Search results
Results From The WOW.Com Content Network
Totally additive is also used in this sense by analogy with totally multiplicative functions. If f is a completely additive function then f(1) = 0. Every completely additive function is additive, but not vice versa.
Other notions of additivity which may be considered as topics in this category include sigma additivity (i.e., properties of additive set functions in measure-theoretic contexts) and additive maps defining a slightly different variant of additivity than is typically considered in number-theoretic contexts.
Multiplicative function; Additive function; Dirichlet convolution; Erdős–Kac theorem; Möbius function. Möbius inversion formula; Divisor function; Liouville function; Partition function (number theory) Integer partition; Bell numbers; Landau's function; Pentagonal number theorem; Bell series; Lambert series
Cauchy's functional equation is the functional equation: (+) = + (). A function that solves this equation is called an additive function.Over the rational numbers, it can be shown using elementary algebra that there is a single family of solutions, namely : for any rational constant .
A related function counts prime powers with weight 1 for primes, 1/2 for their squares, 1/3 for cubes, etc. It is the summation function of the arithmetic function which takes the value 1/k on integers which are the kth power of some prime number, and the value 0 on other integers.
The function is called additive or finitely additive, if whenever and are disjoint sets in , then = + (). A consequence of this is that an additive function cannot take both − ∞ {\displaystyle -\infty } and + ∞ {\displaystyle +\infty } as values, for the expression ∞ − ∞ {\displaystyle \infty -\infty } is undefined.
An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.
A modern, abstract point of view contrasts large function spaces, which are infinite-dimensional and within which most functions are 'anonymous', with special functions picked out by properties such as symmetry, or relationship to harmonic analysis and group representations. See also List of types of functions