Meromorphic Function
A
meromorphic function
is a function that is
holomorphic
on an
open subset
of the
complex number plane
C
(or on some other connected
Riemann surface
)
except
at points in a set of isolated
poles
, which are certain well-behaved
singularities
. Every meromorphic function can be expressed as the ratio between two
holomorphic functions
(with the denominator not constant 0): the poles then occur at the zeroes of the denominator. Examples of meromorphic functions are all
rational functions
such as
f
(
z
) = (
z
3
− 2
z
+ 1)/(
z
5
+ 3
z
− 1), the functions
f
(
z
) = exp(
z
)/
z
and
f
(
z
) = sin(
z
)/(
z
− 1)
2
as well as the
gamma function
and the
Riemann zeta function
. The functions
f
(
z
) = ln(
z
) and
f
(
z
) = exp(1/
z
) are not meromorphic. By using
analytic continuation
to eliminate
removable singularities
, meromorphic functions can be added, subtracted, multiplied, and the quotient
f
/
g
can be formed unless
g
(
z
) = 0 for all
z
. Thus, the meromorphic functions form a
field
, in fact a
field extension
of the
complex numbers
. In the language of
Riemann surfaces
, a meromorphic function is the same as a holomorphic function that maps to the
Riemann sphere
and which is not constant ∞. The poles correspond to those complex numbers which are mapped to ∞.
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