Welcome to an introduction to Fourier coefficients. These coefficients are the building blocks of Fourier series, which allow us to represent periodic functions as sums of sines and cosines. The key idea is that any periodic function, no matter how complex, can be decomposed into a sum of simple sinusoidal waves. On the right, you can see a square wave in red, and how adding more sine terms with specific coefficients creates better approximations of the target function. This powerful mathematical tool has applications in signal processing, physics, and engineering.
The Fourier series represents a periodic function as an infinite sum of sines and cosines. For a function with period 2L, the series is given by: f(x) equals a-zero over 2, plus the sum from n equals 1 to infinity of a-n times cosine of n pi x over L plus b-n times sine of n pi x over L. The coefficients a-zero, a-n, and b-n determine the amplitude of each component. On the right, you can see the individual components: the constant term a-zero over 2 in gray, the first cosine term in blue, the first sine term in red, and their combination in purple. Each coefficient controls how much of each frequency component contributes to the overall function.
Now let's look at how to calculate the Fourier coefficients. These coefficients are determined by specific integral formulas. For a function with period 2L, the coefficient a-zero equals 1 over L times the integral from negative L to L of f(x) dx. This represents the average value of the function. The coefficient a-n equals 1 over L times the integral from negative L to L of f(x) times cosine of n pi x over L dx. And b-n equals 1 over L times the integral from negative L to L of f(x) times sine of n pi x over L dx. These integrals essentially measure how much of each frequency component is present in the original function. On the right, you can see a square wave function in red, and the cosine and sine functions in blue and green. When we multiply the function by cosine or sine and integrate, we get the corresponding Fourier coefficients.
Let's examine a specific example: the Fourier series for a square wave with period 2π. The function equals 1 for x between 0 and π, and equals negative 1 for x between π and 2π. When we calculate the Fourier coefficients, we find that a-zero equals zero, and all a-n coefficients are zero. For the b-n coefficients, we get 4 over n π when n is odd, and zero when n is even. This gives us the Fourier series: f(x) equals 4 over π times the sum of sine of x plus 1 over 3 times sine of 3x plus 1 over 5 times sine of 5x, and so on. On the right, you can see how adding more terms improves the approximation. With just the first term, we get a basic sine wave. Adding the third and fifth harmonics makes the approximation much closer to the square wave. Adding even more terms would make the approximation even better, especially at the discontinuities.
To summarize what we've learned about Fourier coefficients: They represent the frequency components of a periodic function, allowing us to decompose complex signals into simple sinusoidal waves. The coefficients determine how much each frequency contributes to the original function. Fourier coefficients have numerous applications across science and engineering. In signal processing, they help filter out unwanted frequencies. In image compression, they allow us to represent images more efficiently by keeping only the most significant frequency components. They're also essential for solving certain types of differential equations in physics and engineering. In quantum mechanics, they help describe wave functions and energy states. The beauty of Fourier analysis is that it provides a universal language for understanding periodic phenomena across different fields of science and mathematics.