In the applied mathematics textbooks or papers, I often see the phrase 'in the first approximation'. For example, substitution of Eq.(1) into the boundary condition (2) results in Eq.(3) describing ... in the first approximation.
2026-03-27 01:47:48.1774576068
What is the exact meaning of 'in the first approximation' in the context of applied mathematics?
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Depending on context, this can vary from a very formal usage to a very informal one. Commonly, this means "to one term of the Taylor series expansion", but it can also mean one term or one iteration of whatever expansion/recursive method is relevant. Less formally, it tends to mean the crudest approximation that the speaker believes has credibility and utility. Informally, it can mean a naïve approximation, or one using a "quick and dirty" methods, without regard to exactly where that fits into any series of approximations. At a similar level of informality, "second approximation" may indicate that the speaker has taken some care to refine the estimate, but greater care is probably needed for firm conclusions.