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In mathematics, the floor function (or greatest integer function) is the function that takes as input a real number x, and gives as output the greatest integer less than or equal to x, denoted ⌊x⌋ or floor (x). Similarly, the ceiling function maps x to the smallest integer greater than or equal to x, denoted ⌈x⌉ or ceil (x).
Identity function. In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unchanged. That is, when f is the identity function, the equality f(x) = x is true for all values of x to which f can be applied.
In number theory, Euler's totient function counts the positive integers up to a given integer n that are relatively prime to n. It is written using the Greek letter phi as or , and may also be called Euler's phi function. In other words, it is the number of integers k in the range 1 ≤ k ≤ n for which the greatest common divisor gcd (n, k ...
Laplace transform. In mathematics, the Laplace transform, named after Pierre-Simon Laplace ( / ləˈplɑːs / ), is an integral transform that converts a function of a real variable (usually , in the time domain) to a function of a complex variable (in the complex-valued frequency domain, also known as s-domain, or s-plane ).
In number theory, the partition function p(n) represents the number of possible partitions of a non-negative integer n. For instance, p(4) = 5 because the integer 4 has the five partitions 1 + 1 + 1 + 1, 1 + 1 + 2, 1 + 3, 2 + 2, and 4 . No closed-form expression for the partition function is known, but it has both asymptotic expansions that ...
Using tangent half-angle formulas, it follows immediately that α = sin A = 2r / (1 + r2) and β = cos A = (1 − r2) / (1 + r2) are both rational and that α2 + β2 = 1. Multiplying up by the smallest integer that clears the denominators of α and β recovers the original primitive Pythagorean triple.
In number theory, an arithmetic, arithmetical, or number-theoretic function [1] [2] is generally any function f ( n) whose domain is the positive integers and whose range is a subset of the complex numbers. [3] [4] [5] Hardy & Wright include in their definition the requirement that an arithmetical function "expresses some arithmetical property ...
The existence of the inverse Möbius transformation and its explicit formula are easily derived by the composition of the inverse functions of the simpler transformations. That is, define functions g 1, g 2, g 3, g 4 such that each g i is the inverse of f i.