Why is arctan a transcendental function?
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From: hagman on 15 Jul 2010 16:53 On 15 Jul., 16:59, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Tim Little <t...(a)little-possums.net> writes: > > On 2010-07-15, Jesse F. Hughes <je...(a)phiwumbda.org> wrote: > >> Perhaps you can help clean up Wikipedia by moving to delete > >>http://en.wikipedia.org/wiki/Transcendental_function. > > > Or at least fixing one statement on it! > > > "If f(z) is an algebraic function and alpha is an algebraic number > > then f(alpha) will also be an algebraic number." > > Well, they could at least cite your recent post, if only Usenet counted > as a reliable source, right? It sure isn't - for a reason. > > (I'm taking your word for it that this claim is false. I don't know > doodlysquat about transcendental functions.) The definition suffices for this: If f is algebraic and P in Z[X,Y] is an irreducible non-zero integer coefficient polynomial such that P(z,f(z))=0 for all z in C and alpha is algebraic then P(alpha,X) is a polynomial with coefficients in the algebraic closure of Q. It is not the zero polynomial as that would imply that the irreducible polynomial of alpha is a factor of it, which can't be the case. Since the algebraic closure is algebraically closed, any root of P(alpha,X) is algebraic, esp. f(alpha) is algebraic.
From: Bill Dubuque on 15 Jul 2010 19:11 Fred Richman <richman(a)FAU.EDU> wrote: > > Thanks. I don't know how I missed the argument that the inverse of an >algebraic function is algebraic---I even wondered if that were true. You might also enjoy learning about hypotranscendental functions, and transcendentally transcendental functions - really! See this very interesting paper: Lee A. Rubel. Some Research Problems about Algebraic Differential Equations Trans. of the American Math. Society, Vol. 280, No. 1, pp. 43-52 http://www.jstor.org/stable/pdfplus/1999601.pdf
From: Dave L. Renfro on 16 Jul 2010 14:03
Bill Dubuque wrote: > You might also enjoy learning about hypotranscendental functions, > and transcendentally transcendental functions - really! See > this very interesting paper: > > Lee A. Rubel. Some Research Problems about Algebraic Differential > Equations Trans. of the American Math. Society, Vol. 280, No. 1, > pp. 43-52 > http://www.jstor.org/stable/pdfplus/1999601.pdf More accessible mathematically (and both accessible or not accessible, depending on access to JSTOR) is Rubel's "A survey of transcendentally transcendental functions", American Mathematical Monthly 96 #9 (November 1989), 777-788. Dave L. Renfro |