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Nevanlinna function

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

- 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} - 1 + i + z and i + \operatorname{e}^{i z} 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 :: \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*. ::callout[type=info title="Wikipedia Source"] This article was imported from [Wikipedia](https://en.wikipedia.org/wiki/Nevanlinna_function) and is available under the [Creative Commons Attribution-ShareAlike 4.0 License](https://creativecommons.org/licenses/by-sa/4.0/). Content has been adapted to SurfDoc format. Original contributors can be found on the [article history page](https://en.wikipedia.org/wiki/Nevanlinna_function?action=history). ::

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