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t. e. In mathematics, function composition is an operation ∘ that takes two functions f and g, and produces a function h = g ∘ f such that h(x) = g(f(x)). In this operation, the function g is applied to the result of applying the function f to x. That is, the functions f : X → Y and g : Y → Z are composed to yield a function that maps x ...
Multiplicative function. In number theory, a multiplicative function is an arithmetic function f ( n) of a positive integer n with the property that f (1) = 1 and whenever a and b are coprime . An arithmetic function f ( n) is said to be completely multiplicative (or totally multiplicative) if f (1) = 1 and f ( ab) = f ( a) f ( b) holds for all ...
t. e. In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions f and g in terms of the derivatives of f and g. More precisely, if is the function such that for every x, then the chain rule is, in Lagrange's notation , or, equivalently, The chain rule may also be expressed in ...
Dirichlet convolution is a special case of the convolution multiplication for the incidence algebra of a poset, in this case the poset of positive integers ordered by divisibility. The Dirichlet hyperbola method computes the summation of a convolution in terms of its functions and their summation functions.
A function is often denoted by a letter such as f, g or h. The value of a function f at an element x of its domain (that is the element of the codomain that is associated to x) is denoted by f(x); for example, the value of f at x = 4 is denoted by f(4).
The term convolution refers to both the result function and to the process of computing it. It is defined as the integral of the product of the two functions after one is reflected about the y-axis and shifted. The integral is evaluated for all values of shift, producing the convolution function.
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.
Definition. A completely multiplicative function (or totally multiplicative function) is an arithmetic function (that is, a function whose domain is the natural numbers ), such that f (1) = 1 and f ( ab) = f ( a) f ( b) holds for all positive integers a and b. [ 1]
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