Cauchy Repeated Integration Explanation?

In summary: In the third step of the proof by induction, you are allowed to choose the lower limit as t and the upper limit as x, because this gives you the desired result. The proof is still valid even if you don't choose these limits, but it is easier to work with the limits that are given.
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TheDemx27 said:
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##.
 
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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.
 
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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)
 
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1. What is Cauchy repeated integration?

Cauchy repeated integration is a mathematical technique used to evaluate definite integrals. It involves breaking down the integral into smaller intervals and integrating each interval separately.

2. How does Cauchy repeated integration work?

Cauchy repeated integration works by dividing the interval of integration into smaller intervals and integrating each interval separately. The results are then summed together to get the final value of the integral.

3. What is the purpose of using Cauchy repeated integration?

The purpose of using Cauchy repeated integration is to evaluate complex or improper integrals that cannot be solved using traditional integration methods. It allows for a more accurate and efficient way of finding the value of an integral.

4. What are the advantages of using Cauchy repeated integration?

One advantage of using Cauchy repeated integration is that it can be used to evaluate a wide range of integrals, including those that cannot be solved using other methods. It also allows for more accurate results, as the smaller intervals used in the process help to minimize error.

5. Are there any limitations to using Cauchy repeated integration?

One limitation of using Cauchy repeated integration is that it can be time-consuming and labor-intensive, especially for integrals with a large number of intervals. It also requires a good understanding of integration techniques and may not always provide exact solutions for certain integrals.

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