Hurwitz's theorem (complex analysis)

The set of Hurwitz (after Adolf Hurwitz named) is a set of function theory.

The set of Hurwitz

Let be a sequence of functions which are holomorphic in a field and the sequence converges compactly. Also is the number of zeros of the functions limited (as an index ) by M, then:

The limit function has a maximum of M zeros on or on. (The zero function)

Conclusion

Let be a sequence of functions that are holomorphic in a field and are injective and the sequence converges to compact.

Then is injective on or constant on.