What is Fourier series

Fourier series expresses the periodic function as the sum of trigonometric functions, the specific form is: f(x) = a_0 Σ(a_n cos(nωx) b_n sin(nωx)). where a_n and b_n are Fourier coefficients, ω is the angular frequency, n is the summation index, and a_0 is the constant term. This series can be used to calculate Fourier coefficients through integration and is widely used in signal processing, vibration analysis, heat conduction, electromagnetics and other fields.

Fourier series: a mathematical description of periodic functions

The Fourier series is a A mathematical tool that allows you to express periodic functions as sums of trigonometric functions. A periodic function is a function that appears repeatedly within a specific period.

Fourier's theorem states that any periodic function can be expressed as a sum of trigonometric functions in the following form:

<code>f(x) = a_0 + Σ(a_n cos(nωx) + b_n sin(nωx))</code>

where:

• ##a_0 is the constant term
• a_n and b_n is the Fourier coefficient
• ω is the angular frequency (2π/ period)
• n is the summation index

The Fourier coefficients are calculated by integrating:

• a_n = (2/period) ∫[0,period] f(x) cos(nωx) dx
• b_n = (2/period) ∫[0,period ] f(x) sin(nωx) dx

Applications of Fourier series:

Fourier series in mathematics, science and engineering It has a wide range of applications, including:

Signal processing: analyzing and processing waveforms, sounds and images
• Vibration analysis: predicting the vibration frequency and amplitude of mechanical components
• Heat Conduction: Solving the Unsteady-State Heat Conduction Equation
• Electromagnetics: Calculating Antenna and Propagation Characteristics

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