Multiplicity

From Wikipedia, the free encyclopedia

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. For example, the term is used to refer to the number of times a given polynomial equation has a root at a given point.

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

[edit] 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. Thus, 60 has 4 prime factors, but only 3 distinct prime factors.

[edit] 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. 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). 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.

[edit] Geometric behavior

Let f(x) be a polynomial function. 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 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.

[edit] 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.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 $f$ .

[edit] 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. 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). 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.