1

I'm trying to decipher what general principals generalize all elementary functions.

Does the following hold?:

All elementary functions $y = f(x)$ are a solution an equation of the form $P(x,e^x,\ln(x),y) = Q(x,e^x,\ln(x))$?

Because if you have just $x,$ you have all algebraic equations to start with encapsulated as being solutions to $P(y) = Q(x).$

StackQuest
  • 579
  • Elementary functions also allow compositions of things like this, as well as trigonometric functions, don't they? – A. Thomas Yerger Jan 06 '22 at 01:25
  • I'm less sure of composition in the general case. I'm just trying to figure out what generalizes the concept of elementary-ness. The thing is, not all solutions to such an equation are elementary, so I'm not sure if I have it right. I think though, the statement is closer to correct at least for algebraic functions, though I don't remember the theorem. – StackQuest Jan 06 '22 at 01:27
  • 1
    Maybe I don't understand the question. The definition of an elementary function from Wikipedia https://en.wikipedia.org/wiki/Elementary_function "a function of a single variable (typically real or complex) that is defined as taking sums, products, and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and exponential functions, including possibly their inverse functions". What do you mean exactly by generalizing the concept of elementary-ness? – Snaw Jan 06 '22 at 01:29
  • Perhaps one up to the extent that elementary functions form a field. – StackQuest Jan 06 '22 at 01:32
  • I don't know if this makes sense, but perhaps the most minimal subfield of which all other possible elementary functions can also be comprised, perhaps those are the ones of the equation I present. – StackQuest Jan 06 '22 at 02:01
  • See https://math.stackexchange.com/questions/361969/what-does-elementary-function-mean and https://math.stackexchange.com/questions/118113/what-makes-elementary-functions-elementary and https://math.stackexchange.com/questions/2358050/what-constitutes-the-classification-of-functions-into-elementary-and-non-e and https://math.stackexchange.com/questions/628414/is-there-anything-special-about-elementary-functions and probably many others. – Gerry Myerson Jan 06 '22 at 03:41
  • Did those links help? – Gerry Myerson Jan 07 '22 at 03:57
  • Not quite. The first link did, but I wish there was a less clunky definition than saying "algebraic functions, logarithms and exponentials over a complex field". – StackQuest Jan 07 '22 at 05:11
  • Well, there's more where those came from. If you just type elementary functions into the search box on this page, many older (and possibly helpful) questions will turn up. – Gerry Myerson Jan 07 '22 at 16:10

0 Answers0