Fundamental normality test
Encyclopedia
In complex analysis
, a mathematical discipline, the fundamental normality test gives sufficient conditions to test the normality of a family of analytic functions.
be a family of analytic functions defined on a domain 
. If there are two fixed complex numbers a and b that are omitted from the range of every ƒ ∈ 
, then 
is a normal family
on
.
The proof relies on properties of the elliptic modular function and can be found here:
Complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
, a mathematical discipline, the fundamental normality test gives sufficient conditions to test the normality of a family of analytic functions.
Statement of theorem
Let be a family of analytic functions defined on a domain . If there are two fixed complex numbers a and b that are omitted from the range of every ƒ ∈ , then is a normal familyNormal family
In mathematics, with special application to complex analysis, a normal family is a pre-compact family of continuous functions. Informally, this means that the functions in the family are not exceedingly numerous or widely spread out; rather, they stick together in a relatively "compact" manner...
on
.The proof relies on properties of the elliptic modular function and can be found here: