# Looking for references on this form of a Taylor series

• I
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.

Homework Helper
2022 Award
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.

Hiero
Gold Member
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
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