- #1

- 2,361

- 338

Specifically...

We define the finite difference and the difference quotient of a function for some fixed non-zero parameter [itex]h[/itex] as...

[tex] \Delta f(x) = f(x+h)-f(x)[/tex] For the identity function in particular: [itex] \Delta x = h[/itex]

Then the difference quotient is:[tex]\frac{\Delta f(x)}{\Delta x} = \frac{f(x+h)-f(x)}{h}[/tex]

We then define the "preternatural" (any ideas on a better name?) base [itex] \tilde{e} = (1+h)^{1/h}[/itex]

and the [itex]n^{th}[/itex]-degree polynomials (generalized powers):[tex] p_n(x,h) = x(x-h)(x-2h)\cdots (x-nh+h)[/tex]

and for [itex] n<0[/itex] the rational functions: [tex]p_n(x) = \frac{1}{(x+h)(x+2h)\cdots (x-nh)}[/tex]

The idea is to show the usual derivative formulas, but for the difference quotients:

[tex] \frac{\Delta \tilde{e}^x}{\Delta x} = \tilde{e}^x \quad \text{ and } \quad \frac{\Delta p_n(x)}{\Delta x} = n p_{n-1}(x)[/tex] we also have a product rule:

[tex]\frac{\Delta [u\cdot v]}{\Delta x} = \frac{\Delta u}{\Delta x}\cdot v + u \frac{\Delta v}{\Delta x} + h\frac{\Delta u}{\Delta x}\frac{\Delta v}{\Delta x}[/tex]

Then "in the limit as [itex]h\to 0[/itex]:

[tex]\frac{\Delta}{\Delta x}\to \frac{d}{dx}, \quad p_n(x,h)\to x^n,\quad \tilde{e}^x \to e^x[/tex]

And the power rule, product rule, and derivative of the exponential function all manifest.

I'd be interested in hearing from anyone if they think this would be conceptually useful approach, or a horrible idea, or if there were any suggestions. At the very least I thought it might be a good undergraduate special topics course.

There are, further extensions as we can express the perturbed versions of any other analytic functions using the generalized powers in their usual power series expansion, e.g.[tex] \widetilde{\sin}(x) = p_1(x)-\frac{p_3(x)}{3!} + \frac{p_5(x)}{5!} + \cdots; \quad \frac{\Delta \widetilde{\sin}(x)}\Delta{x} = \widetilde{\cos}(x) [/tex]

Too much? Comments encouraged!