# Identitätssatz

Identity theorem In real analysis and complex analysis, Zweige der Mathematik, the identity theorem for analytic functions states: given functions f and g analytic on a domain D (open and connected subset of {Anzeigestil mathbb {R} } oder {Anzeigestil mathbb {C} } ), if f = g on some {displaystyle Ssubseteq D} , wo {Anzeigestil S} has an accumulation point, then f = g on D.

Thus an analytic function is completely determined by its values on a single open neighborhood in D, or even a countable subset of D (provided this contains a converging sequence). This is not true in general for real-differentiable functions, even infinitely real-differentiable functions. In comparison, analytic functions are a much more rigid notion. Informell, one sometimes summarizes the theorem by saying analytic functions are "hard" (as opposed to, sagen, continuous functions which are "soft").

The underpinning fact from which the theorem is established is the expandability of a holomorphic function into its Taylor series.

The connectedness assumption on the domain D is necessary. Zum Beispiel, if D consists of two disjoint open sets, {Anzeigestil f} kann sein {Anzeigestil 0} on one open set, und {Anzeigestil 1} on another, während {Anzeigestil g} ist {Anzeigestil 0} on one, und {Anzeigestil 2} on another.

Inhalt 1 Lemma 2 Nachweisen 3 Full characterisation 3.1 Claim 3.2 Nachweisen 4 Siehe auch 5 References Lemma If two holomorphic functions {Anzeigestil f} und {Anzeigestil g} on a domain D agree on a set S which has an accumulation point {Anzeigestil c} in {Anzeigestil D} , dann {displaystyle f=g} on a disk in {Anzeigestil D} centered at {Anzeigestil c} .

Um dies zu beweisen, it is enough to show that {Anzeigestil f^{(n)}(c)=g^{(n)}(c)} für alle {Anzeigestil ngeq 0} .

If this is not the case, Lassen {Anzeigestil m} be the smallest nonnegative integer with {Anzeigestil f^{(m)}(c)neq g^{(m)}(c)} . By holomorphy, we have the following Taylor series representation in some open neighborhood U of {Anzeigestil c} : {Anzeigestil {Start{ausgerichtet}(f-g)(z)&{}=(zc)^{m}cdot links[{frac {(f-g)^{(m)}(c)}{m!}}+{frac {(zc)cdot (f-g)^{(m+1)}(c)}{(m+1)!}}+cdots richtig]\[6Punkt]&{}=(zc)^{m}cdot h(z).Ende{ausgerichtet}}} By continuity, {Anzeigestil h} is non-zero in some small open disk {Anzeigestil B} um {Anzeigestil c} . But then {displaystyle f-gneq 0} on the punctured set {displaystyle B-{c}} . This contradicts the assumption that {Anzeigestil c} is an accumulation point of {Anzeigestil {f=g}} .

This lemma shows that for a complex number {displaystyle ain mathbb {C} } , the fiber {Anzeigestil f^{-1}(a)} is a discrete (and therefore countable) einstellen, unless {displaystyle fequiv a} .

Proof Define the set on which {Anzeigestil f} und {Anzeigestil g} have the same Taylor expansion: {displaystyle S=left{zin Dmid f^{(k)}(z)=g^{(k)}(z){Text{ für alle }}kgeq 0right}=bigcap _{k=0}^{unendlich }links{zin Dmid left(f^{(k)}-g^{(k)}Rechts)(z)=0right}.} We'll show {Anzeigestil S} is nonempty, open, and closed. Then by connectedness of {Anzeigestil D} , {Anzeigestil S} must be all of {Anzeigestil D} , which implies {displaystyle f=g} an {displaystyle S=D} .

Durch das Lemma, {displaystyle f=g} in a disk centered at {Anzeigestil c} in {Anzeigestil D} , they have the same Taylor series at {Anzeigestil c} , Also {displaystyle cin S} , {Anzeigestil S} is nonempty.

Wie {Anzeigestil f} und {Anzeigestil g} are holomorphic on {Anzeigestil D} , {displaystyle forall win S} , the Taylor series of {Anzeigestil f} und {Anzeigestil g} bei {Anzeigestil m} have non-zero radius of convergence. Deswegen, the open disk {Anzeigestil B_{r}(w)} also lies in {Anzeigestil S} für einige {Anzeigestil r} . So {Anzeigestil S} is open.

By holomorphy of {Anzeigestil f} und {Anzeigestil g} , they have holomorphic derivatives, so all {Anzeigestil f^{(n)},g^{(n)}} are continuous. Das bedeutet, dass {Anzeigestil {zin Dmid (f^{(k)}-g^{(k)})(z)=0}} is closed for all {Anzeigestil k} . {Anzeigestil S} is an intersection of closed sets, so it's closed.

Full characterisation Since the Identity Theorem is concerned with the equality of two holomorphic functions, we can simply consider the difference (which remains holomorphic) and can simply characterise when a holomorphic function is identically {textstyle 0} . The following result can be found in.[1] Claim Let {textstyle Gsubseteq mathbb {C} } denote a non-empty, connected open subset of the complex plane. Zum {textstyle h:Gto mathbb {C} } die folgenden sind äquivalent.

{textstyle hequiv 0} an {textstyle G} ; der Satz {textstyle G_{0}={zin Gmid h(z)=0}} contains an accumulation point, {textstyle z_{0}} ; der Satz {textstyle G_{Ast }=bigcap _{nin mathbb {N} _{0}}G_{n}} is non-empty, wo {textstyle G_{n}:={zin Gmid h^{(n)}(z)=0}} . Proof The directions (1 {textstyle Rightarrow } 2) und (1 {textstyle Rightarrow } 3) hold trivially.

Zum (3 {textstyle Rightarrow } 1), by connectedness of {textstyle G} it suffices to prove that the non-empty subset, {textstyle G_{Ast }subseteq G} , is clopen (since a topological space is connected if and only if it has no proper clopen subsets). Since holomorphic functions are infinitely differentiable, d.h. {textstyle hin C^{unendlich }(G)} , it is clear that {textstyle G_{Ast }} is closed. To show openness, consider some {textstyle uin G_{Ast }} . Consider an open ball {textstyle Usubseteq G} enthält {textstyle u} , in which {textstyle h} has a convergent Taylor-series expansion centered on {textstyle u} . By virtue of {textstyle uin G_{Ast }} , all coefficients of this series are {textstyle 0} , whence {textstyle hequiv 0} an {textstyle U} . It follows that all {textstyle n} -th derivatives of {textstyle h} sind {textstyle 0} an {textstyle U} , whence {textstyle Usubseteq G_{Ast }} . So each {textstyle uin G_{Ast }} lies in the interior of {textstyle G_{Ast }} .

Towards (2 {textstyle Rightarrow } 3), fix an accumulation point {textstyle z_{0}in G_{0}} . We now prove directly by induction that {textstyle z_{0}in G_{n}} für jeden {textstyle nin mathbb {N} _{0}} . To this end let {textstyle rin (0,unendlich )} be strictly smaller than the convergence radius of the power series expansion of {textstyle h} um {textstyle z_{0}} , gegeben von {textstyle sum _{kin mathbb {N} _{0}}{frac {h^{(k)}(z_{0})}{k!}}(z-z_{0})^{k}} . Fix now some {textstyle ngeq 0} and assume that {textstyle z_{0}in G_{k}} für alle {textstyle k

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