Identity theorem for holomorphic functions

The identity theorem for holomorphic functions is an important theorem in function theory . It states that due to the strong restrictions on holomorphic functions, the local equality of two such functions is often sufficient to infer them globally.

Identity set

Let and be holomorphic functions on a neighborhood of and be an accumulation point of the coincidence set , then there exists a neighborhood of with on whole . ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle U}$ ${\displaystyle z_{0}}$ ${\displaystyle z_{0}}$ ${\displaystyle \{z\in U\mid f(z)=g(z)\}}$ ${\displaystyle V}$ ${\displaystyle z_{0}}$ ${\displaystyle f(z)=g(z)}$ ${\displaystyle V}$

Identity set for areas

For domains, especially since they are connected, the statement of the identity theorem can easily be tightened and is also called the fundamental theorem of function theory.

statement

Let be a domain and and in this domain holomorphic functions. Then the following statements are equivalent: ${\displaystyle G\subseteq \mathbb {C} }$ ${\displaystyle f}$ ${\displaystyle g}$

1. ${\displaystyle f(z)=g(z)}$for all , that is, the functions are the same in the whole area. ${\displaystyle z\in G}$
2. The coincidence set has an accumulation point in . ${\displaystyle \{z\in G\mid f(z)=g(z)\}}$ ${\displaystyle G}$
3. There is one such that for all , that is, in one point of the functions and all their derivatives agree. ${\displaystyle z\in G}$ ${\displaystyle f^{(n)}(z)=g^{(n)}(z)}$ ${\displaystyle n\in \mathbb {N} _{0}}$ ${\displaystyle G}$

proof

Holomorphic functions are analytical ; H. locally represented by their Taylor series.

• 2. follows immediately from 1., since every point in is an accumulation point of . ${\displaystyle G}$ ${\displaystyle G}$
• 3. follows from 2. by evidence of contradiction. Let be an accumulation point of the coincidence set. We can assume without restriction . Assumption: There is a with . Be the smallest such. Then in a neighborhood the zero is with and the zero set of is equal to the concidence set, since is continuous. In particular, in contradiction to the minimality of . ${\displaystyle z_{0}}$ ${\displaystyle z_{0}=0}$ ${\displaystyle n\in \mathbb {N} _{0}}$ ${\displaystyle f^{(n)}(0)\neq g^{(n)}(0)}$ ${\displaystyle N}$ ${\displaystyle f(z)-g(z)=z^{N}h(z)}$ ${\displaystyle h(z)=\sum \limits _{n=0}^{\infty }{\frac {f^{(N+n)}(0)-g^{(N+n)}(0)}{(N+n)!}}z^{n}}$ ${\displaystyle h}$ ${\displaystyle h}$ ${\displaystyle 0=h(0)={\frac {f^{(N)}(0)-g^{(N)}(0)}{N!}}}$ ${\displaystyle N}$
• 1. follows from the third because coherently is. It suffices to show that the set is non-empty, open and closed in . The former is, by assumption, the latter is clear, as is the as steady archetypes of the agreed quantity are complete again and the average closed sets is complete again. Finally, it is open: Is , then as an analytic function in a neighborhood of equal to its Taylor series , i.e. , identically zero. This environment also belongs to . ${\displaystyle G}$ ${\displaystyle A=\{z\in G|\forall n\in \mathbb {N} _{0}:f^{(n)}(z)=g^{(n)}(z)\}}$ ${\displaystyle G}$ ${\displaystyle \textstyle A=\bigcap _{n\in \mathbb {N} _{0}}A_{n}}$ ${\displaystyle A_{n}=\{z\in G|f^{(n)}(z)=g^{(n)}(z)\}=(f^{(n)}-g^{(n)})^{-1}(\{0\})}$ ${\displaystyle \{0\}}$ ${\displaystyle A}$ ${\displaystyle z\in A}$ ${\displaystyle f-g}$ ${\displaystyle z}$ ${\displaystyle A}$

example

With the second point it is essential that the cluster point is in the area and not on its edge. Consider the following example: ${\displaystyle G}$

The function is holomorphic on , the sequence lies in it and converges to 0. So 0 is an accumulation point of the sequence and it applies , but of course also applies . So on the set of (which has the accumulation point 0) agrees with the null function , but obviously not completely . ${\displaystyle \sin({\tfrac {1}{z}})}$ ${\displaystyle \mathbb {C} \setminus \{0\}}$ ${\displaystyle z_{n}={\tfrac {1}{n\pi }}}$ ${\displaystyle (z_{n})}$ ${\displaystyle \sin({\tfrac {1}{z_{n}}})=\sin(n\pi )=0}$ ${\displaystyle \sin({\tfrac {1}{z}})\not \equiv 0}$ ${\displaystyle \sin({\tfrac {1}{z}})}$ ${\displaystyle z_{n}}$ ${\displaystyle \mathbb {C} \setminus \{0\}}$

Inferences

Clear continuability of real functions

An essential consequence of the identity theorem is the unambiguous continuability of real functions :
If a real function can be holomorphically extended to the complex level (this is generally not possible), then this continuation is unambiguous.
The complex sine is therefore really the only holomorphic continuation of the real sine. In particular, the addition theorems also apply to the complex sine.

