Search results
Results from the WOW.Com Content Network
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial.According to the theorem, the power (+) expands into a polynomial with terms of the form , where the exponents and are nonnegative integers satisfying + = and the coefficient of each term is a specific positive integer ...
This proof of the multinomial theorem uses the binomial theorem and induction on m.. First, for m = 1, both sides equal x 1 n since there is only one term k 1 = n in the sum. For the induction step, suppose the multinomial theorem holds for m.
The binomial approximation for the square root, + + /, can be applied for the following expression, + where and are real but .. The mathematical form for the binomial approximation can be recovered by factoring out the large term and recalling that a square root is the same as a power of one half.
The dotted vector, in this case B, is differentiated, while the (undotted) A is held constant. The utility of the Feynman subscript notation lies in its use in the derivation of vector and tensor derivative identities, as in the following example which uses the algebraic identity C⋅(A×B) = (C×A)⋅B:
In mathematics, an expansion of a product of sums expresses it as a sum of products by using the fact that multiplication distributes over addition. Expansion of a polynomial expression can be obtained by repeatedly replacing subexpressions that multiply two other subexpressions, at least one of which is an addition, by the equivalent sum of products, continuing until the expression becomes a ...
Define e t (z) ≡ e tz, and n ≡ deg P. Then S t (z) is the unique degree < n polynomial which satisfies S t (k) (a) = e t (k) (a) whenever k is less than the multiplicity of a as a root of P. We assume, as we obviously can, that P is the minimal polynomial of A. We further assume that A is a diagonalizable matrix.
Get AOL Mail for FREE! Manage your email like never before with travel, photo & document views. Personalize your inbox with themes & tabs. You've Got Mail!
A Laurent series is a generalization of the Taylor series, allowing terms with negative exponents; it takes the form = and converges in an annulus. [6] In particular, a Laurent series can be used to examine the behavior of a complex function near a singularity by considering the series expansion on an annulus centered at the singularity.