Final answer:
A Fourier series expands a periodic function into a sum of sines and cosines with coefficients an and bn calculated through specific integrals over a period L, providing a way to analyze functions' frequency components.
Step-by-step explanation:
The Fourier series expansion of a function f(x) is a way to represent a periodic function using sums of sines and cosines. The standard form of the Fourier series is:
f(x) = a0/2 + ∑ (an cos(nx) + bn sin(nx))
where the sum is from n = 1 to infinity, and the Fourier coefficients an and bn are given by:
an = (2/L) ∫0L f(x)cos(nx)dx
bn = (2/L) ∫0L f(x)sin(nx)dx
The coefficient a0 is the average value of the function over the interval, and it's computed as a0 = (1/L) ∫0L f(x)dx.
The limits of integration for these coefficients are typically over the period L of the function f(x). This series expansion allows functions to be analyzed in terms of their frequency components.