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Power series. In mathematics, a power series (in one variable) is an infinite series of the form where an represents the coefficient of the n th term and c is a constant. Power series are useful in mathematical analysis, where they arise as Taylor series of infinitely differentiable functions.
In calculus, the power rule is used to differentiate functions of the form , whenever is a real number. Since differentiation is a linear operation on the space of differentiable functions, polynomials can also be differentiated using this rule. The power rule underlies the Taylor series as it relates a power series with a function's derivatives .
Symmetric power. In mathematics, the n-th symmetric power of an object X is the quotient of the n -fold product by the permutation action of the symmetric group . More precisely, the notion exists at least in the following three areas: In linear algebra, the n -th symmetric power of a vector space V is the vector subspace of the symmetric ...
e. In mathematics, exponentiation is an operation involving two numbers: the base and the exponent or power. Exponentiation is written as bn, where b is the base and n is the power; this is pronounced as " b (raised) to the (power of) n ". [ 1]
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial.According to the theorem, it is possible to expand the polynomial (x + y) n into a sum involving terms of the form ax b y c, where the exponents b and c are nonnegative integers with b + c = n, and the coefficient a of each term is a specific positive integer depending ...
ln (r) is the standard natural logarithm of the real number r. Arg (z) is the principal value of the arg function; its value is restricted to (−π, π]. It can be computed using Arg (x + iy) = atan2 (y, x). Log (z) is the principal value of the complex logarithm function and has imaginary part in the range (−π, π].
The symmetric algebra is a graded algebra. That is, it is a direct sum. where called the n th symmetric power of V, is the vector subspace or submodule generated by the products of n elements of V. (The second symmetric power is sometimes called the symmetric square of V ). This can be proved by various means.
The k th exterior power of ... is the projection for the exterior (quotient) algebra onto the r-homogeneous alternating tensor subspace. On the ...