MATH528

MATH528 Lesson19: Fourier series

The Euler formulas for periodic functions with period $2 L$ , $f (x) = f (x + 2 L)$ are

a_{0} = \frac{1}{2 L} \int_{- L}^{L} f (x) d x,

a_{n} = \frac{1}{L} \int_{- L}^{L} f (x) \cos (\frac{n π x}{L}) d x,

b_{n} = \frac{1}{L} \int_{- L}^{L} f (x) \sin (\frac{n π x}{L}) d x .

f (x) = a_{0} + \sum_{n = 1}^{\infty} (a_{n} \cos (\frac{n π x}{L}) + b_{n} \sin (\frac{n π x}{L}))

Even, functions, $f (x) = f (- x)$ are represented by a Fourier cosine series

f (x) = a_{0} + \sum_{n = 1}^{\infty} a_{n} \cos (\frac{n π x}{L})

Odd, functions, $f (x) = - f (- x)$ are represented by a Fourier sine series

f (x) = \sum_{n = 1}^{\infty} b_{n} \sin (\frac{n π x}{L})