Identity theorem explained

In real analysis and complex analysis, branches of mathematics, the identity theorem for analytic functions states: given functions f and g analytic on a domain D (open and connected subset of

R or C ), if f = g on some S \subseteq D , where S has an accumulation point in D, then f = g on D.^[1]

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 together with its limit). 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. Informally, one sometimes summarizes the theorem by saying analytic functions are "hard" (as opposed to, say, 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. For example, if D consists of two disjoint open sets,

f can be 0 on one open set, and 1 on another, while g is 0 on one, and 2 on another.

Lemma

If two holomorphic functions

f and g on a domain D agree on a set S which has an accumulation point c in D , then f = g on a disk in D centered at c .

To prove this, it is enough to show that

f ^{(n)} (c) = g ^{(n)} (c) for all n\geq 0 .

If this is not the case, let

m be the smallest nonnegative integer with f ^{(m)} (c)\ne g ^{(m)} (c) . By holomorphy, we have the following Taylor series representation in some open neighborhood U of c :\begin{align} (f - g)(z) &{} = (z - c) ^{m} ⋅ \left[ ((f - g) ^{(m)} (c))/(m!) + ((z - c) ⋅ (f - g) ^{(m+1)} (c))/((m +1 )!) + … \right] \\[ 6 pt] &{} = (z - c) ^{m} ⋅ h(z). \end{align}

By continuity,

h is non-zero in some small open disk B around c . But then f - g ≠ 0 on the punctured set B - \{c\} . This contradicts the assumption that c is an accumulation point of \{f = g\} .

This lemma shows that for a complex number

a \in C , the fiber f ^{-1} (a) is a discrete (and therefore countable) set, unless f \equiv a .

Proof

Define the set on which

f and g have the same Taylor expansion:$$S\; =\; \backslash left\backslash \; =\; \backslash bigcap\_^\backslash infty\; \backslash left\backslash .$$

We'll show

S is nonempty, open, and closed. Then by connectedness of D , S must be all of D , which implies f = g on S = D .

By the lemma,

f = g in a disk centered at c in D , they have the same Taylor series at c , so c\in S , S is nonempty.

As

f and g are holomorphic on D , \forall w\in S , the Taylor series of f and g at w have non-zero radius of convergence. Therefore, the open disk B _{r(w)} also lies in S for some r . So S is open.

By holomorphy of

f and g , they have holomorphic derivatives, so all f ^{(n)} , g ^{(n)} are continuous. This means that \{z \in D \mid (f ^{(k)} - g ^{(k)} )(z) = 0 \} is closed for all k . 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 $0$. The following result can be found in.^[2]

Claim

Let $G\backslash subseteq\backslash mathbb$ denote a non-empty, connected open subset of the complex plane.For $h:G\backslash to\backslash mathbb$ the following are equivalent.

1. $h\backslash equiv\; 0$ on $G$;
2. the set $G\_=\backslash$ contains an accumulation point, $z\_$;
3. the set $G\_=\backslash bigcap\_\; G\_$ is non-empty, where $G\_\; :=\; \backslash$.

Proof

The directions (1 $\backslash Rightarrow$ 2) and (1 $\backslash Rightarrow$ 3) hold trivially.

For (3 $\backslash Rightarrow$ 1), by connectedness of $G$ it suffices to prove that the non-empty subset, $G\_\backslash subseteq\; G$, is clopen (since a topological space is connected if and only if it has no proper clopen subsets).Since holomorphic functions are infinitely differentiable, i.e. $h\backslash in\; C^(G)$, it is clear that $G\_$ is closed. To show openness, consider some $u\; \backslash in\; G\_$.Consider an open ball $U\backslash subseteq\; G$ containing $u$, in which $h$ has a convergent Taylor-series expansion centered on $u$.By virtue of $u\backslash in\; G\_$, all coefficients of this series are $0$, whence $h\backslash equiv\; 0$ on $U$.It follows that all $n$-th derivatives of $h$ are $0$ on $U$, whence $U\backslash subseteq\; G\_$.So each $u\backslash in\; G\_$ lies in the interior of $G\_$.

Towards (2 $\backslash Rightarrow$ 3), fix an accumulation point $z\_\backslash in\; G\_$.We now prove directly by induction that $z\_\backslash in\; G\_$ for each $n\; \backslash in\; \backslash N\_0$.To this end let $r\backslash in(0,\backslash infty)$ be strictly smaller than the convergence radius of the power series expansion of $h$ around $z\_$, given by $\backslash sum\_\backslash frac(z-z\_)^$. Fix now some $n\backslash geq\; 0$ and assume that $z\_\backslash in\; G\_$ for all . Then for $z\; \backslash in\; \backslash bar\_(z\_)\; \backslash setminus\; \backslash$ manipulation of the power series expansion yieldsNote that, since $r$ is smaller than radius of the power series, one can readily derive that the power series $R(\backslash cdot)$ is continuous and thus bounded on $\backslash bar\_(z\_)$.

Now, since $z\_$ is an accumulation point in $G\_$, there is a sequence of points $(z^)\_\backslash subseteq\; G\_\backslash cap\; B\_(z\_)\backslash setminus\backslash$ convergent to $z\_$.Since $h\backslash equiv\; 0$ on $G\_$ and since each $z^\backslash in\; G\_\backslash cap\; B\_(z\_)\backslash setminus\backslash$, the expression in yieldsBy the boundedness of $R(\backslash cdot)$ on $\backslash bar\_(z\_)$,it follows that $h^(z\_)=0$, whence $z\_\backslash in\; G\_$.Via induction the claim holds.Q.E.D.

See also

• Analytic continuation
• Identity theorem for Riemann surfaces

References

1. For real functions, see Book: Krantz, Steven G. . Harold R. . Parks . A Primer of Real Analytic Functions . Second . Birkhäuser . Boston . 2002 . 0-8176-4264-1 . Corollary 1.2.7 .
2. Book: Lexikon der Mathematik. de. 2. Springer Spektrum Verlag. Guido Walz. Mannheim. 2017. 476. 978-3-662-53503-5.

• Book: Ablowitz, Mark J.. Fokas A. S.. en. Complex variables: Introduction and applications. Cambridge University Press. 122. Cambridge, UK. 1997. 0-521-48058-2.