This post is about an early definition I made, which I call infinitieth derivatives, or more precisely -th derivative. The -th derivative of a smooth function from to is the pointwise limit of the sequence of -th derivatives etc, assuming, of course, that the pointwise limit does in fact exist for every in .
To give some examples, the -th derivative of every polynomial is the function. The -th derivative of the exponential function is . Slightly more interestingly, the -th derivative of the exponential function is the function. The case of is interesting because the -th derivatives of converge pointwise but not uniformly to the function.
However, it seems that “most” smooth functions do not have a -th derivative, like sine or cosine or an exponential function with .
The reason this definition is interesting, is because we can take the derivative of the -th derivative itself, to get the -th derivative, and keep iterating this process thoroughout the transfinite ordinals, where, at limit ordinals, we take the limit of the sequence of derivatives thus far.
However, I have noticed that in every case I have examined so far, the -th derivative is a scalar multiple of the exponential function , which forces all further derivatives to be the same as the -th derivative, thus trivializing the notion of ordinal derivatives. What I really want to know is whether the -th derivative of every real function that has a -th derivative is a scalar multiple of the exponential function .
Comments
Very interesting; I can’t easily understand it. Can you provide the link to the post where you introduce -th derivative, please? I can’t seem to find it in the list of your FN posts.
This post is in fact the post where I introduce my definition.
I see. The first sentence in your post confused me, where you mention “early definition” - that’s why I decided there was another post.
I am wondering if there is a chance you can be so kind as to elaborate on what “the pointwise limit of the sequence of -th derivatives” means, please. I am not a mathematician, it’s been difficult for me to understand how this definition looks mathematically and how it can be interperted geometrically. I would greatly appreciate if you could provide the formula for this definition, please. Thank you in advance.
So you are swapping to a net for bigger than countable ordinals, the limit is still unique by pts wise unicity via Hausdorffness. Consider trying a non abakytic smooth function? Perhaps that will break your conundrum.
No, I am not swapping to a net for bigger than countable ordinals, I am talking about limits of transfinite limit ordinal sequences. There is a notion of “ordinal limits”, which I came up with independently, although some mathematicians came up with that concept several decades ago.