FOURIER SERIES
FOURIER SERIES, Even and Odd Function, Sine and Cosine Function and Exponential Function.
Course Description
A Fourier series is a way to represent a periodic function as the sum of simple sine and cosine functions. It explores the idea that any periodic waveform can be expressed as an infinite sum of harmonically related sinusoidal functions. This mathematical tool is widely used in signal processing, electrical engineering, and various branches of physics to analyze and synthesize periodic signals. The course typically covers the theory behind Fourier series, its applications, and techniques for solving problems related to signal decomposition and reconstruction.
In Fourier series, even and odd functions play distinct roles.
1. **Even Function:**
– An even function, denoted as \( f(x) = f(-x) \), is symmetric with respect to the y-axis.
– In the context of Fourier series, when a function is even, only cosine terms appear in its expansion.
– The Fourier coefficients for an even function are given by \( a_n = \frac{2}{T} \int_{0}^{T} f(x) \cos\left(\frac{2\pi n x}{T}\right) \,dx \), and \( b_n = 0 \) for all \( n \).
2. **Odd Function:**
– An odd function, denoted as \( f(x) = -f(-x) \), exhibits rotational symmetry with respect to the origin.
– For odd functions in Fourier series, only sine terms appear in the expansion.
– The Fourier coefficients for an odd function are \( a_n = 0 \) for all \( n \), and \( b_n = \frac{2}{T} \int_{0}^{T} f(x) \sin\left(\frac{2\pi n x}{T}\right) \,dx \).
Understanding whether a function is even, odd, or neither helps simplify the computation of Fourier series coefficients and provides insight into the function’s symmetry.
