A periodic function is a function that repeats its values in regular intervals or periods. Mathematically, a function f(x) is said to be periodic with a period T if:
f(x + T) = f(x) for all x.
Common examples in physics include the sine and cosine functions, which describe oscillations and waves. In the context of Fourier series, we often look at functions with a period of 2π, meaning f(x + 2π) = f(x).
The concept of orthogonality is central to finding Fourier coefficients. Two functions are orthogonal over an interval [a, b] if the integral of their product over that interval is zero.
For the set of functions {sin(nx), cos(nx)}, the following properties hold over the interval [-π, π]:
When m = n (and not zero):
Not every periodic function can be represented as a Fourier series. The Dirichlet Conditions are the sufficient conditions for a function f(x) to be expanded into a convergent Fourier series:
A periodic function f(x) with period 2π can be expanded as:
where the sum Σ goes from n = 1 to infinity. The Fourier Coefficients are determined using Euler's Formulas:
One powerful application of Fourier series is the summation of numerical infinite series. By evaluating a Fourier series at a specific point (like x = 0 or x = π/2), we can often derive values for series like the Basel problem (sum of 1/n²).
Example: Expanding f(x) = x² in [-π, π] and setting x = π leads to the conclusion that Σ (1/n²) = π²/6.
The symmetry of f(x) greatly simplifies the calculation of coefficients:
| Function Type | Property | Resulting Coefficients |
|---|---|---|
| Even Function | f(-x) = f(x) | bn = 0; only a0 and an exist (Cosine Series) |
| Odd Function | f(-x) = -f(x) | a0 = 0, an = 0; only bn exists (Sine Series) |
Using Euler's identity [exp(inx) = cos(nx) + i*sin(nx)], we can write the Fourier series in a more compact complex form:
where n ranges from -infinity to +infinity. The complex coefficient cn is given by:
If the function has a period 2L instead of 2π, we transform the variable. The series becomes:
The coefficients are adjusted by replacing π with L in the limits and the denominator.
A function defined only on a finite interval [0, L] can be expanded as a Fourier series by creating a periodic extension of it. We can create:
Term-by-term integration: This is generally safe. If f(x) is piecewise continuous, the integral of its Fourier series converges to the integral of f(x).
Term-by-term differentiation: This requires stricter conditions. The function f(x) must be continuous everywhere, and f(-π) must equal f(π) for the derivative of the series to converge to f'(x).
Check Symmetry First: Before calculating integrals, check if the function is even or odd. It will save you 50% of the work! Mnemonics: "Even is Cosine" (E-C), "Odd is Sine" (O-S).