Definition
Formally, given complex-valued functions f and g of a natural number variable n, one writes
to express the fact, stated terms of little-o notation, that
or equivalently
Explicitly this means that for every positive constant ε there exists a constant N such that
- .
Unless g(n) is infinitely often zero (which would make the limit below undefined), this statement is also equivalent to
This relation is an equivalence relation on the set of functions of n. The equivalence class of f informally consists of all functions g which are approximately equal to f in a relative sense, in the limit.
Read more about this topic: Asymptotic Analysis
Famous quotes containing the word definition:
“Beauty, like all other qualities presented to human experience, is relative; and the definition of it becomes unmeaning and useless in proportion to its abstractness. To define beauty not in the most abstract, but in the most concrete terms possible, not to find a universal formula for it, but the formula which expresses most adequately this or that special manifestation of it, is the aim of the true student of aesthetics.”
—Walter Pater (1839–1894)
“Scientific method is the way to truth, but it affords, even in
principle, no unique definition of truth. Any so-called pragmatic
definition of truth is doomed to failure equally.”
—Willard Van Orman Quine (b. 1908)
“Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
—Elaine Heffner (20th century)