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Math Help - Using Integration to evaluate the Bernoulli Polynomials

  1. #1
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    Using Integration to evaluate the Bernoulli Polynomials

    I'm given the following information:

    Bo(x) = 1

    & for n greater than or equal to one:

    B'n(x) = nBn-1(x)
    The integral from 0 to 1 of Bn(x) = 1

    I'm asked to find Bn(x) for n = 1,2,3, and 4 and express the coefficients as fractions...

    I've tired everything that I can think of, I have a whole page worth of "notes" for this problem and I think that I'm way over thinking it...

    Any suggestions on where I can start (I just need that first shove in the right direction...)

    -Ben
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  2. #2
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    Quote Originally Posted by TyrsFromAbove37 View Post
    I'm given the following information:

    B_0(x) = 1

    & for n greater than or equal to one:

    B'_n(x) = nB_{n-1}(x),\quad\int_0^1\!\!\!B_n(x)\,dx = 1

    I'm asked to find B_n(x) for n = 1,2,3, and 4 and express the coefficients as fractions...
    You have to take them one at a time, calculating each one from the previous one.

    So B_1'(x) = B_0(x) = 1, which you integrate to get B_1(x) = x+c_1. To find the constant, use the condition on the integral: 1 = \int_0^1\!\!\!B_1(x)\,dx = \Bigl[\tfrac12x^2 + c_1\Bigr]_0^1 = \tfrac12+c_1. Therefore c_1=\tfrac12, and B_1(x) = x+\tfrac12.

    Next, B_2'(x) = 2B_1(x) = 2x+1, which you integrate to find B_2(x), using the condition on the integral to find the constant. And so on.

    The polynomials that you get in this way are slightly different from the usual Bernoulli polynomials, because the signs of some of the coefficients are different.
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  3. #3
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    Thanks for the help, it's much appreciated. I was able to get all of the rest of the 4 that I needed.

    -Ben
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