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