gross@spp2.UUCP (Howard E. Gross) (01/11/86)
Is tan 1 transcendental? Does there exist a direct proof that it
is (without using a big name theorem)?
--
gross (Howard Gross) {decvax,hplabs,ihnp4,sdcrdcf}!trwrb!trwspp!spp2lambert@boring.UUCP (Lambert Meertens) (01/16/86)
> Is tan 1 transcendental? Does there exist a direct proof that it > is (without using a big name theorem)? If we put t = tan 1 and u = exp i, then t = (u^2-1)/(u^2+1). So the transcendence of t follows from that of u, which, I think, is an immediate consequence of a theorem of Gel'fond. However, that is a big name. If we let s = sin 1, then t = s/sqrt(1-s^2). So proving that s is transcendental should suffice. This can probably be done by "replaying" Hermite's proof for e. That proof is not simple at all, unfortunately. (The proof that s is irrational is in comparison trivial.) -- Lambert Meertens ...!{seismo,okstate,garfield,decvax,philabs}!lambert@mcvax.UUCP CWI (Centre for Mathematics and Computer Science), Amsterdam
steve@anasazi.UUCP (Steve Villee) (01/17/86)
> Is tan 1 transcendental? Does there exist a direct proof that it > is (without using a big name theorem)? > > > -- > gross (Howard Gross) {decvax,hplabs,ihnp4,sdcrdcf}!trwrb!trwspp!spp2 You may rest assured that tan(1) is transcendental. In fact, given that e**(2*i*x) = (1 - i * tan(x)) / (1 + i * tan(x)) and applying the Lindemann-Weierstrass theorem, it should be clear that tan(x) is transcendental whenever x is algebraic and nonzero. I don't know of any way to prove this without using something like Lindemann-Weierstrass, though. --- Steve Villee (ihnp4!terak!anasazi!steve) International Anasazi, Inc. 7500 North Dreamy Draw Drive, Suite 120 Phoenix, Arizona 85020 (602) 870-3330