Variable power as an axis, nonsense? please comment

  • Context: Undergrad 
  • Thread starter Thread starter elegysix
  • Start date Start date
  • Tags Tags
    Axis Power Variable
Click For Summary

Discussion Overview

The discussion revolves around the concept of using powers of a variable as the x-axis in a graph, specifically in the context of representing coefficients in polynomial or power series. Participants explore the implications and usability of this idea, particularly in relation to the power series of functions like sine and cosine.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant proposes a graph where the x-axis consists of powers of x (x^0, x^1, x^2, etc.) instead of traditional numerical values, with the y-axis representing coefficients of a polynomial or power series.
  • Another participant suggests that this concept resembles a logarithmic scale, but questions whether it would be a logarithm base x, as typical logarithmic plots use fixed bases like 10.
  • A participant seeks clarification on whether the x-axis values are indeed variables rather than numbers, expressing confusion about the representation of the x-axis.
  • One participant illustrates the idea by using the power series of sin(x), detailing how the coefficients correspond to specific powers of x and suggesting that the plotted coefficients resemble a sine function.
  • The same participant speculates that this method might help determine the function to which a power series converges, assuming convergence occurs.

Areas of Agreement / Disagreement

Participants express varying levels of understanding and interpretation of the proposed concept. There is no consensus on the usability or implications of using powers of x on the x-axis, and the discussion remains unresolved regarding its mathematical validity and potential applications.

Contextual Notes

Some participants note the need for clarification on the definitions and interpretations of the proposed graphing method, as well as the mathematical implications of using variable powers on the x-axis.

elegysix
Messages
404
Reaction score
15
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
 
Physics news on Phys.org
elegysix said:
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

Yes, it's called a log scale.

http://en.wikipedia.org/wiki/Logarithmic_scale
 
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?
 
elegysix said:
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?

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.
 
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
 
Last edited:

Similar threads

Undergrad Quintic Polynomial Tschirnhaus Transform: Possible complex Bring Radicals
  • · Replies 0 ·
Replies
0
Views
3K
Undergrad Can one find a matrix that's 'unique' to a collection of eigenvectors?
  • · Replies 33 ·
2
Replies
33
Views
3K
Undergrad Factoring question - generalized factoring in integers
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad Did I figure-out z-translation of 2D-coordinates correctly?
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad Linear Models vs Nonlinear Models
  • · Replies 22 ·
Replies
22
Views
4K
Undergrad Linear regression and random variables
  • · Replies 30 ·
2
Replies
30
Views
5K
Undergrad Count and Categorical Variables...
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad What is the difference between these two power series theorems?
  • · Replies 5 ·
Replies
5
Views
3K
Undergrad Showing ##k[x_1,\ldots,x_n]/\mathfrak{a}## is finite dimensional
  • · Replies 40 ·
2
Replies
40
Views
5K
Undergrad Nonlinear Least Squares or OLS for Nonlinear Models?
  • · Replies 7 ·
Replies
7
Views
3K