leadingorder

Leading order refers to the dominant contribution to a mathematical expression or model in a specified limit. It is the term that determines the overall behavior of a function when a parameter approaches a particular value, such as infinity or zero, and is the first term kept in an asymptotic expansion.

In practice, when a function f(x) is analyzed as x approaches a limit, one writes f(x) ~ c x^p plus lower-order terms. The leading order is the c x^p term, and the remainder is of smaller order, often written as o(x^p) in the sense of little o notation. The concept generalizes to multiple terms with different growth rates: the term with the fastest growth (or slowest decay) in the chosen limit is the leading order.

Determining the leading order involves comparing the growth rates of terms. For example, as x → ∞, a function like f(x) = 3x^2 + 2x + 1 has leading order 3x^2. In a sum with an exponential term, such as e^x + x^2, the leading order as x → ∞ is e^x. For a small parameter ε → 0, a function f(ε) = a0 + a1 ε + a2 ε^2 has leading order a0.

In physics and applied mathematics, the term often appears in perturbation theory as the lowest nontrivial order in a small parameter, with higher orders representing corrections (next-to-leading order, etc.). Leading order is context-dependent and can change with the limit or the variables being considered.

See also: asymptotic expansion, big-O notation, matched asymptotics, perturbation theory.