meromorphic

Etymology

mero- + -morphic

adj

1. (complex analysis, of a function) That is the ratio of two holomorphic functions (and so possibly infinite at a discrete set of points).
   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. 1993, Joel L. Schiff, Normal Families, Springer, page 71
   Note that such a transformation is holomorphic at the origin, but is essentially singular at infinity. However, since T(z) commutes with A(z), the transformed system has coefficient matrix A(z)-zq'(z)I and hence is again meromorphic at infinity. 2000, Werner Balser, Formal Power Series and Linear Systems of Meromorphic Ordinary Differential Equations, Springer, page 39
   {{quote-book|en|year=2012|author=Marius van der Put; Michael F. Singer|title=Galois Theory of Linear Differential Equations|pageurl=https://books.google.com.au/books?id=cuj1CAAAQBAJ&pg=PA147&dq=%22meromorphic+transformation%22&hl=en&sa=X&ved=0ahUKEwj_lOqAkLXWAhUIEbwKHVW_DgwQ6AEIUzAH#v=onepage&q=%22meromorphic%20transformation%22&f=false|page=147|publisher=Springer