Nevanlinna function

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See also Nevanlinna theory

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. They map the upper half-plane to itself (or to a real constant), but are 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

 N(z) = C + Dz + \int_{\mathbb{R}} \left(\frac{1}{\lambda - z} - \frac{\lambda}{1+\lambda^2} \right) d\mu(\lambda), \quad z\in\mathbb{H},

where C is a real constant, D is a non-negative constant and μ is a Borel measure on R satisfying the growth condition

 \int_{\mathbb{R}} \frac{d\mu(\lambda)}{1+\lambda^2} < \infty.

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 = \mathrm{Re}(N(i)) \qquad\text{and}\qquad D = \lim_{y\rightarrow\infty} \frac{N(iy)}{iy}

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((\lambda_1,\lambda_2]) = \lim_{\delta\rightarrow0} \lim_{\varepsilon\rightarrow 0} \frac{1}{\pi} \int_{\lambda_1+\delta}^{\lambda_2+\delta} \mathrm{Im}(N(\lambda+i\varepsilon))d\lambda.

A very similar representation of functions is also called the Poisson representation.[1]

Examples[edit]

  • Some elementary examples of Nevanlinna functions follow (with appropriately chosen branch cuts in the first three). (z can be replaced by z-a for some real number a.)
z^p\text{ with }0\le p\le 1
-z^p\text{ with }-1\le p\le 0
The above examples can also be rotated to some extent around the origin, such as i(z/i)^p\text{ with }-1\le p\le 1.
\ln(z)
\tan(z) (an example that is not injective)
z \mapsto  \frac{az+b}{cz+d}
is a Nevanlinna function if (but not only if) a^\ast d - bc^\ast is a positive real number and \mathrm{Im}(b^\ast d) = \mathrm{Im}(a^\ast 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{iz+i-2}{z+1+i}
 \langle(S-z)^{-1} f,f\rangle
is a Nevanlinna function.
  • If M and N are non-constant Nevanlinna functions, then their composition is a Nevanlinna function as well.

References[edit]

  1. ^ 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.