A family of meromorphic functions on a domain is a normal family if every sequence from has a subsequence converging uniformly on every compact subset, measured in the chordal metric on the Riemann sphere. The limit is a meromorphic function or the constant infinity. For holomorphic functions, a finite limit is holomorphic; a sequence may also tend locally uniformly to infinity.
The constant functions form a normal family, because any sequence has either a constant subsequence or a subsequence tending uniformly to infinity in the chordal metric. In contrast is not a normal family on any neighbourhood of zero. Every subsequence tends to infinity at every nonzero point but remains zero at zero; this pointwise pattern cannot be a continuous Riemann sphere valued limit.
For a rational map , the Fatou set is precisely where its iterations of a map form a normal family. Normality asks for convergent subsequences, so it allows stable rotations as well as convergence to an attracting periodic point.[ref]
Dynamics in one complex variable: introductory lectures by John Milnor. Section 2, normal families and Montel theory
For a holomorphic normal family, limits are holomorphic or identically infinity. A uniform bound at one interior point excludes infinity on its connected domain. Every sequence therefore has a finite locally uniform subsequential limit. The Cauchy integral formula makes the derivatives converge locally uniformly too, so the derivative family is normal. Without the point bound this can fail: tends uniformly to infinity on the unit disk, whereas its derivatives are not normal at zero.
Suppose a family of meromorphic functions is a normal family on a neighbourhood of every point of a planar domain. Choose a countable cover by open discs with that property. From any sequence extract successive subsequences converging locally uniformly in the chordal metric on the first disc, second disc, and so on; take the diagonal subsequence argument.
The local Riemann sphere valued limits agree on overlaps, since the same subsequence converges there. They therefore define one continuous global limit. Locally it is meromorphic or identically infinity, so on a connected domain it is globally meromorphic or identically infinity. A finite cover of each compact subset transfers the local uniform estimates to that subset. Thus the original family is a normal family on the entire domain. The argument handles poles using the chordal metric; it does not require every local limit to be a finite holomorphic function.
If a bounded holomorphic function on the complex upper half-plane has as , the rescaled functions form a normal family. Every subsequential limit equals on the imaginary axis, hence everywhere by the identity theorem for holomorphic functions. The entire sequence converges locally uniformly to . For a nontangential sequence , put ; then remains in a fixed compact subset of the half-plane. This gives . The bounded example shows that unrestricted boundary convergence can fail.
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A family of meromorphic functions on a domain that omits three fixed points of the Riemann sphere is normal. For holomorphic functions with values in the complex plane, omitting two fixed values suffices.
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The term "normal family" typically refers to a family structure that aligns with widely accepted societal norms and expectations regarding family dynamics, roles, and relationships. However, the definition of what constitutes a "normal" family can vary greatly depending on cultural, social, and individual perspectives. In many Western cultures, a "normal family" often implies a nuclear family consisting of two parents (a mother and a father) and their biological children.