Talk:Transcendental function

Use in Calculus section

I just removed the section entitled "Use in Calculus". Not that transcendentals aren't used in calculus, but the section discussed the use of the logarithm, as opposed to anything related to transcendentals. I would fill in the section myself, but I don't know anything about transcendentals; perhaps a mathematician might do so? Ourai ^тʃ_с 00:17, 15 October 2007 (UTC)

Next time, leave it blank, don't delete it. That way, people reading the article know that a section on use in calculus is needed. LokiClock (talk) 11:17, 26 May 2009 (UTC)

Field extensions

Is it a good idea to mention field extensions? Adding an algebraic number is a finite field extension, but a transcendental one gives an infinite one. Is there a parallel for algebraic/transcendental functions? Paxinum (talk) 07:53, 27 April 2008 (UTC)

Common examples?

Aren't exp(x), sin(x), and cos(x) transcendental functions? And any infinite Taylor series?

Do you have any idea of the appropriate level for a general encyclopedia?

This article is way way too technicalm, at least in the intro (and yes, I can spell and use correct grammar, I'm too annoyed) — Preceding unsigned comment added by 50.49.131.230 (talk) 01:05, 4 January 2015 (UTC)

Trivial transcendental functions

If all functions which are not algebraic are transcendental (per the article), then many simple functions are transcendental, e.g. the constant function $\pi$, any non-differentiable function, e.g., the absolute value, the step function, the complex conjugate, the indicator function of any non-trivial set, e.g., $\mathbf {1} _{\{0\}}$, etc. Indeed, some of these functions take only algebraic values for all arguments (not just algebraic arguments).

I don't think I've ever seen those functions talked about as transcendental. Should the definition should be defined as a subset of the analytic functions? --Macrakis (talk) 21:02, 17 April 2015 (UTC)

A constant function can be considered algebraic; the absolute value and step function are piecewise algebraic. The complex conjugate is not analytic. Think about this from the pedagogical view: algebraic and transcendental can be discussed in early courses, the analytic function idea requires more student maturity. Your suggestion does not appear to be helpful.Rgdboer (talk) 21:25, 18 April 2015 (UTC)

WP is an encyclopedia, not an "early course". Our definitions should not be bowdlerized, making them incomplete or incorrect.

I have now had the time to do a little more research, and it seems clear that the literature (including the existing reference!) is unanimous that "transcendental functions" are a defined as a subset of "analytic functions", not of all functions. I will include that (with sources) in the lead. --Macrakis (talk) 00:01, 19 April 2015 (UTC)

Eisenstein / Heine

The following was removed:

• Transcendental functions were first defined by Euler in his Introductio (1748) as functions either not definable by the "ordinary operations of algebra", or defined by such operations "repeated infinitely often". But this definition is unsatisfactory, since some functions defined with infinitely many operations remain algebraic or even rational. The theory was further developed by Gotthold Eisenstein (Eisenstein's theorem), Eduard Heine, and others.<ref Amy Dahan-Dalmédico, Jeanne Peiffer, History of Mathematics: Highways and Byways, 2010, p. 240 /ref>

The "unsatisfactory" explanation is unsatisfactory. The contributions of Eisenstein and Heine may be appended to the article with sufficient introduction and context. Rgdboer (talk) 02:00, 21 January 2017 (UTC)

Dimensional Analysis

The last (as of 8.25.2022) sentence in the section is wrong-headed. DA does NOT allow you to 'split' the number from its units of measure. That is, you can no more claim log(5 meters) = log(5)+log(meters) than you can claim 5meters +5centimeters = 5(meters+centimeters). Neither is correct. And both are incorrect for the same reason.The solution mentioned in the article, to use tf's with only dimensionless quantities seems to be best. If the section needs to explain something, then it needs to explain why the units of log(5m) isn't log(meters). It also seems to me that no explanation is needed with the process of taking sine(4 degrees) to be a dimensionless number, rather than claiming its units are Sum(radians^n) where n = 1...infinity (taylor series expansion). 4meters*sin(30°)=2 meters, after all. BTW, when I do DA, I always place units of measure in parentheses to keep them isolated from the numbers and magnitudes. The mathematical operations performed on the magnitudes aren't suitable for operations on units of measure.98.21.245.38 (talk) 02:34, 26 August 2022 (UTC)