Why is arctan a transcendental function?
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From: Fred Richman on 14 Jul 2010 18:51 How does the proof go? --Fred
From: Tim Little on 15 Jul 2010 01:33 On 2010-07-15, Fred Richman <richman(a)FAU.EDU> wrote: [ Why is arctan a transcendental function? ] > How does the proof go? It doesn't take much more than noting that atan(1) = pi/4, and 1 is algebraic but pi/4 is transcendental. - Tim
From: Gerry Myerson on 15 Jul 2010 01:42 In article <slrni3t7d3.jrj.tim(a)soprano.little-possums.net>, Tim Little <tim(a)little-possums.net> wrote: > On 2010-07-15, Fred Richman <richman(a)FAU.EDU> wrote: > [ Why is arctan a transcendental function? ] > > How does the proof go? > > It doesn't take much more than noting that atan(1) = pi/4, and 1 is > algebraic but pi/4 is transcendental. The function f(x) = pi x has f(1) = pi, but it's not a transcendental function - it's not transcendental over the field C(x) (heck, it's in the field C(x)). -- Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for email)
From: Tim Little on 15 Jul 2010 02:19 On 2010-07-15, Gerry Myerson <gerry(a)maths.mq.edi.ai.i2u4email> wrote: > The function f(x) = pi x has f(1) = pi, but it's not a > transcendental function - it's not transcendental over the field > C(x) (heck, it's in the field C(x)). That's what I get for somehow misremembering a "theorem" that an algebraic function of an algebraic number always yields an algebraic result. I have no idea where that came from. - Tim
From: Ludovicus on 15 Jul 2010 06:10
On 15 jul, 02:19, Tim Little <t...(a)little-possums.net> wrote: > On 2010-07-15, Gerry Myerson <ge...(a)maths.mq.edi.ai.i2u4email> wrote: > > > The function f(x) = pi x has f(1) = pi, but it's not a > > transcendental function - it's not transcendental over the field > > C(x) (heck, it's in the field C(x)). > > That's what I get for somehow misremembering a "theorem" that an > atlgebraic function of an algebraic number always yields an algebraic > result. I have no idea where that came from. > > - Tim There are not trascendental funtions but trascendenal numbers. A number is trascendental when it cannot be a solution of an algebraic equation. An algebraic equation is a sum of powers of x in C, with integer coeficients, and that sum equaled to zero. pi.x = 0 is not an algebraic equation. The number 4arctg(1) is not a solution of an algebraic equation. |