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I Cauchy Repeated Integration Explanation?

  1. Mar 11, 2016 #1

    TheDemx27

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  2. jcsd
  3. Mar 11, 2016 #2

    andrewkirk

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    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##.
     
  4. Mar 11, 2016 #3

    FactChecker

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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.
     
  5. Mar 12, 2016 #4

    TheDemx27

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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)
     
    Last edited: Mar 12, 2016
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