Talk:Cauchy formula for repeated integration

This entire page needs a lot of work, particularly in the scalar "proof" section. KlappCK (talk) 17:49, 27 July 2011 (UTC)

Well, that's overstating things a bit. Is there a lot to say about the Cauchy formula for repeated integration? :P I can fill in the rest of the proof sometime today, but it's a straightforward application of the multidimensional chain rule. Hell, I'll put that much in right now, that takes no work... Sniffnoy (talk) 06:31, 28 July 2011 (UTC)

Huh, now that I look closer, it's worse than I realized -- it pretty much totally ignores basepoint issues, concerning itself solely with the antiderivative part of the result. OK, I'll fix that later. Sniffnoy (talk) —Preceding undated comment added 06:37, 28 July 2011 (UTC).

Agreed Sniffnoy, I will give you another 48 hours to improve upon this (arguably) more elegant proof by induction before I go in and simply demonstrate that the result (Cauchy's formula) is correct by working "backwards" via repeated differentiation. I think that this section would also benefit from a rigorous explanation of when this rule applies. KlappCK (talk) 12:49, 28 July 2011 (UTC)

Uh... how does that differ from the approach currently there? I'm not following. This *is* a proof by repeated differentiation. Sniffnoy (talk) 17:23, 28 July 2011 (UTC)

Saying "Applying the chain rule, we can determine that ${\frac {d}{;dx}}f^{(-n)}(x)={\frac {d}{dx}}{\frac {1}{(n-1)!}}\int _{a}^{x}\left(x-y\right)^{n-1}f(y)\,dy=f^{-(n-1)}(x).$" is NOT repeated differentiation. If any thing it is a pointer to repeated differentiation, which is not satisfactory for a proof. Furthermore, this proof starts with a single integral and purportedly uses proof by induction to show the result comes from repeated used of the formula, ${\frac {d}{dx}}f^{(-1)}(x)={\frac {d}{dx}}\int _{a}^{x}f(y)\,dy=f(x).$, except it fails to use the base case to demonstrate the result for every subsequent case. This is fundamentally different than starting with the assertion: $\int _{x_{0}}^{x}\int _{x_{0}}^{x_{1}}\dots \int _{x_{0}}^{x_{n-1}}f(x_{n})\,dx_{n}\dots \,dx_{2}\,dx_{1}=\int _{x_{0}}^{x}f(t){\frac {(x-t)^{n-1}}{(n-1)!}}\,dt.$ and repeatedly differentiating until you arrive at an explicitly equivalent result.KlappCK (talk) 13:10, 29 July 2011 (UTC)

Repetition is just a special case of induction. Every proof by "repeated" something, if written out formally and explicitly, is actually a proof by induction, because the whole definition of "doing something n times" is recursive (inductive). There isn't a meaningful distinction. I personally find the explicit induction easier to understand in this case, but regardless, they're the same. Sniffnoy (talk) 18:45, 29 July 2011 (UTC)

Disagreements over what constitutes a proof by induction aside, I believe your changes are a good step in the right direction. However, I am uncertain of the purpose of $f^{(-1)}(a)=\int _{a}^{a}f(t)\,dt=0$ and $f^{-(n+1)}(a)={\frac {1}{n!}}\int _{a}^{a}\left(x-t\right)^{n}f(t)\,dt=0$ in the proof. KlappCK (talk) 14:44, 2 August 2011 (UTC)

Ah, simple. Without that, it would be an nth antiderivative, but it would not necessarily be the nth integral based at a. We don't just want it to be any nth antiderivative, we want it to be the particular one given by the original repeated integral. Sniffnoy (talk) 15:23, 2 August 2011 (UTC)

Broken equation, but only when logged out

The equation before Proof follows, is broken, but only when I am logged out. I dont know why. It looks like,

Failed to parse(unknown function '\begin'): {\begin{aligned}f^{{-(n+1)}}(x)&=\int _{a}^{x}\int _{a}^{{\sigma _{1}}}\cdots \int _{a}^{{\sigma _{{n}}}}f(\sigma _{{n+1}})\,{\mathrm {d}}\sigma _{{n+1}}\cdots \,{\mathrm {d}}\sigma _{2}\,{\mathrm {d}}\sigma _{1}\\&={\frac {1}{(n-1)!}}\int _{a}^{x}\int _{a}^{{\sigma _{1}}}\left(\sigma _{1}-t\right)^{{n-1}}f(t)\,{\mathrm {d}}t\,{\mathrm {d}}\sigma _{1}\\&={\frac {1}{(n-1)!}}\int _{a}^{x}\int _{t}^{x}\left(\sigma _{1}-t\right)^{{n-1}}f(t)\,{\mathrm {d}}\sigma _{1}\,{\mathrm {d}}t\\&={\frac {1}{n!}}\int _{a}^{x}\left(x-t\right)^{n}f(t)\,{\mathrm {d}}t\end{aligned}}

203.3.133.17 (talk) 08:23, 10 February 2014 (UTC)

Confusing Base Case

The Base Case part doesn't seem to do its job. An easy way is to write out the definition of f^-n(x) for n = 1 and then note that (x-t)^n-1/(n-1)! = 1. Or so it seems to me. I'm not sure, so I'll wait for comment before I attempt a correction.Stony Lonesome (talk) 17:02, 16 March 2021 (UTC)

So, to be clear, what you find confusing is that the factor of (x-t)^n-1/(n-1)! isn't explicitly noted to be 1, but rather is simply omitted as 1? Or what? Note that the base case follows the same recipe as the rest -- take the definition, check the derivative, and check the basepoint. Is it just confusing because that factor isn't written out explicitly? Yes, adding that in might be a good idea then. Do you think it needs further reorganization beyond that? Sniffnoy (talk) 17:02, 17 March 2021 (UTC)

Well I went ahead and added that in, FWIW. Sniffnoy (talk) 17:05, 17 March 2021 (UTC)