Looking for references on this form of a Taylor series

In summary, the conversation discusses the Taylor series and its usefulness, particularly in regards to expressing a function as a 3D Fourier transform. The formula for the Taylor series is provided and its validity is confirmed, along with a resource that explains its use in multi-index notation. The conversation also mentions the need for the function to be analytic for the formula to make sense.
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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.
 
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  • #2
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.
 
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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.
 
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Infrared said:
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
 

1. What is a Taylor series?

A Taylor series is a mathematical representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.

2. How is a Taylor series useful in scientific research?

Taylor series are useful in approximating complex functions, making it easier to analyze and understand their behavior. They also allow for the estimation of values beyond the given data points, making it a valuable tool in scientific research and data analysis.

3. What is the purpose of looking for references on a Taylor series?

Looking for references on a Taylor series allows scientists to build upon existing knowledge and findings in the field. It also ensures accuracy and credibility in their research by referencing reputable sources.

4. Are there different forms of a Taylor series?

Yes, there are different forms of a Taylor series such as the Maclaurin series (centered at x=0) and the general Taylor series (centered at any point). These forms differ in the way the terms of the series are calculated.

5. How can I find references on a specific form of a Taylor series?

You can find references on a specific form of a Taylor series by searching in scientific databases or journals, consulting textbooks on mathematical analysis, or seeking guidance from experts in the field.

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