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 {\displaystyle \,{\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 + ∫ R ( 1 λ − z − λ 1 + λ 2 ) d μ ( λ ) , z ∈ H , {\displaystyle N(z)=C+Dz+\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, H {\displaystyle {\mathcal {H}}} is the upper half-plane, and μ is a Borel measure on ℝ satisfying the growth condition
∫ R d μ ( λ ) 1 + λ 2 < ∞ . {\displaystyle \int _{\mathbb {R} }{\frac {\operatorname {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 = ℜ ( N ( i ) ) and D = lim y → ∞ N ( i y ) i y {\displaystyle C=\Re {\big (}N(i){\big )}\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):
μ ( ( λ 1 , λ 2 ] ) = lim δ → 0 lim ε → 0 1 π ∫ λ 1 + δ λ 2 + δ ℑ ( N ( λ + i ε ) ) d λ . {\displaystyle \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 {\displaystyle z} can be replaced by z − a {\displaystyle z-a} for any real number a {\displaystyle a} .)
z p with 0 ≤ p ≤ 1 {\displaystyle z^{p}{\text{ with }}0\leq p\leq 1}
− z p with − 1 ≤ p ≤ 0 {\displaystyle -z^{p}{\text{ with }}-1\leq p\leq 0}
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