varepsilon\rightarrow 0} \frac{1}{\pi} \int_{\lambda_1+\delta}^{\lambda_2+\delta} \Im \big( N(\lambda + i \varepsilon) \big) \operatorname{d} \lambda.

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). (z can be replaced by z - a for any real number a.)

  • z^p\text{ with } 0 \le p \le 1

  • -z^p\text{ with } -1 \le p \le 0

:::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 i(z/i)^p \text{ with }-1\le p\le 1.

  • A sheet of \ln(z) such as the one with f(1)=0.

  • \tan(z) (an example that is surjective but not injective).

  • A Möbius transformation

::z \mapsto \frac{az+b}{cz+d}

: is a Nevanlinna function if (sufficient but not necessary) \overline{a} d - b \overline{c} is a positive real number and \Im (\overline{b} d ) = \Im (\overline{a} c) = 0. 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: \frac{i z + i - 2}{z + 1 + i}

:: \langle (S-z)^{-1} f, f \rangle

: is a Nevanlinna function.

  • If M(z) and N(z) are both Nevanlinna functions, then the composition M \big( N(z) \big) is a Nevanlinna function as well.

Importance in operator theory

Nevanlinna functions appear in the study of Operator monotone functions.

References

General

References

  1. A real number is not considered to be in the upper half-plane.
  2. [[Louis de Branges]]. (1968). "Hilbert Spaces of Entire Functions". Prentice-Hall.

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