Multiplicity
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- For other senses of this word, see multiplicity (disambiguation).
The common reason to consider notions of multiplicity is to count right, without specifying exceptions (for example, double roots counted twice). Hence the expression counted with (sometimes implicit) multiplicity.
Multiplicity of a prime factor
In the prime factorization
- 60 = 2 × 2 × 3 × 5
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 a called a simple root.
For instance, the polynomial [p(x)=x^3+2x^2-7x+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 (multiplicity [1]).
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 [l] ≠ [0] such that
- [\lim_\frac
>
=\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] ∈ [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 [l] ≠ [0] such that - [\lim_\frac}^k}=l.]
- [\lim_\frac}^k}=0.]
- [\lim_\frac
>
>
=1,] 0 is a zero of multiplicity 1 for the function sine function. Example 2. Since
- [\lim_\frac
>
=\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] ≠ [0]. Then, since
- [\lim_\frac
>
=0] for each [k] ∈ 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 [m] such that, the [m]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).See also
- Zero (complex analysis)
- Collection
- Fundamental theorem of algebra
- Fundamental theorem of arithmetic
- Multiset
- Algebraic multiplicity and geometric multiplicity of an eigenvalue
References
- Krantz, S. G. Handbook of Complex Variables. Boston, MA: Birkhäuser, 1999. ISBN 0817640118.
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