Special case g = 0

A special case of identity set for areas which is very often used, results with : Has the zero set of in an area has an accumulation point, applies to the whole . ${\displaystyle g=0}$
${\displaystyle f}$ ${\displaystyle G}$ ${\displaystyle f\equiv 0}$ ${\displaystyle G}$

Zero divisor of the ring of holomorphic functions

The ring of holomorphic functions in a domain is zero-divisor-free , i. H. from always follows or . For this, be holomorphic with and . Then there is a point in and an environment of with for everyone . But then applies , and thus after the special case. ${\displaystyle G}$ ${\displaystyle fg\equiv 0}$ ${\displaystyle f\equiv 0}$ ${\displaystyle g\equiv 0}$ ${\displaystyle f,g\colon G\to \mathbb {C} }$ ${\displaystyle f\not \equiv 0}$ ${\displaystyle fg\equiv 0}$ ${\displaystyle z_{0}}$ ${\displaystyle G}$ ${\displaystyle U}$ ${\displaystyle z_{0}}$ ${\displaystyle f(z)\neq 0}$ ${\displaystyle z\in U}$ ${\displaystyle g|_{U}\equiv 0}$ ${\displaystyle g\equiv 0}$

Identity theorem for power series

Be there

${\displaystyle \sum _{\nu =0}^{\infty }a_{\nu }(x-x_{0})^{\nu }}$  and   ${\displaystyle \sum _{\nu =0}^{\infty }b_{\nu }(x-x_{0})^{\nu }}$

two power series around the same expansion point with real or complex coefficients or and a common nontrivial convergence domain . If the values ​​for all of a sequence ( ) agree with and , then the series are identical, i.e. H. ${\displaystyle x_{0}}$ ${\displaystyle a_{\nu }}$ ${\displaystyle b_{\nu }}$ ${\displaystyle I}$ ${\displaystyle x_{n}}$ ${\displaystyle x_{n}}$ ${\displaystyle I\ni x_{n}\neq x_{0}}$ ${\displaystyle x_{n}\rightarrow x_{0}}$

${\displaystyle a_{\nu }=b_{\nu }\qquad (\nu \in \mathbb {N} _{0})}$

The proof is obtained inductively by differentiating a power series from the identity theorem for holomorphic functions.

Identity theorem for polynomials

The identity theorem for polynomials is a special case of the identity theorem for power series and is the basis for the coefficient comparison .

Several variables

In the function theory of several variables, sets of zeros with accumulation points occur. The holomorphic function vanishes on the straight line without being the null function itself. In the function theory of several variables, an identity theorem applies in the following form: ${\displaystyle \mathbb {C} ^{2}\rightarrow \mathbb {C} ,(z_{1},z_{2})\mapsto z_{1}-z_{2}}$ ${\displaystyle \{(z,z)\mid z\in \mathbb {C} \}}$

• If there is a domain and there are two holomorphic functions that match on a non-empty open subset of , then is on whole . ${\displaystyle G\subset \mathbb {C} ^{n}}$ ${\displaystyle f,g:G\rightarrow \mathbb {C} }$ ${\displaystyle G}$ ${\displaystyle f=g}$ ${\displaystyle G}$

literature

• E. Freitag, R. Busam: Function theory 1 . 4th edition. Springer-Verlag, ISBN 3-540-67641-4 .

Individual evidence

1. ↑ ^{a b c} Guido Walz (Ed.): Lexicon of Mathematics . tape 2 (Eig-Inn). Springer Spektrum Verlag, Mannheim 2017, ISBN 978-3-662-53503-5 , p. 476 , doi : 10.1007 / 978-3-662-53504-2 .
2. ↑ Guido Walz (Ed.): Lexicon of Mathematics . tape 3 (Inp-Mon). Springer Spektrum Verlag, Mannheim 2017, ISBN 978-3-662-53501-1 , p. 131 , doi : 10.1007 / 978-3-662-53502-8 .
3. ^ Gunning - Rossi : Analytic functions of several complex variables . Prentice-Hall 1965, chap. IA, Theorem 6 (Identity Theorem)