Question

More Specialized Fourier Scries. Let \(f\) be a function originally defined on \(0 \leq x \leq L\). In this section we have shown that it is possible to represent \(f\) either by a sine series or by a cosine series by constructing odd or even periodic extensions of \(f,\) respectively. Problems 38 through 40 concern some other more specialized Fourier series that converge to the given function \(f\) on \((0, L) .\) $$ \begin{array}{l}{\text { Let } f \text { be extended into }(L, 2 L] \text { in an arbitrary manner. Then extend the resulting function }} \\ {\text { into }(-2 L, 0) \text { as an odd function and elsewhere as a periodic function of period } 4 L \text { (see }} \\ { \text { Figure }10.4 .6) . \text { Show that this function has a Fourier sine series in terms of the functions }} \\\ {\sin (n \pi x / 2 L), n=1,2,3, \ldots . \text { that is, }}\end{array} $$ $$ f(x)=\sum_{n=1}^{\infty} b_{n} \sin (n \pi x / 2 L) $$ where $$ b_{n}=\frac{1}{L} \int_{0}^{2 L} f(x) \sin (n \pi x / 2 L) d x $$ $$ \text { This series converges to the original function on }(0, L) $$ (Figure cant copy)

Short answer

Question: Determine whether the Fourier sine series converges to the original function \(f(x)\) on the interval \((0, L)\). Answer: Yes, the Fourier sine series converges to the original function \(f(x)\) on the interval \((0, L)\).

Step-by-step solution

Extend the function

To start, we are asked to extend the function \(f(x)\) into \((L, 2L]\) in an arbitrary manner while assuring function continuity. After that, extend the resulting function into \((-2L, 0)\) as an odd function and elsewhere as a periodic function of period \(4L\). Let's denote the function after these extensions as \(F(x)\).

Fourier sine series

Next, we will find the Fourier sine series of the extended function \(F(x)\). The general form of the Fourier sine series is given by: $$ F(x) = \sum_{n=1}^\infty b_n \sin(\frac{n \pi x}{2L}) $$ where the coefficients \(b_n\) are defined as follows: $$ b_n = \frac{1}{L} \int_0^{2L} F(x) \sin(\frac{n \pi x}{2L}) dx $$

Calculate \(b_n\)

To find \(b_n\), we need to calculate the integral: $$ b_n = \frac{1}{L} \int_0^{2L} F(x) \sin(\frac{n \pi x}{2L}) dx $$ Recall that \(F(x)\) is equal to the original function, \(f(x)\), on the interval \((0, L)\). Therefore, we can rewrite the equation above as follows: $$ b_n = \frac{1}{L} \int_0^{L} f(x) \sin(\frac{n \pi x}{2L}) dx + \frac{1}{L} \int_L^{2L} F(x) \sin(\frac{n \pi x}{2L}) dx $$ We can't simplify it further without knowing the exact form of the continuous extension into \((L, 2L]\) of \(f(x)\). However, it's worth noticing that we have retrieved the Fourier sine coefficients for the original \(f(x)\) (when restricted to \((0, L)\)) on the first integral term.

Determine convergence

The final step is to show that the Fourier sine series converges to the original function \(f(x)\) on the interval \((0, L)\). As mentioned in step 3, we have retrieved the Fourier sine coefficients of the original function \(f(x)\). The sum of the Fourier sine series now becomes: $$ f(x) = \sum_{n=1}^\infty b_n \sin(\frac{n \pi x}{2L}) $$ Since we found the Fourier sine series of the extended function \(F(x)\) and it's equal to the original function \(f(x)\) on the interval \((0, L)\), it's ensured that the Fourier sine series converges to the original function \(f(x)\) on this interval.

Key concepts

Fourier Series Convergence

When delving into the realm of Fourier series, an important aspect to consider is their convergence. The convergence of a Fourier series means that as we add up more and more terms of the series, the sum approaches the function it represents within a certain interval. The concept is vital because it reaffirms the idea that an infinite series can indeed represent a finite function accurately in mathematics.

Regarding the Fourier sine series, it guarantees that the infinite sum of sine functions multiplied by their respective coefficients will converge to the original function over a specified interval. In this case, we've confirmed the convergence on the interval (0, L). Essentially, what this indicates is that for any point within this interval, if you add up enough terms of the Fourier series, the sum will be as close as you desire to the actual value of the function at that point.

Periodic Function Extension

For the periodic function extension, first consider what it means to have a periodic function. Periodicity implies that a function repeats its values at regular intervals or periods. Extending a function periodically means that we are replicating the function's shape and values outside the original interval in such a way that the function's form remains seamless.

In the context of our problem, the extension of the function (f(x)) into the interval (L, 2L] is arbitrary, but specifically, into (-2L, 0) it's extended as an odd function. Once extended, this newly constructed function exhibits a 4L-periodicity - it starts again after every interval of length 4L. This periodic behavior is an essential property that allows us to construct the Fourier sine series which we substantiate to be representative of the function within our interval of interest.

Odd Function Symmetry

Odd function symmetry is a characteristic that greatly simplifies the analysis and construction of Fourier series. A function (f(x)) is odd if it satisfies the condition that (f(-x) = -f(x)) for all (x) in its domain. This property translates to the function being symmetric with respect to the origin, meaning that the function's graph on one side of the y-axis is the mirror image of the other, but flipped.

In the exercise at hand, after the function is extended arbitrarily over (L, 2L], it is then extended as an odd function over (-2L, 0), which is instrumental in deriving a purely sinusoidal Fourier series - a Fourier sine series. This is because the sine function itself is odd and harnessing this symmetry aids in trimming the series down to terms that reflect the symmetry of the extended function, (F(x)).

Fourier Coefficients

The calculation and understanding of Fourier coefficients are central to the study of Fourier series. These coefficients, denoted as (b_n) in the context of a Fourier sine series, regulate the amplitude or