Meromorphic function: Difference between revisions

Revision as of 17:08, 16 June 2019

Revision as of 14:10, 17 June 2019

Line 4:

[[File:Gamma abs 3D.png|thumb|right|The [[gamma function]] is meromorphic in the whole complex plane.]]

Intuitively, a meromorphic function is a ratio of two well-behaved (holomorphic) functions. Such a function will still be well-behaved, except possibly at the points where the denominator of the fraction is zero. If the denominator has a zero at ''z'' and the numerator does not, then the value of the function will approach infinity; if both parts have a zero at ''z'', then one must compare the [[zero Multiplicity_(complex analysismathematics)#Multiplicity_of_a_root_of_a_polynomial|multiplicitiesmultiplicity]] of these zeros.

From an algebraic point of view, if ''D'' is [[connected set|connected]], then the set of meromorphic functions is the [[field of fractions]] of the [[integral ___domain]] of the set of holomorphic functions. This is analogous to the relationship between the [[rational number]]s and the [[integer]]s.