# Thread: am i going right on this fourier series

1. ## am i going right on this fourier series

how will i integrate this by parts? what is u and dv?

2. Originally Posted by llinocoe

how will i integrate this by parts? what is u and dv?
It should be $\displaystyle b_n = 2 \int_{\frac{-1}{2}}^{\frac{1}{2}} (x^3+x)\sin(nx) dx$

Where n = 1, 2, 3, 4, 5, 6, 7...

Do it using parts, yes.

$\displaystyle b_n = 2 \int_{\frac{-1}{2}}^{\frac{1}{2}} (x^3+x)\sin(nx) dx = 2 \int_{\frac{-1}{2}}^{\frac{1}{2}}\bigg(x^3\sin(nx) +x\sin(nx)\bigg) dx$

$\displaystyle = 2 \int_{\frac{-1}{2}}^{\frac{1}{2}} x^3\sin(nx) dx+2 \int_{\frac{-1}{2}}^{\frac{1}{2}}x\sin(nx) dx$

3. Originally Posted by llinocoe

how will i integrate this by parts? what is u and dv?
Originally posted by HallsofIvy in one of the other threads this question was asked (reply was not moved because it pre-dated this thread):

Originally Posted by HallsofIvy
Strictly speaking only periodic functions have Fourier series. Do you intend that this will be repeated in exactly the same way over intervals of length 1? If so, do you know the formulas for the coefficients of a Fourier series. This is an odd function so its Fourier series will involve only sine functions.

4. Originally Posted by llinocoe

how will i integrate this by parts? what is u and dv?
Originally posted by HallsofIvy in one of the other threads this question was asked (reply was not moved because it pre-dated this thread):

Originally Posted by HallsofIvy
Strictly speaking only periodic functions have Fourier series. Do you intend that this will be repeated in exactly the same way over intervals of length 1? If so, do you know the formulas for the coefficients of a Fourier series. This is an odd function so its Fourier series will involve only sine functions.
And strictly speaking he was wrong, the trig functions (with appropriate scalling) form a complete orthonormal basis for $\displaystyle L^2(a,b)$ , and Fourier series theory does not refer to anything outside the interval in the definition of the function space.

(other function and generalised function spaces may be used instead, but the principle is the same, we are only discussing the given interval, if you have a periodic function and its Fourier series is also periodic, when treated as a formal series, is also periodic with the same period then fine you can use that but the development of the theory does not require it)

The functions are only defined on the interval $\displaystyle (a,b)$, to consider what they might do outside this interval is irrelevant, they are not defined there.

CB