Identity theorem

The identity theorem for holomorphic functions is an important set of function theory. He says that because of the severe restrictions on holomorphic functions often the local equality of two such functions is sufficient to conclude this globally.

Identity theorem

Be and holomorphic functions on a neighborhood of and be a cluster point of the coincidence set, then there exists a neighborhood of with completely.

Identity theorem for areas

For areas in particular because they are contiguous to the statement of identity set can easily worsen.

Statement

Be a territory and and holomorphic functions in this area. Then the following statements are equivalent:

• For all, that is, the functions are the same in the whole area.
• The coincidence set has an accumulation point in.
• There is a such that for all, that is a point of vote functions and all their derivatives match.

Example

On the second point, it is essential that the limit point is in the field and not on its edge. Consider the following example:

The function is holomorphic on, the result is that converges to 0 and so 0 is a cluster point of the sequence and it is, but of course also applies. So true on the set of ( 0 provides that the accumulation point ) correspond to the zero function, but obviously not at all.

Conclusions

An important consequence of the identity theorem is the unique continuability real functions:

Can you build a real-valued function holomorphic on the complex plane to continue ( this is not possible in general), so this continuation is unique. The complex sine is therefore really the only holomorphic continuation of the real sine. In particular, the addition theorems for the complex sine apply.

A special case of the identity theorem for areas, which is very often used, results with:

If the set of zeros of in an area has an accumulation point, applies to the whole.

The ring of holomorphic functions in a field is zero divisors, ie follows or always. Be this holomorphic with and. Then there is a point in and around with for all. But then applies, and thus according to the special situation.