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Identity theorem

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From Wikipedia, the free encyclopedia

In complex analysis, the identity theorem states that if any two functions, holomorphic on some domain $D$ of the complex plane, are equal throughout some neighbourhood of a point in $D$, then the functions are identical throughout $D$. This is equivalent to saying that if such a holomorphic function is zero throughout some neighbourhood of a point in $D$, then it is zero everywhere in $D$, which is best summarised as the concept that "zeroes of holomorphic functions are isolated". The theorem is also referred to as the uniqueness theorem.

The theorem is established on the principal that holomorphic functions are analytic, and can therefore be expanded as a convergent series called the Taylor series. It can be extended to include another condition; that if all derivatives of two holomorphic functions are equal at some point, then they are identical through the domain. The theorem does not hold in general for real-differentiable functions, which are not necessarily holomorphic.

Theorem

Let $D$ be a domain in $\mathbb {C}$ and let $f,g:D\to \mathbb {C}$ be holomorphic functions. Then the following are equivalent:

1. $f\equiv g$ on $D$.
2. The set $\{z\in D:f(z)=g(z)\}$ has an accumulation point in $D$.
3. There exists $a\in D$ such that $f^{(k)}(a)=g^{(k)}(a)$ for all $k\in \mathbb {N} \cup \{0\}$.

This is the most generalised version of the identity theorem. Here $\mathbb {C}$ and $\mathbb {N}$ represent the sets of complex and natural numbers respectively, and $f^{(k)}(a)$ is the $k$^th derivative of $f$ evaluated at $a$. For the purposes of this theorem a domain is defined as an open connected subset of the complex plane.

References

• Priestley, H. A. (2003). Introduction to Complex Analysis (2nd ed.). Oxford, UK: Oxford University Press. pp. 179–180. ISBN 0-19-852562-1.
• Ablowitz, Mark J.; Fokas, Athanasios S. (1997). Complex Variables: Introduction and Applications. Cambridge, UK: Cambridge University Press. p. 123. ISBN 0-521-48058-2.