variable power as an axis, nonsense? please comment

elegysix

Hello,
Does anyone know if this makes sense or is usable?

I've only been able to describe it through a graph.
Instead of the x axis being numbers, being powers of x.

Let me clarify - where normally would be x=0,1,2,3... would now be x^0, x^1, x^2, x^3
and the y axis would be values of coefficients in a polynomial / coefficients in a power series.

Does anyone know of anything like this?
I thought it up yesterday and have been intrigued by it.
I don't know how to work with it though.

thanks
austin

SteveL27

Yes, it's called a log scale.

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

elegysix

this would be a log base x though right? The log plots on wiki are in base 10 and other values, but not variables. Does that make a difference?

SteveL27

Maybe I'm not clear on what you meant. The values on the x-axis aren't numbers? They're variables? Can you give a specific example of what you mean?

'x' is just a dummy variable that ranges over the real numbers when you're graphing a function, for example. But the x-axis represents the real numbers. I'm not sure I understand what you mean by saying it consists of variables like x^n.

elegysix

The idea is like this: write out a few of the first terms in the power series of sin(x),

then mark your x axis as [itex] x^{0}, x^{1}, x^{2}, x^{3}... [/itex] in place of the integers

The coefficients of the power series, [itex]a_{n}[/itex], are the y coordinates.

Ordered pairs would be [itex](x^{0},a_{0}), (x^{1},a_{1}), (x^{2},a_{2}) [/itex] and so forth.

doing this for sin(x) -- the coefficients of the power series are 0, 1, 0, -1/6, 0, 1/120...
Where I have used 0's for the coefficients of even powers of x.

Plotting this looks like [itex] \frac {sin()}{n!} [/itex] but I have no clue how to interpret what I've done. it does look like if we tried to fit those points, that would be an exact fit.

I find it curious that just plotting these coefficients resembles a sine function. Perhaps this could be used somehow to determine the function a power series converges to? (assuming it does converge)
The same can be done for the cosine, and the plot looks like cos/n!.

any idea about this?

thanks
austin