Let's work through each part of this problem step-by-step.

1(a). Writing Fourier Series Coefficients in Terms of $g(x)$

The Fourier series for an odd function $g(x)$ over the interval $[-L, L]$ can be represented using sine terms only because sine functions are odd functions. The Fourier sine series is given by: $g(x) = \sum_{n=1}^{\infty} b_n \sin\left(\frac{n \pi x}{L}\right),$ where $b_n$ are the Fourier sine series coefficients, defined as: $b_n = \frac{2}{L} \int_0^L g(x) \sin\left(\frac{n \pi x}{L}\right) \, dx.$

1(b). Simplifying the Coefficients in Terms of $f(x)$

Since $g(x)$ is the odd extension of $f(x)$, we have: [ g(x) = \begin{cases} f(x) & \text{for } x \in [0, L], \ -f(-x) & \text{for } x \in [-L, 0]. \end{cases} ]

Using this definition of $g(x)$, we can rewrite $b_n$ in terms of $f(x)$ on the interval $[0, L]$: $b_n = \frac{2}{L} \int_0^L f(x) \sin\left(\frac{n \pi x}{L}\right) \, dx.$ This simplification is possible because $g(x)$ is odd, so the integral over $[-L, L]$ simplifies to twice the integral over $[0, L]$ for an odd function $g(x)$.

1(c). Why is this Series Called the Fourier Sine Series?

This series is called the Fourier sine series because it represents the function $f(x)$ using only sine terms. This is due to the fact that $g(x)$, the odd extension of $f(x)$, has only sine terms in its Fourier expansion (since sine functions are odd and match the odd symmetry of $g(x)$). As a result, only sine terms appear in the series, capturing the behavior of $f(x)$ over $[0, L]$ without requiring cosine terms, which are associated with even functions.

Would you like more details on any part of this explanation? Here are some related questions for further exploration:

1. How would the Fourier series differ if $f(x)$ were extended as an even function?
2. How does the sine series relate to boundary conditions in solving differential equations?
3. What is the impact of the choice of $L$ on the Fourier sine series of $f(x)$?
4. Can a function with both even and odd components be represented by a Fourier sine series?
5. What properties of odd and even functions make them useful in Fourier series expansions?

Tip: In Fourier analysis, remember that odd functions use sine terms, while even functions use cosine terms in their expansions. This distinction is crucial when constructing series for functions with specific symmetries.