Zero function

Cosine zeros on the interval [-2π, 2π] (red dots)

The zero of a function in mathematics is an element from the domain of definition of a function in which it takes a zero value. For example, for a function $f$ ${\ displaystyle f}$ $f$ given by the formula

f(x)=x^{2}-6x+9\,.

{\ displaystyle f (x) = x ^ {2} -6x + 9 \ ,.}

f (x) = x ^ {2} -6x + 9 \ ,.

$x=3$ ${\ displaystyle x = 3}$ $x = 3$ is zero because

f(3)=3^{2}-6\cdot 3+9=0

{\ displaystyle f (3) = 3 ^ {2} -6 \ cdot 3 + 9 = 0}

f (3) = 3 ^ {2} -6 \ cdot 3 + 9 = 0

.

The concept of zeros of a function can be considered for any functions whose range of values contains a zero or zero element of the corresponding algebraic structure .

For the function of a real variable $f:\mathbb {R} \to \mathbb {R}$ ${\ displaystyle f: \ mathbb {R} \ to \ mathbb {R}}$ $f: \ mathbb {R} \ to \ mathbb {R}$ zeros are the values at which the function graph crosses the abscissa axis .

Finding function zeros often requires the use of numerical methods (for example, Newton's method , gradient methods ).

One of the unsolved mathematical problems is finding the zeros of the Riemann zeta function .

Polynomial Root

The problem of finding the zeros of a quadratic trinomial led to the concept of complex numbers .

The basic theorem of algebra states that every polynomial of degree n has n complex roots, given their multiplicity. Complex roots always come in conjugate pairs. Each odd degree polynomial has at least one real root. The connection between the roots of the polynomial and its coefficients is established by the Vieta theorem .

Comprehensive Analysis

A simple zero analytic in some area $G\subset \mathbb {C}$ ${\ displaystyle G \ subset \ mathbb {C}}$ the functions $f$ ${\ displaystyle f}$ - point $z_{0}\in G$ ${\ displaystyle z_ {0} \ in G}$ , in some neighborhood of which the representation $f(z)=(z-z_{0})g(z)$ ${\ displaystyle f (z) = (z-z_ {0}) g (z)}$ where $g$ ${\ displaystyle g}$ analytic in $z_{0}$ ${\ displaystyle z_ {0}}$ and does not vanish at this point.

Zero order $k$ ${\ displaystyle k}$ analytical in some area $G\subset \mathbb {C}$ ${\ displaystyle G \ subset \ mathbb {C}}$ the functions $f$ ${\ displaystyle f}$ - point $z_{0}\in G$ ${\ displaystyle z_ {0} \ in G}$ , in some neighborhood of which the representation $f(z)=(z-z_{0})^{k}g(z)$ ${\ displaystyle f (z) = (z-z_ {0}) ^ {k} g (z)}$ where $g$ ${\ displaystyle g}$ analytic in $z_{0}$ ${\ displaystyle z_ {0}}$ and does not vanish at this point.

Zeros of the analytic function are isolated .

Other specific properties of zeros of complex functions are expressed in various theorems:

Roucher's theorem ,
Marden's theorem ,
Gauss - Luke Theorem

Literature

Zero functions - an article from the Great Soviet Encyclopedia .
Weisstein, Eric W. Root on the Wolfram MathWorld website.

Zero function

Polynomial Root

Comprehensive Analysis

Literature

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