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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 or ), if f = g on some , where has an accumulation point in D, then f = g on D.
Identity Theorem Identity Theorem: Let D ˆC be a domain and f : D !C is analytic. If there exists an in nite sequence fz kgˆD, such that f(z k) = 0; 8k 2N and z k!z 0 2D, f(z) = 0 for all z 2D: Proof. Case I: If D = fz 2C : jz z 0j<rgthen f(z) = X1 n=0 a n(z z 0) n; for all z 2D: We will show that fn(z 0) = 0 for all n:If possible assume that ...
If U is a domain, and f, g are two real-analytic functions defined on U, and if V ⊂ U is a nonempty open set with f |V ≡ g|V, then f ≡ g. If the domain is one-dimensional (an interval in R), then it suffices that f |M ≡ g|M for some M ⊂ U that has an accumulation point in U. Claim.
This theorem provides a very powerful and useful tool to test whether two analytic functions, whose values coincide in some points, are indeed the same function. Namely, unless the points in which they are equal are isolated, they are the same function.
Prove the identity theorem for real-analytic functions. That is, if \(U \subset \mathbb{R}^n\) is a domain, \(f \colon U \to \mathbb{R}\) a real-analytic function and \(f\) is zero on a nonempty open subset of \(U\), then \(f\) is identically zero.
The identity theorem significantly impacts complex analysis by providing a foundational principle for understanding analytic functions and their properties. Its implications extend into fields like physics and engineering, where complex functions model real-world phenomena.
Remark 3.3 The significance of the Identity Theorem is that an analytic function on a connected open GˆCis determined on all of Gby its behaviour near a single point. Thus if an analytic function is given on one part of Gby a formula like f(z) = 1
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not generally hold for real analytic functions.
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 or ), if f = g on some , where has an accumulation point in D, then f = g on D.
Theorem 1.2. If F1 : D0 ! C and F2 : D0 ! C are two analytic continuations of f : D ! C then F1(z) = F2(z) for all z 2 D0. Proof. Notice that F1(z) F2(z) vanishes on D D0, which is an open set. By the identity theorem F1 and F2 must be the same. Remark 1.3.