Algebraic function

[AnnD]

Hello

 

I'm curious about something about the function f(x) = (cos(x),sin(x)). Is it a transcendental or an algebraic curve ? It looks like an algebraic curve, but I don't know if you can have cos and sin in an algebraic curve.

[shadowacct]

Assuming that you mean this is a function from R to R², then this indeed is a transcendental function. Sin(x) and cos(x) cannot be represented as a finite series of powers of x.

 

If, however, you mean the parameterized curve x=cos(t), y=sin(t), then you have an algebraic curve. This is the unit circle, and it can be written as x² + y² = 1.

 

You can have sin() and cos() function in algebraic functions, but then they need to be combined with proper functions, which 'cancel' the transcendentence.

 

A trivial example is f(x) = cos(arccos(x)), but there are less trivial examples,

e.g. f(x) = sin(arccos(x)) is algebraic (being part of the unit circle).

[Country Boy]

The FUNCTION, from R to R^2, f(x)= (cos(x),sin(x)) is indeed a "transcendental function". It's GRAPH happens to be the same as the graph of the (algebraic) relation x²+ y²= 1. Many different functions or relations, whether "algebraic", "transcendental" or whatever, may have exactly the same graph.

[AnnD]

Yes but on wikipedia; "In mathematics, a transcendental curve is a curve that is not an algebraic curve. Here for a curve C what matters is the point set (typically in the plane) underlying C, not a given parametrisation."

 

Clearly; x = cos(t), y = sin(t) can be translated to x^2+y^2-1=0, so it's an algebraic curve.

[Country Boy]

Your quote says "what matters is the point set (typically in the plane) underlying C. Do they give a definition of "algebraic SET" or "transcendental SET"?

 

(edit: Yes, it does:

"In mathematics, an algebraic set over a field K is the set of solutions in Kn (n-tuples of elements of K) of a set of simultaneous equations

 

P1(X1, ...,Xn) = 0

P2(X1, ...,Xn) = 0 "

 

That definition, and yours, is in the field of "algebraic varieties" and has nothing to do with the question here.

[AnnD]

But the question is if the curve is algebraic or not. Like shadow said, f(x) = cos(arccos(x)) is algebraic, despite having cosinus in it. So a cercle should also be algebric, despite being written with a cosinus.

[Martin]

Hello

 

I'm curious about something about the function f(x) = (cos(x),sin(x)). Is it a transcendental or an algebraic curve ? It looks like an algebraic curve, but I don't know if you can have cos and sin in an algebraic curve.

 

Hi AnnD,

I am a retired mathematician. Your simple question causes me pleasure, owing to the memories it evokes.

 

The only time in my life I remember hearing the term algebraic curve was in the context of Algebraic Geometry. A wonderful subject.

 

The definition of an algebraic curve is what Wikipedia gives

http://en.wikipedia.org/wiki/Algebraic_curve

"An algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections."

 

It seems obvious to me that your question puts us immediately in the context of Algebraic Geometry and, as you have clarified, it concerns the point set described by the function you mention----its range---namely the circle. Indeed without question this is an algebraic variety of dimension one!

 

Perhaps I am overlooking something? Halls, I do not understand your objection.

[Country Boy]

There was no "objection". What I said was that the FUNCTION f(x)= (cos(x), sin(x)) is transcendental but that the GRAPH (which is what I think was meant by "curve") is algebraic. Everyone seems to be agreeing with that.