Deriving the Fourier Coefficients Formulas with a Manual Procedure

Question

I already know that there are built-in functions to compute the Fourier coefficients but I want to derive their formulas manually. In fact, I want to obtain the final well-known formulas for the coefficients with Mathematica carrying a step by step approach. Like the way we do on paper by hand.

So I consider an arbitrary function $f(x)$. I want to start with the known equation

$$f(x)=a_0+\sum _{i=1}^n \left[a_i \cos \frac{i\,\pi}{L} x +b_i \sin \frac{i\, \pi}{L} x \right] \tag{1}$$

and then use the orthogonality relations of trigonometric functions over $[-L,L]$ to derive $a_i$ and $b_i$. Specifically, the steps are

1. Multiplying by $\cos \frac{j\,\pi}{L} x$ and $\sin \frac{j\,\pi}{L} x$.
2. Integrating over $[-L,L]$.
3. Using orthogonality relations.
4. Finding coefficients.

As I am new to Mathematica I do not know that what is the best way to proceed.

This is my little effort. But I don't know how to simplify further. I need to distribute the integration over summations and get any constant out of the integrals. Then I should use orthogonality and finally I should find the coefficients $a_i$.

Any hint or help is appreciated. :)

Eq = f[x] == Subscript[a, 0] + Sum[Subscript[a, i]*Cos[((i*Pi)/L)*x] + 
      Subscript[b, i]*Sin[((i*Pi)/L)*x], {i, 1, n}]
For[i = 1, i <= 2, i++,
Eq[[i]] = Inactive[Integrate][Cos[(j \[Pi])/L x] Eq[[i]], {x, -L, L}];
]

Answer 1

How do you want to "use" orthogonality relation? There is a straightforward formula for a and b Here is a formula for a

A[f_, n_, L_] := 1/L Integrate[(f[x]) Cos[Pi x n /L], {x, -L, L}];

So here are your first ten coefficients for an "arbitrary" function:

f[x_] := Cos[Pi x/L] + 8 Cos[4 Pi x/L];
Table[A[f, i, L], {i, 0, 10}]
(*{0, 1, 0, 0, 8, 0, 0, 0, 0, 0, 0}*)