# Using Integration to evaluate the Bernoulli Polynomials

• Feb 1st 2009, 05:58 PM
TyrsFromAbove37
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
• Feb 2nd 2009, 04:43 AM
Opalg
Quote:

Originally Posted by TyrsFromAbove37
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.
• Feb 2nd 2009, 06:41 AM
TyrsFromAbove37
Thanks for the help, it's much appreciated. I was able to get all of the rest of the 4 that I needed.

-Ben