Faà di Bruno's formula - Wikipedia, the free encyclopedia
(to the composition )
Let for and .
Show that for all and show that the Taylor series for g about 0 agrees with only at .
So yeah, I am completely lost now in my real analysis class and really don't understand much of anything going on now. It says to use induction to prove that there exist polynomials , so that for , where . Don't really now where to begin, thanks.
Sorry, I don't understand. We have not learned anything that advanced yet. I don't understand what Bell polynomials are and how that relates to getting a polynomial of one variable. I don't understand how two superscripts come into play or what it means to have a polynomial called .
So, which part are you having trouble with?
So, we can prove by induction that . To do this, we prove as was suggested that .
To, see this we first note that which clearly satisfies the conditions.
Next, we suppose that and so , but a little factoring shows this is just a polynomial in times
I had an idea like that but I guess I got too caught up with the suggested hint because it says to consider that where and then show that is actually a series of products, where each product is . That notation of completely through me off, never seen that before and not sure where to begin, but if it can be done more simply as a single polynomial of times , I much rather do that.