Looking for references on this form of a Taylor series

[Hiero]

I was trying to find this form of the Taylor series online:
$$\vec f(\vec x+\vec a) = \sum_{n=0}^{\infty}\frac{1}{n!}(\vec a \cdot \nabla)^n\vec f(\vec x)$$
But I can’t find it anywhere. Can someone confirm it’s validity and/or provide any links which mention it? It seems quite powerful to be so obscure; maybe I’m just bad at googling.

Thanks.

 

[hutchphd]

It is clear that expressing f(x) as a 3D Fourier transform shows this to be true but I don't know whether that is unnecessarily restrictive.

 

[Infrared]

I found your formula here in the first line of page 3: https://sites.math.washington.edu/~folland/Math425/taylor2.pdf. You should of course assume that your function is actually analytic for this to make sense (not using a finite sum and error term).

It's equivalent to the usual formulation of Taylor's theorem once you expand out what $(a\cdot\nabla)^n$ means and do the combinatorics.

 

[Hiero]

I found your formula here in the first line of page 3: https://sites.math.washington.edu/~folland/Math425/taylor2.pdf. You should of course assume that your function is actually analytic for this to make sense (not using a finite sum and error term).

It's equivalent to the usual formulation of Taylor's theorem once you expand out what $(a\cdot\nabla)^n$ means and do the combinatorics.

Thank you! Great resource. I had never seen “multi-index” notation before; it’s very useful for this purpose!

I’m impressed by how quickly you found that. I guess I am just bad with google

Take care