Multiplicity

For other senses of this word, see Multiplicity (disambiguation).

In mathematics, the multiplicity of a member of a multiset is how many memberships in the multiset it has. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a multiset (or bag) is a generalization of a set. For example, the term is used to refer to the number of times a given polynomial equation has a root at a given point. Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with

The common reason to consider notions of multiplicity is to count correctly, without specifying exceptions (for example, double roots counted twice). Hence the expression counted with (sometimes implicit) multiplicity.

When mathematicians wish to ignore multiplicity they will refer to the number of distinct elements of a set.

1 Multiplicity of a prime factor
2 Multiplicity of a root of a polynomial
- 2.1 Geometric behavior
3 Multiplicity of a zero of a function
4 In complex analysis
5 See also
6 References

Multiplicity of a prime factor

Main article: p-adic order

In the prime factorization

60 = 2 × 2 × 3 × 5

the multiplicity of the prime factor 2 is 2, while the multiplicity of the prime factors 3 and 5 is 1. In Number theory, for a given Prime number p, the p -adic order or additive p -adic valuation of a number n is Thus, 60 has 4 prime factors, but only 3 distinct prime factors.

Multiplicity of a root of a polynomial

Let F be a field and p(x) be a polynomial in one variable and coefficients in F. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations An element a ∈ F is called a root of multiplicity k of p(x) if there is a polynomial s(x) such that s(a) ≠ 0 and p(x) = (x − a)^ks(x). This article is about the zeros of a function which should not be confused with the value at zero. If k = 1, then a is called a simple root.

For instance, the polynomial p(x) = x³ + 2x² − 7x + 4 has 1 and −4 as roots, and can be written as p(x) = (x + 4)(x − 1)². This means that 1 is a root of multiplicity 2, and −4 is a 'simple' root (of multiplicity 1).

The discriminant of a polynomial is zero if and only if the polynomial has a multiple root. In Algebra, the discriminant of a Polynomial with real or complex Coefficients is a certain expression in the coefficients of the

Geometric behavior

Let f(x) be a polynomial function. In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations Then, if f is graphed on a Cartesian coordinate system, its graph will cross the x-axis at real zeros of odd multiplicity and will not cross the x-axis at real zeros of even multiplicity. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane In addition, if f(x) has a zero with a multiplicity greater than 1, the graph will be tangent to the x-axis and will have slope 0.

Multiplicity of a zero of a function

Let $I$ be an interval of R, let $f$ be a function from $I$ into R or C be a real (resp. complex) function, and let $c$ ∈ $I$ be a zero of $f$ , i. e. a point such that $f (c) = 0$ . The point $c$ is said a zero of multiplicity $k$ of $f$ if there exist a real number $\ell\neq 0$ such that

$\lim_{x\to c}\frac{|f(x)|}{|x-c|^k}=\ell.$

In a more general setting, let $f$ be a function from an open subset $A$ of a normed vector space $E$ into a normed vector space $F$ , and let $c \in A$ be a zero of $f$ , i. In Mathematics, with 2- or 3-dimensional vectors with real -valued entries the idea of the "length" of a vector is intuitive and can easily be extended to e. a point such that $f (c) = 0$ . The point $c$ is said a zero of multiplicity $k$ of $f$ if there exist a real number $\ell \neq 0$ such that

$\lim_{x\to c}\frac{\|f(x)\|_{\mathcal F}}{\|x-c\|_{\mathcal E}^k}=\ell.$

The point $c$ is said a zero of multiplicity ∞ of $f$ if for each $k$ , it holds that

$\lim_{x\to c}\frac{\|f(x)\|_{\mathcal F}}{\|x-c\|_{\mathcal E}^k}=0.$

Example 1. Since

$\lim_{x\to 0}\frac{|\sin x|}{|x|}=1,$

0 is a zero of multiplicity 1 for the function sine function.

Example 2. Since

$\lim_{x\to 0}\frac{|1-\cos x|}{|x|^2}=\frac 12,$

0 is a zero of multiplicity 2 for the function $1 - cos$ .

Example 3. Consider the function $f$ from R into R such that $f (0) = 0$ and that $f (x) = exp(1 / x 2)$ when $x \neq 0$ . Then, since

$\lim_{x\to 0}\frac{|f(x)|}{|x|^k}=0 \mbox{ for each }k \in \mathbb{N}$

0 is a zero of multiplicity ∞ for the function $f$ .

In complex analysis

Let $z 0$ be a root of a holomorphic function $f$ , and let $n$ be the least positive integer such that, the $n$ ^th derivative of $f$ evaluated at $z 0$ differs from zero. Holomorphic functions are the central object of study of Complex analysis; they are functions defined on an open subset of the complex number plane Then the power series of $f$ about $z 0$ begins with the $n$ ^th term, and $f$ is said to have a root of multiplicity (or “order”) $n$ . If $n = 1$ , the root is called a simple root (Krantz 1999, p. 70).

We can also define the multiplicity of the zeroes and poles of a meromorphic function thus: If we have a meromorphic function $f = \dfrac{g}{h}$ , take the Taylor expansions of g and h about a point z₀, and find the first non-zero term in each (denote the term numbers m and n respectively). In Complex analysis, a meromorphic function on an open subset D of the Complex plane is a function that is holomorphic In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives if m = n, then the point has non-zero value. If m > n, then the point is a zero of multiplicity m - n. If m < n, then the point has a pole of multiplicity n - m.

References

Krantz, S. In Complex analysis, a zero of a Holomorphic function f is a Complex number a such that f ( a) = 0 In Mathematics, the Fundamental theorem of algebra states that every non-constant single-variable Polynomial with complex coefficients has at In Number theory, the fundamental theorem of arithmetic (or unique-prime-factorization theorem) states that every Natural number greater than 1 can be written In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes G. Handbook of Complex Variables. Boston, MA: Birkhäuser, 1999. ISBN 0-8176-4011-8.

Dictionary

-noun

the state of being made of multiple diverse elements
(mathematics) the number of values for which a given condition holds
a large indeterminate number

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