To find the coefficients $a_0$, $a_n$, $b_n$, and $c_n$, we need more context. Usually, these coefficients are part of a Fourier series representation of the function $f(x)$ over a given interval.

Given the function:

$f(x) = 2 - x, \quad 0 < x < 4$

we would typically expand this function as a Fourier series on a specified interval, such as $[0, 4]$ or $[0, 2\pi]$. The Fourier coefficients are calculated as follows:

Fourier Coefficients

1. $a_0$ (the average or DC component): $a_0 = \frac{1}{L} \int_0^L f(x) \, dx$ where $L$ is the length of the interval. If the interval is $[0, 4]$, then $L = 4$.

2. $a_n$ (cosine coefficients): $a_n = \frac{2}{L} \int_0^L f(x) \cos\left(\frac{n \pi x}{L}\right) \, dx$

3. $b_n$ (sine coefficients): $b_n = \frac{2}{L} \int_0^L f(x) \sin\left(\frac{n \pi x}{L}\right) \, dx$

4. $c_n$ (complex form coefficients): $c_n = \frac{1}{L} \int_0^L f(x) e^{-i \frac{2 \pi n x}{L}} \, dx$

To proceed with the exact calculation of these coefficients, I need to confirm:

1. The type of Fourier series (e.g., complex, sine, cosine, or general form).
2. The interval over which we are calculating these coefficients (usually given or assumed to be $[0, 4]$ or $[0, 2\pi]$).

Please confirm or provide more context, so I can help you compute these coefficients accurately.

Relative Questions

1. Should I consider the interval $[0, 4]$ for computing the Fourier coefficients?
2. Do you need the Fourier series expansion in terms of sine and cosine, or in the complex exponential form?
3. Are there any specific boundary conditions to consider for this function $f(x)$?
4. Would you like a step-by-step calculation for each coefficient?
5. Do you have any preferred method for calculating these coefficients (e.g., by hand, using a particular software)?

Tip

When dealing with Fourier series, always ensure that the function is periodic over the interval, or use piecewise extensions to make it periodic before calculating the coefficients.