Lowerorder

Lower-order term, in mathematics, refers to any term that grows more slowly than the leading term of a function as its variable becomes large, or, in the context of differential equations, to terms involving lower-order derivatives relative to the highest derivative present. The exact meaning depends on the comparison used, such as a polynomial degree or the order of differentiation, and is often clarified by the surrounding terms being considered.

In polynomials, the leading term is the one with the highest degree, and all other terms are lower-order. For example, in p(n) = 4n^5 + 3n^3 + n, the leading term is 4n^5, while 3n^3 and n are lower-order terms. In asymptotic analysis, these lower-order terms become insignificant compared to the leading term as n grows, and the leading term typically determines the growth rate. Thus f(n) = n^3 + n^2 log n + 7 has n^3 as the dominant term, with the remaining parts considered lower-order.

In differential equations, the notion also applies to derivatives: in an equation of order m, the term containing the m-th derivative is the highest-order term, while terms with derivatives of order less than m are lower-order terms. This distinction helps in simplifying analysis and in constructing approximations, such as retaining only the leading-order behavior for large inputs or fast-growing solutions.

Overall, the concept of lower-order terms is relative and context-dependent, serving as a useful way to distinguish the main growth behavior from smaller, often negligible contributions in both algebraic and analytic settings.