- #1

- 325

- 3

I have this translation operator T(a) that acts on a function y(x) and causes the transformation T(a)y(x) = y(x+a).

I am supposed to be "expanding y(x+a) as a taylor series in a" to show that T(a)=e

So, I've started out with the general equation for the Taylor expansion, to expand, say, f(x) about the point x=a:

[tex] f(x) = \sum^{\infty}_{0} \frac{\frac{d^n}{dx^n}f(x=a)}{n!} (x-a)^n[/tex]

I've used this plenty of times before but only when I have been expanding something like f(x) rather than, say, f(x+c), so I'm making mistakes and messing up when I try and work through this problem.

Anyway, back to the original thing I'm trying to expand out. What I've got so far is:

[tex] y(x+a) = \sum^{\infty}_{0} \frac{\frac{d^n}{dx^n}y(x+a = ?)}{n!}[(x+a)- ?]^n [/tex]

Where I've left the question marks is where I'm not certain what to use. My y(x) expansion is otherwise the same way as the general formula for f(x) just with x+a replacing f(x).

I know I'm supposed to end up with something of the same form as [tex] e^x = \sum^{\infty}_{0} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2} + ...[/tex] but I just can't seem to get there.

I am supposed to be "expanding y(x+a) as a taylor series in a" to show that T(a)=e

^{ipa}, where p is the operator p = -i.d/dx]So, I've started out with the general equation for the Taylor expansion, to expand, say, f(x) about the point x=a:

[tex] f(x) = \sum^{\infty}_{0} \frac{\frac{d^n}{dx^n}f(x=a)}{n!} (x-a)^n[/tex]

I've used this plenty of times before but only when I have been expanding something like f(x) rather than, say, f(x+c), so I'm making mistakes and messing up when I try and work through this problem.

Anyway, back to the original thing I'm trying to expand out. What I've got so far is:

[tex] y(x+a) = \sum^{\infty}_{0} \frac{\frac{d^n}{dx^n}y(x+a = ?)}{n!}[(x+a)- ?]^n [/tex]

Where I've left the question marks is where I'm not certain what to use. My y(x) expansion is otherwise the same way as the general formula for f(x) just with x+a replacing f(x).

I know I'm supposed to end up with something of the same form as [tex] e^x = \sum^{\infty}_{0} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2} + ...[/tex] but I just can't seem to get there.

Last edited: