# persistence of analytic relations

The principle of persistence of analytic relations states that any algebraic relation between several analytic functions^{} which holds on a sufficiently large set also holds wherever the functions are defined.

A more explicit statement of this principle is as follows: Let ${f}_{1},{f}_{2},\mathrm{\dots}{f}_{n}$ be complex analytic functions. Suppose that there exists an open set $D$ on which all these functions are defined and that there exists a polynomial $p$ of $n$ variables such that $p({f}_{1}(z),{f}_{2}(z),\mathrm{\dots},{f}_{n}(z))=0$ whenever $z$ lies in a subset $X$ of $D$ which has a limit point^{} in $D$. Then $p({f}_{1}(z),{f}_{2}(z),\mathrm{\dots}{f}_{n}(z))=0$ for all $z\in D$.

This fact is a simple consequence of the rigidity theorem for analytic functions. If ${f}_{1},{f}_{2},\mathrm{\dots}{f}_{n}$ are all analytic in $D$, then $p({f}_{1}(z),{f}_{2}(z),\mathrm{\dots}{f}_{n}(z))$ is also analytic in $D$. Hence, if $p({f}_{1}(z),{f}_{2}(z),\mathrm{\dots},{f}_{n}(z))=0$ when $z$ in $X$, then $p({f}_{1}(z),{f}_{2}(z),\mathrm{\dots},{f}_{n}(z))=0$ for all $z\in D$.

This principle is very useful in establishing identites involving analytic functions because it means that it suffices to show that the identity holds on a small subset. For instance, from the fact that the familiar identity ${\mathrm{sin}}^{2}x+{\mathrm{cos}}^{2}x=1$ holds for all real $x$, it automatically holds for all complex values of $x$. This principle also means that it is unnecessary to specify for which values of the variable an algebraic relation between analytic functions holds since, if such a relation holds, it will hold for all values for which the functions appearing in the relation are defined.

Title | persistence of analytic relations |
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Canonical name | PersistenceOfAnalyticRelations |

Date of creation | 2013-03-22 14:44:17 |

Last modified on | 2013-03-22 14:44:17 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 9 |

Author | rspuzio (6075) |

Entry type | Theorem |

Classification | msc 30A99 |

Related topic | ComplexSineAndCosine |