Periodic functions are those functions which are used to represent periodic motion. A function f(t) is said to be periodic, if
f(t) = f(t + T)
= f(t + 2T) ...(i)
Where T is called the period of periodic function, sin θ and cos θ are the exponents of periodic functions with period equal to 2π radians because
sin θ = sin (θ + 2π)
= sin (θ + 4π) …(ii)
And cos θ = cos (θ + 2π)
= cos (θ + 4π) …(iii)
The periodic function T (t) can also be represented as
f(t) = sin \(\frac{2\pi t}{T}\)…(iv)
Fourier theorem: According to this theorem, any periodic function f(t) of the period T, however complex it may, can be represented by a unique combination of the functions fc(t) and gn(t).
Here, fn(t) = sin \(\frac{2\pi n t}{T} \)
And gn(t) = cos \(\frac{2\pi nt}{T}\)
Where n = 0, 1, 2, 3.
Mathematically, Fourier theorem can be written as:
F(t) = [ansin \(\frac{2\pi n t}{T}+b_n\,cos\frac{2\pi nt}{T}\)]
Or F(t) = a1 sin t + a2 sin( t + T) + a3 sin(t + 2T) . . . . +b0 + b1 cos t + b2 cos(t + T) + b3 cos(t + 2T)+. . … (i)
The relation (i) is called Fourier series where a1, a2, a3, . . , b0, b1, b2, b3, . . … are amplitudes of Fourier series and cosines terms are called Fourier constants.
A periodic motion for which only the Fourier coefficients a1 and b1 are non-zero, is called simple harmonic motion.
Similarly other periodic function of t can be
g(t) = cos \(\frac{2\pi t}{T}\) …(iv)
f(t) = sin \(\frac{2\pi t}{T}\)
In order to check that these two functions 4 and 5 has a period T1 can be tested by substituting (t + T) in place of t in those relations.
f(t + T) = sin \(\frac{2\pi}{T}\) (T + t)
= sin ( \(\frac{2\pi t}{T}+2\pi\))
= sin \(\frac{2\pi t}{T}\) = F(t)
g(t + T) = cos \(\frac{2\pi}{T}\) (t + T)
= cos ( \(\frac{2\pi t}{T}+2\pi\) ) = cos \(\frac{2\pi t}{T} \) = g(t)
It can easily verified that:
f(t + nt) = f(t)
And g(t + nt) = g(t)
Therefore, the infinite sets of periodic functions of period T may be represented as
fn(t) = \(\frac{sin\,2\pi nt}{T}\)
gn(t) = \(\frac{cos\,2\pi nt}{T}\)
