Looking for references on this form of a Taylor series

  • I
  • Thread starter Hiero
  • Start date
  • #1
321
68
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.
 

Answers and Replies

  • #2
hutchphd
Science Advisor
Homework Helper
2,324
1,619
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.
 
  • Like
Likes Hiero
  • #3
Infrared
Science Advisor
Gold Member
859
469
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.
 
  • Like
Likes Hiero
  • #4
321
68
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 o:)

Take care
 

Related Threads on Looking for references on this form of a Taylor series

  • Last Post
Replies
8
Views
2K
Replies
7
Views
2K
Replies
1
Views
1K
Replies
1
Views
3K
Replies
14
Views
3K
Replies
5
Views
13K
Replies
2
Views
4K
  • Last Post
Replies
2
Views
8K
Replies
4
Views
2K
Replies
2
Views
2K
Top