Taylor Expansion: Do Assumptions Apply?

[Niles]

Homework Statement

Hi

Say I want to Taylor-expand
$$f(\omega + m\sin(\Omega t))$$
where ω and Ω are frequencies, m is some constant and t denotes time. Then I would get
$$f(\omega + m\sin(\Omega t)) = f(\omega) + (m\sin(\Omega t)\frac{dI}{d\omega} + \ldots$$
Is it necessary to make any assumptions on the sizes of m and Ω in order to make the above expansion?


Niles.

 

[Dick]

If you are Taylor expanding in the quantity you are adding to ω and I means f(ω) that's fine. If you are going to truncate the expansion there and wondering if it's a accurate expansion that's going to depend on the size of the quantity you are adding to ω and the behaviour of f''(ω). You'll want to look at Taylor series remainder terms if you are concerned about how good it is.

 

[Niles]

Thanks!Niles.

 

[Niles]

If you are Taylor expanding in the quantity you are adding to ω ...

Actually, isn't this an expansion around the point ω rather than msin(Ωt) if we use the definition here http://en.wikipedia.org/wiki/Taylor_series#Definition?

 

[Dick]

Actually, isn't this an expansion around the point ω rather than msin(Ωt) if we use the definition here http://en.wikipedia.org/wiki/Taylor_series#Definition?

Sure it is. It's 'around' ω. I say the expansion is 'in' msin(Ωt) because that's the thing that appears in all the powers. Just terminology.

 

[Niles]

Ah, I see. Thanks for clarifying.Niles.