Fourier Series

Introduction

Fourier series is a mathematical tool used to represent a periodic function as the sum of simple sine and cosine functions. It was introduced by the French mathematician Joseph Fourier in the early 19th century. The Fourier series has applications in various fields such as signal processing, physics, engineering, and music.

Representation of a Function

Consider a periodic function $f(x)$ with period $2L$ defined on the interval $[-L, L]$. The Fourier series representation of $f(x)$ is given by:

where

• $a_0$ is the average value of $f(x)$ over one period, given by $a_0 = \frac{1}{L} \int_{-L}^{L} f(x) dx$
• $a_n$ and $b_n$ are coefficients determined by the formulas $a_n = \frac{1}{L} \int_{-L}^{L} f(x) \cos\left(\frac{n\pi x}{L}\right) dx$ and $b_n = \frac{1}{L} \int_{-L}^{L} f(x) \sin\left(\frac{n\pi x}{L}\right) dx$

Connecting with Calculus

To find the coefficients $a_n$ and $b_n$, we need to evaluate the integrals involving trigonometric functions. This requires the use of calculus techniques such as integration by parts, trigonometric identities, and properties of definite integrals.

The convergence of a Fourier series is also a topic that involves calculus concepts. It can be shown that under certain conditions, the Fourier series converges to the original function. These conditions are often related to the integrability and continuity of the function $f(x)$.

Conclusion

Fourier series is a powerful tool in mathematics and has wide applications in physical and engineering problems. Understanding and applying Fourier series involves the use of calculus concepts such as integration, trigonometric functions, and convergence. It provides a rich area for exploration and application in calculus and beyond.

Overall, the study of Fourier series provides a deeper understanding of the connection between calculus and real-world phenomena.