# I Cauchy Repeated Integration Explanation?

1. Mar 11, 2016

### TheDemx27

2. Mar 11, 2016

### andrewkirk

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 $\sigma_2=\sigma_1$.

3. Mar 11, 2016

### FactChecker

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

4. Mar 12, 2016

### 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)

Last edited: Mar 12, 2016
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