Mathematical Physics Vol 1

6. Trigonometric Fourier series. Fourier integral

6.1 Periodic functions

In the natural sciences and technology, we often encounter processes such as: rotations of individual machine parts (Watt regulator, engine rocker,...), oscillations (clock pendulum ,...). These processes are repeated over time, i.e. they are periodic. Such processes are mathematically described by periodic functions.

Definition Afunction f ( x ) of one variable is periodic , if there exists a constant T̸ = 0 such that

f ( x + T )= f ( x ) ,

for ∀ x .

(6.1)

The constant T is called the period of the function f ( x ) .

Theorem19 If f ( x ) is a periodic function with a period T , then its period is also nT , where n is a whole number.