To find the Fourier series of the function $f(x) = 1$ defined on the interval $(- \pi, \pi]$, we follow these steps:

1. Determine the Fourier coefficients $a_0$, $a_n$, and $b_n$.

The Fourier series of a function $f(x)$ defined on $(- \pi, \pi]$ is given by: $f(x) = a_0 + \sum_{n=1}^{\infty} \left( a_n \cos(nx) + b_n \sin(nx) \right)$

The coefficients are calculated as follows:

$a_0 = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) \, dx$ $a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos(nx) \, dx$ $b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin(nx) \, dx$

2. Calculate $a_0$:

$a_0 = \frac{1}{2\pi} \int_{-\pi}^{\pi} 1 \, dx$

Since $f(x) = 1$ over the entire interval: $a_0 = \frac{1}{2\pi} \cdot \left[ x \right]_{-\pi}^{\pi} = \frac{1}{2\pi} \cdot ( \pi - (-\pi)) = \frac{1}{2\pi} \cdot 2\pi = 1$

3. Calculate $a_n$:

$a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} 1 \cos(nx) \, dx$

Since $\cos(nx)$ is an odd function when $n \neq 0$ and we are integrating over a symmetric interval: $a_n = \frac{1}{\pi} \cdot 0 = 0 \quad \text{for} \quad n \geq 1$

4. Calculate $b_n$:

$b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} 1 \sin(nx) \, dx$

Since $\sin(nx)$ is an odd function and we are integrating over a symmetric interval: $b_n = \frac{1}{\pi} \cdot 0 = 0 \quad \text{for} \quad n \geq 1$

Therefore, the Fourier coefficients $a_n$ and $b_n$ for $n \geq 1$ are all zero.

5. Construct the Fourier series:

Given that $a_0 = 1$ and $a_n = b_n = 0$ for $n \geq 1$, the Fourier series for $f(x) = 1$ is: $f(x) = 1 + \sum_{n=1}^{\infty} \left( 0 \cdot \cos(nx) + 0 \cdot \sin(nx) \right) = 1$

Thus, the Fourier series of the function $f(x) = 1$ defined on the interval $(- \pi, \pi]$ is simply:

$f(x) = 1$

Summary:

The Fourier series of the function $f(x) = 1$ on the interval $(- \pi, \pi]$ is:

$f(x) = 1$

Would you like any further details or have any questions?

Here are some related questions to expand on the topic:

1. How do you find the Fourier series of a function defined on an interval other than $(- \pi, \pi]$?
2. What is the physical significance of the Fourier coefficients?
3. How does the Fourier series converge for different types of functions?
4. Can you provide an example of a function with a more complex Fourier series?
5. How are Fourier series used in signal processing?
6. What is the relationship between the Fourier series and the Fourier transform?
7. How do Gibbs phenomena appear in Fourier series?
8. What are the conditions for a function to be represented by a Fourier series?

Tip: When calculating Fourier coefficients, taking advantage of symmetry (odd/even functions) can simplify the integrals significantly.