Stemming from a comment thread in another question I got curious about why exponential and trig functions are considered elementary but there are so very many other non-algebraic functions which are not. Are there any particular motivations or is it something that becomes obvious when one has studied enough analysis? Is it the exponential functions relation to being eigenfunction to differentiation that is central to this choice or something else?
2026-03-27 03:41:58.1774582918
What constitutes the classification of functions into **elementary** and **non-elementary**?
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To qualify as being elementary, a function must be (1) well known to mathematicians for centuries, (2) taught in mathematics courses in most secondary schools, and (3) indispensable in a wide range of disciplines that use mathematics. The elementary functions pretty much coincide with the functions that feature on the more basic and popular models of "scientific" calculator.
The above "definition" might seem rather arbitrary, and unattractive to those who would prefer a more conceptual definition. However, the concept of elementary, as applied to functions, is not a mathematical one, and lies within the realm of common (albeit specialist) language, which is always subject to evolution and varying opinion.