# meromorphic

## English

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### Etymology

From mero- +‎ -morphic.

meromorphic (not comparable)

1. (complex analysis, of a function) That is the ratio of two holomorphic functions (and so possibly infinite at a discrete set of points).
• 1993, Joel L. Schiff, Normal Families, Springer, page 71:
Normal families of meromorphic functions are most naturally studied using the spherical metric (§1.2), an approach initiated by Ostrowski [1926]. Some results for meromorphic functions, such as the FNT, are immediate extensions from the analytic case, whereas others, such as Landau's or Julia's theorem are set in a much broader context than their analytic counterparts. Normality criteria pertinent to families of meromorphic functions, such as Marty's theorem, have not yet been encountered.
• 2000, Werner Balser, Formal Power Series and Linear Systems of Meromorphic Ordinary Differential Equations, Springer, page 39:
Note that such a transformation is holomorphic at the origin, but is essentially singular at infinity. However, since ${\displaystyle T(z)}$  commutes with ${\displaystyle A(z)}$ , the transformed system has coefficient matrix ${\displaystyle A(z)-zq'(z)I}$  and hence is again meromorphic at infinity.
• 2012, Marius van der Put, Michael F. Singer, Galois Theory of Linear Differential Equations, Springer, page 147:
A point ${\displaystyle p\in P^{1}}$  is singular for ${\displaystyle \textstyle {\frac {d}{dz}}-A}$  if the equation cannot be made regular at ${\displaystyle p}$  with a local meromorphic transformation.