Question

You are given \(f(x)\) on an interval, say \(0< x< b\). Sketch several periods of the even function \(f_{c}\) of period \(2 b,\) the odd function \(f_{s}\) of period \(2 b,\) and the function \(f_{p}\) of period \(b\), each of which equals \(f(x)\) on \(0< x< b\). Expand each of the three functions in an appropriate Fourier series. $$f(x)=\left\\{\begin{array}{cl} x, & 0< x< 1 \\ 2-x, & 1< x< 2 \end{array}\right.$$

Short answer

Sketch even, odd, periodic function, calculate appropriate Fourier series.

Step-by-step solution

Sketch the Functions

First, identify the intervals and sketch the given function for each type of extension:1. Even function of period 2b: Reflects symmetrically about y-axis. For the interval 0 < x < b,repeat and reflect the function.2. Odd function of period 2b: Reflects antisymmetrically about y-axis. On same interval, reflect function.3. Periodic function of period b: Repeat the function in each period.

Determine Fourier Series Expansion for Period \(2b\) - Even Function

For the even function,compute the Fourier cosine series.Fourier series of even: $$f_c(x)=a_0/2+\sum_{n=1}^{\infty} a_n\cos\left(\frac{\pi nx}{2}\right),$$where coefficients are $$a_0=\frac{2}{b}\int_0^b f(x) dx,$$$$a_n=\frac{2}{b}\int_0^b f(x)\cos\left(\frac{\pi nx}{b}\right)dx.$$

Determine Fourier Series Expansion for Period \(2b\) - Odd Function

For the odd function,compute Fourier sine series.Fourier series is $$f_s(x)=\sum_{n=1}^{\infty} b_n\sin\left(\frac{\pi nx}{2}\right),$$where coefficients are $$b_n=\frac{2}{b}\int_0^b f(x)\sin\left(\frac{\pi nx}{b}\right) dx.$$

Determine Fourier Series Expansion for Period \(b\) - Periodic Function

Use the periodic function,compute complete Fourier series.Fourier series is $$f_p(x)=a_0/2+\sum_{n=1}^{\infty}\left[a_n\cos\left(\frac{2a_n x}{b}\right)+b_new\frac{2n x}{b}\right]\sin\left(\frac{2n x}{b}\right),$$coefficients are,$$a_0=\frac{2}{b}\int_0^b f(x)dx,$$$$a_n=\frac{2}{b}\int_0^b f(x)\cos\left(\frac{2n x}{b}\right) dx,$$\$$b_n=\frac{2}{b}\int_0^b f(x)\sin\(x\).$$

Finalize the Series

Compute integrals for fhe coefficients.Combine identified Fourier series.Use standard integration methods. Delivery final expressions for expansions.

Key concepts

Even Function

An even function is a function where the graph is symmetric about the y-axis. This means if you fold the graph along the y-axis, both sides would match perfectly. Mathematically, an even function holds the property that \(f(x) = f(-x)\).
When extending a function to make it even on the interval \(0 < x < b\) to \(2b\), you reflect the function across the y-axis and duplicate it in the opposite direction. This results in a periodic pattern where each cycle is double the original interval.
The Fourier series for an even function uses only cosine terms, because cosine functions are also even.

Odd Function

An odd function is one where the graph is symmetric about the origin. That is, if you rotate the graph 180 degrees about the origin, the resulting graph would be the same. Mathematically, an odd function satisfies \(f(-x) = -f(x)\).
To extend a function to make it odd on the interval \(0 < x < b\) to \(2b\), you reflect the function across both the y-axis and the x-axis. This creates an antisymmetric pattern.
The Fourier series for an odd function consists only of sine terms, because sine functions are inherently odd.

Periodic Function

A periodic function is essentially a function that repeats its values in regular intervals or periods. Formally, a function \(f(x)\) is said to be periodic with period \(T\) if: \[f(x + T) = f(x)\] for all values of \(x\).
In the given exercise, extending \(f(x)\) with the period \(b\) simply means repeating the basic pattern of the function over each new interval of length \(b\). The complete Fourier series for a periodic function combines both cosine and sine terms, effectively representing the periodicity of the function.

Fourier Cosine Series

The Fourier Cosine Series is used to represent even functions. Since even functions are symmetric about the y-axis, the series only contains cosine terms. It is expressed as: \[f_c(x) = \frac{a_0}{2} + \sum_{n=1}^{\text{\textbackslashinfty}} a_n \text{cos}\frac{\text{\textbackslashpi}nx}{2}\text{\textbackslashright)}\]
The coefficients \(a_0\) and \(a_n\) are determined using the following integrals: \[a_0 = \frac{2}{b} \text{\textbackslashint}_{0}^{b} f(x) dx \] \[a_n = \frac{2}{b} \text{\textbackslashint}_{0}^{b} f(x) \text{cos}\frac{\text{\textbackslashpi} nx}{b} dx\]
These cosine terms effectively capture the even symmetry of the function across its periodic extension.

Fourier Sine Series

The Fourier Sine Series is used to represent odd functions. Since odd functions are antisymmetric about the origin, the series only contains sine terms. It is represented as: \[f_s(x) = \sum_{n=1}^{\text{\textbackslashinfty}} b_n \text{sin}\frac{\text{\textbackslashpi}nx}{2}\]
The coefficients \(b_n\) are calculated with the integral: \[b_n = \frac{2}{b} \text{\textbackslashint}_{0}^{b} f(x) \text{sin}\frac{\text{\textbackslashpi} nx}{b} dx\]
By using sine terms, which are inherently odd functions, the Fourier Sine Series accurately represents the odd symmetry of the original function.