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The leading-order terms within a mathematical equation, expression or model are the terms with the largest order of magnitude. The sizes of the different terms in the equation will change as the variables change, and hence, which terms are leading-order may also change.
A common and powerful way of simplifying and understanding a wide variety of complicated mathematical models is to investigate which terms are the largest , for particular sizes of the variables and parameters, and analyse the behaviour produced by just these terms. This gives the main behaviour – the true behaviour is only small deviations away from this. This main behaviour may be captured sufficiently well by just the strictly leading-order terms, or it may be decided that slightly smaller terms should also be included. In which case, the phrase leading-order terms might be used informally to mean this whole group of terms. The behaviour produced by just the group of leading-order terms is called the leading-order behaviour of the model.
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