Can a repeated integral be simplified into a single integral?

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Discussion Overview

The discussion revolves around the possibility of simplifying repeated integrals into single integrals and whether nth derivatives can be expressed through a single differentiation process. Participants explore theoretical implications and mathematical formulations related to these concepts.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants reference Cauchy's formula for repeated integration and question if similar simplifications can apply to nth derivatives.
  • There is a suggestion that if the nth antiderivative can be expressed as a single integral, then the nth derivative might also be expressible through a single differentiation, contingent on algebraic manipulation.
  • One participant cites Cauchy's differentiation formula but notes the absence of a general formula for expressing the nth derivative as a single differentiation, suggesting it may be impossible for general functions.
  • Leibniz's rule for the nth derivative of a product of functions is mentioned as a known result, but its applicability to the broader question is unclear.
  • Another participant humorously questions the identity of "General Leibniz" in reference to the rule.
  • There is a specific example provided for expressing the derivative of a product, illustrating the complexity of the nth derivative in such cases.

Areas of Agreement / Disagreement

Participants express differing views on the feasibility of expressing nth derivatives as single differentiations, with no consensus reached on the matter. The discussion includes both supportive and skeptical perspectives regarding the simplification of repeated integrals and derivatives.

Contextual Notes

Participants acknowledge limitations in existing formulas and the need for further exploration of the conditions under which these mathematical expressions hold true.

Jhenrique
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If a repeated integral can be expressed how an unique integral:

e48a88551eb7f5907007df368509cc53.png


https://en.wikipedia.org/wiki/Cauchy_formula_for_repeated_integration

So is possible express the nth derivative with an unique differentiation?
 
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Mark44 said:
What are your thoughts on this?

Given a function f, do you think it would be possible to express f'' as a single differentiation?

Yeah! Maybe, if is possible to find the nth antiderivative with an unique integral, so should be possible to find the nth derivative with an unique differentiation through of algebraic manipulation.
 
There is Cauchy's differentiation formula
$$\mathrm{f}^{(n)}(x)=
\frac{n!}{2\pi \imath}\oint \frac{\mathrm{f}(z) \mathrm{d}z } {(z-x)^{n+1}}$$
and some other related formula, but I do not recall any of the form
$$\left( \dfrac{d}{dx}\right) ^n \mathrm{f}(x)=\dfrac{d}{dx} \mathrm{g}_n (x)\mathrm{f}(x)$$
which given the rules of differentiation seems impossible for general f
https://en.wikipedia.org/wiki/Cauchy's_integral_formula
 
lurflurf said:
$$\left( \dfrac{d}{dx}\right) ^n \mathrm{f}(x)=\dfrac{d}{dx} \mathrm{g}_n (x)\mathrm{f}(x)$$

Yeah! I'm looking for something like this!
 
bigfooted said:
We only have Leibniz' rule for the nth derivative of a product of functions:
http://en.wikipedia.org/wiki/General_Leibniz_rule

It's more comprehensible express the derivative of a produt like way:
$$(f\times g)^{(2)} = f^{(2)}g^{(0)} + 2f^{(1)}g^{(1)} + f^{(0)}g^{(2)}$$
 

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