Unit 1: Fourier Series

1. Periodic Functions

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).

2. Orthogonality of Sine and Cosine Functions

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]:

• Integral from -π to π of [sin(mx) * cos(nx)] dx = 0 (for all m, n)
• Integral from -π to π of [sin(mx) * sin(nx)] dx = 0 (if m is not equal to n)
• Integral from -π to π of [cos(mx) * cos(nx)] dx = 0 (if m is not equal to n)

When m = n (and not zero):

• Integral from -π to π of [sin²(nx)] dx = π
• Integral from -π to π of [cos²(nx)] dx = π

3. Dirichlet Conditions

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:

1. f(x) must be single-valued and periodic.
2. f(x) must be piecewise continuous (have a finite number of finite discontinuities) in any one period.
3. f(x) must have a finite number of maxima and minima in any one period.
4. The integral of |f(x)| over one period must be finite (absolutely integrable).

4. Expansion and Determination of Fourier Coefficients

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:

a0 Coefficient

an Coefficient

bn Coefficient

5. Summing of Infinite Series

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.

6. Even and Odd Functions

The symmetry of f(x) greatly simplifies the calculation of coefficients:

7. Complex Representation of Fourier 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:

8. Expansion with Arbitrary Periods

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.

9. Expansion of Non-Periodic Functions

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:

• Half-Range Sine Series: Extend the function as an odd function.
• Half-Range Cosine Series: Extend the function as an even function.

10. Differentiation and Integration of Fourier Series

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).