Fundamental normality test
In complex analysis, a mathematical discipline, the fundamental normality test gives sufficient conditions to test the normality of a family of analytic functions. It is another name for the stronger version of Montel's theorem.
Statement
Let [math]\displaystyle{ \mathcal{F} }[/math] be a family of analytic functions defined on a domain [math]\displaystyle{ \Omega }[/math]. If there are two fixed complex numbers a and b such that for all ƒ ∈ [math]\displaystyle{ \mathcal{F} }[/math] and all x ∊ [math]\displaystyle{ \Omega }[/math], f(x) ∉ {a, b}, then [math]\displaystyle{ \mathcal{F} }[/math] is a normal family on [math]\displaystyle{ \Omega }[/math].
The proof relies on properties of the elliptic modular function and can be found here: J. L. Schiff (1993). Normal Families. Springer-Verlag. ISBN 0-387-97967-0.
See also
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