Even/Odd & Half-Range Fourier Series

$f(x) = a_0 + \sum_{n=1}^{\infty} (a_n \cos(nx) + b_n \sin(nx))$

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Even and Odd Functions

Understanding even and odd functions can significantly simplify Fourier series calculations. When a function exhibits symmetry, many of its Fourier coefficients become zero, reducing the amount of computation required.

Definition of Even and Odd Functions

A function $f(x)$ is said to be even if for all $x$ in its domain:

$$f(-x) = f(x)$$

Even functions are symmetric about the y-axis. Examples include $x^2, \cos(x), |x|$.

A function $f(x)$ is said to be odd if for all $x$ in its domain:

$$f(-x) = -f(x)$$

Odd functions are symmetric about the origin. Examples include $x^3, \sin(x), x$.

Note: Not all functions are strictly even or odd. Many functions are a combination of both. Also, checking for even/odd symmetry numerically has limitations.

Fourier Series for Even Functions (on $[-L, L]$)

If $f(x)$ is an even function on the interval $[-L, L]$, its Fourier series simplifies to a Fourier Cosine Series:

$$f(x) = a_0 + \sum_{n=1}^{\infty} a_n \cos\left(\frac{n\pi x}{L}\right)$$

The coefficients are given by:

$$a_0 = \frac{1}{L} \int_{0}^{L} f(x) \,dx$$ $$a_n = \frac{2}{L} \int_{0}^{L} f(x) \cos\left(\frac{n\pi x}{L}\right) \,dx \quad \text{for } n=1, 2, 3, \ldots$$ $$b_n = 0$$

Even Function $f(x)$ (e.g., x^2):

Period $T$ (e.g., 2*pi for cos(x), 2 for x^2 over [-1,1]):

Number of terms $N$ (e.g., 5, higher N gives better approximation):

Enter even function, period, and number of terms, then calculate.

Fourier Series for Odd Functions (on $[-L, L]$)

If $f(x)$ is an odd function on the interval $[-L, L]$, its Fourier series simplifies to a Fourier Sine Series:

$$f(x) = \sum_{n=1}^{\infty} b_n \sin\left(\frac{n\pi x}{L}\right)$$

The coefficients are given by:

$$a_0 = 0$$ $$a_n = 0$$ $$b_n = \frac{2}{L} \int_{0}^{L} f(x) \sin\left(\frac{n\pi x}{L}\right) \,dx \quad \text{for } n=1, 2, 3, \ldots$$

Odd Function $f(x)$ (e.g., x):

Period $T$ (e.g., 2*pi for sin(x), 2 for x over [-1,1]):

Number of terms $N$ (e.g., 5, higher N gives better approximation):

Enter odd function, period, and number of terms, then calculate.

Fourier Half-Range Series (on $[0, L]$)

For a function $f(x)$ defined only on the interval $[0, L]$, we can extend it to the interval $[-L, L]$ as either an even or an odd function. This allows us to use the simplified Fourier cosine or sine series formulas, resulting in either a Half-Range Cosine Series or a Half-Range Sine Series.

The Fourier series will accurately represent the original function only within its defined range $[0, L]$. Outside this range, it will represent the chosen periodic extension.

Fourier Half-Range Cosine Series

To obtain a cosine series for $f(x)$ defined on $[0, L]$, we assume an even extension to $[-L, L]$, i.e., $f(-x) = f(x)$.

$$f(x) = a_0 + \sum_{n=1}^{\infty} a_n \cos\left(\frac{n\pi x}{L}\right)$$

The coefficients are identical to the even function case:

$$a_0 = \frac{1}{L} \int_{0}^{L} f(x) \,dx$$ $$a_n = \frac{2}{L} \int_{0}^{L} f(x) \cos\left(\frac{n\pi x}{L}\right) \,dx \quad \text{for } n=1, 2, 3, \ldots$$

Function $f(x)$ on $[0, L]$ (e.g., x):

Length $L$ of the half-range (e.g., pi for [0,pi]):

Number of terms $N$:

Enter function, half-range length $L$, and number of terms, then calculate.

Fourier Half-Range Sine Series

To obtain a sine series for $f(x)$ defined on $[0, L]$, we assume an odd extension to $[-L, L]$, i.e., $f(-x) = -f(x)$.

$$f(x) = \sum_{n=1}^{\infty} b_n \sin\left(\frac{n\pi x}{L}\right)$$

The coefficients are identical to the odd function case:

$$b_n = \frac{2}{L} \int_{0}^{L} f(x) \sin\left(\frac{n\pi x}{L}\right) \,dx \quad \text{for } n=1, 2, 3, \ldots$$

Function $f(x)$ on $[0, L]$ (e.g., x):

Length $L$ of the half-range (e.g., pi for [0,pi]):

Number of terms $N$:

Enter function, half-range length $L$, and number of terms, then calculate.