Fourier Series - Half Range Expansions

[erok81]

1. Homework Statement [/b]

Find the half range expansion of the given function.

I tried to show f(x) using LaTeX but couldn't figure out how to stack three rows, so I have attached an image instead.

Homework Equations

Fourier series expansions for a0, an, bn - I can post these once I get to this point.

The Attempt at a Solution

In my image you can see how I've tried to graph this in order to get my even and odd functions. I know this isn't correct, but cannot figure out how to do it. The p throws me off, I know it is usually for the period but I can't figure out how that plays into this case. There aren't any examples in my text that are close to this.

Any tips on even how to get my graph started would be appreciated. Then we can go from there extending them odd/even over a period.

 

[kurt.physics]

2. Homework StatementFind the half range expansion of the given function.f(x) = x^3 - 4x^2 + px + 2, -p/2 < x < p/2Homework EquationsFourier series expansions for a0, an, bnThe Attempt at a SolutionTo find the Fourier series expansion, we can first divide the function into its odd and even parts. The even part is given by the equation f_e(x) = x^3 - 4x^2, and the odd part is given by the equation f_o(x) = px + 2.Next, we can use the formula for the Fourier series expansion to find the coefficients a0, an, and bn. For the even part, the coefficient a0 is given by the equation a0 = (2/p)∫f_e(x) dx from -p/2 to p/2. This gives us a0 = (2/p) [(x^4/4) - (2x^3/3) + (4x^2/2)] evaluated from -p/2 to p/2. Using the fundamental theorem of calculus, we can simplify this to a0 = (2/p)[p^4/8 - p^3/4 + 2p^2/2]. Similarly, the coefficients an and bn for the even part can be found using the equations an = (2/p)∫f_e(x)cos(nπx/p) dx from -p/2 to p/2, and bn = (2/p)∫f_e(x)sin(nπx/p) dx from -p/2 to p/2. For the odd part, the coefficient a0 is 0 since the integral of an odd function from -p/2 to p/2 is 0. The coefficients an and bn for the odd part can be found using the equations an = (2/p)∫f_o(x)cos(nπx/p) dx from -p/2 to p/2, and bn = (2