zero of a function
Suppose is a set and a complex (http://planetmath.org/Complex)-valued function . Then a zero of is an element such that . It is also said that vanishes at .
The zero set of is the set
Remark. When is a “simple” space, such as or a zero is also called a root. However, in pure mathematics and especially if is infinite, it seems to be customary to talk of zeroes and the zero set instead of roots.
Examples
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For any , define by . Then and if .
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Suppose is a polynomial (http://planetmath.org/Polynomial) of degree . Then has at most zeroes. That is, .
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If and are functions and , then
where is the function .
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For any , then
where is the defined .
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If and are both real-valued functions, then
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If is a topological space and is a function, then the support (http://planetmath.org/SupportOfFunction) of is given by:
Further, if is continuous, then is closed (http://planetmath.org/ClosedSet) in (assuming that is given the usual topology of the complex plane where is a closed set).
Title | zero of a function |
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Canonical name | ZeroOfAFunction |
Date of creation | 2013-03-22 14:00:58 |
Last modified on | 2013-03-22 14:00:58 |
Owner | mathcam (2727) |
Last modified by | mathcam (2727) |
Numerical id | 30 |
Author | mathcam (2727) |
Entry type | Definition |
Classification | msc 26E99 |
Synonym | zero |
Synonym | vanish |
Synonym | vanishes |
Related topic | SupportOfFunction |
Defines | zero set |