For the first two questions on this assignment, the following will be useful. Suppose h(x)
is even and w(x) is odd. Then, we have:
Z L
−L
h(x) dx = 2
Z L
0
h(x) dx, and
Z L
−L
w(x) dx = 0
1. Definition: The Fourier sine series for (or generated by) f (x) on [0, L] is the Fourier series
generated by the odd extension of f (x) on [−L, L].
(a) Let g(x) be the odd extension of f (x). Write the Fourier series coefficients in terms of
g(x).
(b) Use the fact that g(x) is the odd extension of f (x) to simplify the coefficients from part
(a) and write them in terms of integrals involving f (x).
(c) Why is this series called the Fourier sine series?

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.

For the first two questions on this assignment, the following will be useful. Suppose h(x) is even and w(x) is odd. Then, we have: Z L −L h(x) dx = 2 Z L 0 h(x) dx, and Z L −L w(x) dx = 0 1. Definition: The Fourier sine series for (or generated by) f(x) on [0, L] is the Fourier series generated by the odd extension of f(x) on [−L, L]. (a) Let g(x) be the odd extension of f(x). Write the Fourier series coefficients in terms of g(x). (b) Use the fact that g(x) is the odd extension of f(x) to simplify the coefficients from part (a) and write them in terms of integrals involving f(x). (c) Why is this series called the Fourier sine series?

This page provides a detailed explanation of how to write Fourier series coefficients in terms of an odd extension of a function, simplifying them into integrals involving the original function f(x). Learn why this series is called the Fourier sine series and its applications in Fourier analysis. Suitable for students in Grades 11-12 and undergraduate math courses.

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