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, but is not necessarily injective or surjective. Functions with this property are sometimes also known as Herglotz, Pick or R functions.
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 , where C is a real constant, D is a non-negative constant and μ is a Borel measure on R satisfying the growth condition
∫ R d μ ( λ ) 1 + λ 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 = R e ( N ( i ) ) and D = lim y → ∞ 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):
μ ( ( λ 1 , λ 2 ] ) = lim δ → 0 lim ε → 0 1 π ∫ λ 1 + δ λ 2 + δ I m ( N ( λ + i ε ) ) d λ . A very similar representation of functions is also called the Poisson representation.
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 . )A Möbius transformationis a Nevanlinna function if (but not only if)
a ∗ d − b c ∗ is a positive real number and
I m ( b ∗ d ) = I m ( 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:
i z + i − 2 z + 1 + i 1 + i + z and i + 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 functionis a Nevanlinna function.
If M(z) and N(z) are Nevanlinna functions, then the composition M(N(z)) is a Nevanlinna function as well.