Fourier series confusion

Question

Find the Fourier series of

Question 1

$$ f(x) =\begin{cases} 0&&\text{for $-1 < x < 0$}\\\\ x&&\text{for $0 \leq x \leq 1$} \end{cases} $$

and

Question 2

$$f(x) = x + \pi,\; - \pi < x < \pi$$

Solutions

For Question 1, the solution is $$ \begin{cases} a_0&= \frac{1}{2} \int_{0}^{1} x dx = 1/4\\ a_n& = \int_{0}^{1} x \cos(n \pi x) dx = \frac{(-1)^n - 1}{(\pi n)^2}\\ b_n&= \int_{0}^{1} x \sin(n \pi x) dx = \frac{(-1)^{n+1}}{(\pi n)} \end{cases} $$

Then, just use the formula $$f(x)=a_0 + \sum_{n=1}^{\infty} a_n \cos(n \pi x) + b_n \sin(n \pi x)$$

For Question 2 $$ \begin{cases} a_0&= \frac{1}{2 \pi} \int_{- \pi}^{\pi} (x + \pi) dx = 2 \pi\\ a_n&= \frac{1}{\pi} \int_{- \pi}^{\pi} (x+\pi)\cos(n \pi x) dx = 0\\ b_n&= \frac{1}{\pi} \int_{- \pi}^{\pi} (x+\pi)\sin(n \pi x) dx = \frac{2(-1)^{n+1}}{n} \end{cases} $$ Then, use the same formula again with something different example: $$f(x) = a_0 + \sum_{n=1}^{\infty} a_n\cos(nx) + b_n \sin(nx)$$

Question

My question is how do you get the denominator like for the first one how does $\frac{1}{2}$ come from? And for the second one where does $\frac{1}{\pi}$ come from and why is it $\cos(nx)$ and $\sin(nx)$ instead of $\cos(n \pi x)$?

Answer 1

It's because the general formulas when the half-period is $L$ are

$$\begin{align} a_0 &= \frac{1}{2L}\int_{-L}^{L} f(x)\, dx \\ a_n &= \frac{1}{L}\int_{-L}^{L} f(x)\cos\left(\frac{n\pi x}{L}\right)\, dx, \quad n\ge 1 \\ b_n &= \frac{1}{L}\int_{-L}^{L} f(x)\sin\left(\frac{n\pi x}{L}\right)\, dx, \quad n\ge 1.\end{align}$$

Note that in the first example, the half-period is $L=1$, whereas in the second example, $L=\pi$.