Fourier Series Classroom

103.1 Even & Odd Functions

Understanding even and odd functions is crucial in Fourier series analysis because it can significantly simplify the calculation of coefficients. Functions can exhibit symmetry around the y-axis (even) or symmetry about the origin (odd).

• An **even function** $f(x)$ satisfies $f(-x) = f(x)$ for all $x$ in its domain. Its graph is symmetric with respect to the y-axis. Examples include $x^2$, $\cos(x)$, and $|x|$.
• An **odd function** $f(x)$ satisfies $f(-x) = -f(x)$ for all $x$ in its domain. Its graph is symmetric with respect to the origin. Examples include $x^3$, $\sin(x)$, and $1/x$.

Most functions are neither even nor odd, but any function can be decomposed into an even and an odd part.

103.2 Fourier Cosine & Sine Series

The symmetry of even and odd functions simplifies Fourier series calculations over symmetric intervals like $[-\pi, \pi]$ (or $[-L, L]$):

• If $f(x)$ is an **even function**, then $f(x)\sin(nx)$ is an odd function. The integral $\int_{-\pi}^{\pi} f(x)\sin(nx) \,dx = 0$, which means all $b_n = 0$. The Fourier series becomes a **Fourier Cosine Series**: $$f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos(nx)$$ where $a_0 = \frac{2}{\pi} \int_{0}^{\pi} f(x) \,dx$ and $a_n = \frac{2}{\pi} \int_{0}^{\pi} f(x) \cos(nx) \,dx$.
• If $f(x)$ is an **odd function**, then $f(x)\cos(nx)$ is an odd function, and $\int_{-\pi}^{\pi} f(x)\cos(nx) \,dx = 0$, so all $a_n = 0$ (including $a_0$). The Fourier series becomes a **Fourier Sine Series**: $$f(x) = \sum_{n=1}^{\infty} b_n \sin(nx)$$ where $b_n = \frac{2}{\pi} \int_{0}^{\pi} f(x) \sin(nx) \,dx$.

This property allows us to integrate only over $[0, \pi]$ and multiply by $2/\pi$, simplifying computations.

103.3 Half-Range Fourier Series

When a function $f(x)$ is defined only on a finite interval $[0, \pi]$ (or more generally $[0, L]$), and we want to represent it using a Fourier series, we can extend it to the symmetric interval $[-\pi, \pi]$ (or $[-L, L]$) in two ways:

• **Even periodic extension:** Define $f(-x) = f(x)$ for $x \in (0, \pi]$. This creates an even function over $[-\pi, \pi]$ with period $2\pi$. The Fourier series will be a **Fourier Cosine Series**.
• **Odd periodic extension:** Define $f(-x) = -f(x)$ for $x \in (0, \pi]$. This creates an odd function over $[-\pi, \pi]$ with period $2\pi$. The Fourier series will be a **Fourier Sine Series**.

These are called "half-range" series because the original function is defined over half of the full period interval. The choice depends on the application or the desired behavior at the boundaries.

104.1 Periodic Function (Period L)

The Fourier series definition can be generalized for any periodic function $f(x)$ with an arbitrary period $T = 2L$ (meaning the function is defined on an interval of length $2L$, e.g., $[-L, L]$ or $[c, c+2L]$). The generalized Fourier series is:

The coefficients are given by:

Here, $L$ is half of the period. If the period is given as $T$, then $L = T/2$. This allows us to analyze periodic functions of any period, not just $2\pi$.

104.2 Half-Range Series (Range L)

Similar to the $2\pi$ case, we can define half-range Fourier series for functions defined on an arbitrary interval $[0, L]$.

• **Fourier Cosine Series (for even extension):** $$f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos\left(\frac{n\pi x}{L}\right)$$ $$a_0 = \frac{2}{L} \int_{0}^{L} f(x) \,dx$$ $$a_n = \frac{2}{L} \int_{0}^{L} f(x) \cos\left(\frac{n\pi x}{L}\right) \,dx$$
• **Fourier Sine Series (for odd extension):** $$f(x) = \sum_{n=1}^{\infty} b_n \sin\left(\frac{n\pi x}{L}\right)$$ $$b_n = \frac{2}{L} \int_{0}^{L} f(x) \sin\left(\frac{n\pi x}{L}\right) \,dx$$

In both cases, the extended function has a period of $2L$. These series are particularly useful for solving boundary value problems in partial differential equations where the solution is sought over a finite interval.