Cauchy Repeated Integration Explanation?

[TheDemx27]

https://anhngq.wordpress.com/2013/04/25/the-cauchy-formula-for-repeated-integration/

Could someone please explain, bit by bit, how the formula works? For instance, why are we integrating with respect to sigma, up to sigma in the equation before it?

 

[andrewkirk]

why are we integrating with respect to sigma, up to sigma in the equation before it?

The easiest way to understand this is to look at the case with only two nested integrals.

$$f^{(-2)}(x)=\int_a^x\int_a^{\sigma_1}f(\sigma_2)d\sigma_2d\sigma_1$$

Now draw the square bordered by (a,a),(a,x),(x,x),(x,a) in the number plane and shade the region in which the integral is being performed, where we map $\sigma_1$ to the horizontal axis and $\sigma_2$ to the vertical axis.
[You may find it easier to visualise this if you set $a=0,x=1$, and then generalise it later]

The outer integration is along the horizontal axis.
The inner integration is in the vertical direction and, for a given value of $\sigma_1$, it integrates along the vertical line from $(\sigma_1,a)$ to $(\sigma_1,\sigma_1)$.

The integration region is the triangle with vertices (a,a), (a,x), (x,x). The triangle is bounded by the horizontal and vertical axes and the 45 degree line with equation $\sigma_2=\sigma_1$.

 

[FactChecker]

Start with f(x₁) = ∫₀^x₁f^'(x₂)dx₂.
Now replace f^'(x₂) with its own integral of f^''(x₃):
f(x₁) = ∫₀^x₁[∫₀^x₂f^''(x₃)dx₃]dx₂.
The part within square brackets, [], is that substitution.
You can keep doing this as many times as you wish.

 

[TheDemx27]

Thankyou both. One last thing, could someone explain the change of the integral's limits in the third step of the proof by induction? I get that choosing the lower limit as t and the upper limit as x will give you the desired result, but I don't see why you are allowed to do that.
(edit: figured it out on my own)