Fourier Series

One-Sentence Idea / 一句话

A Fourier series represents a complicated function as a sum of simple sinusoidal modes.

Fourier series 把复杂函数分解为不同 frequencies、amplitudes 和 phases 的简单 sine/cosine modes 之和。

Core Form

对周期函数，典型展开为

$$f(x)=\frac{a_0}{2}+\sum _{n=1}^{\infty }\left(a_n\cos nx+b_n\sin nx\right).$$

有限个 modes 给出 approximation；加入更多 modes，通常能重建更多细节。

Why It Appears Here

Fourier 研究 Heat Equation 时发现：复杂 initial temperature profile 可以拆成简单 spatial modes。由于 heat equation 是 linear 的，每个 mode 可以分别演化，再通过 superposition 相加回完整 solution。

Connections

• Infinite Sequence and Series
• Trigonometric Derivatives
• Heat Equation
• Differential Equations
• 02 - But What Is a Partial Differential Equation > Knowledge Point 3: The Heat Equation Is a Gateway to Diffusion

Search and Source Mentions

• First series preview: 02 - But What Is a Partial Differential Equation