The trigonometric Fourier series of an even periodic function of time does not have

If a function is even and real periodic then its Fourier series contains only cosine term:

f(x) = a0 + a1 cos ω0 t + a2 cos 2 ω0 t + …

If a function is odd and real periodic then its Fourier series contains only sine term.

f(x) = a0 + b1 sin ω0 t + b2 sin 2 ω0 t + b3 sin 3 ω0 t + …

Important Point:

• For a periodic function, Fourier series is given as
  \(f\left( x \right) = {a_0} + \mathop \sum \nolimits_{n = 1}^\infty ({a_n}\cos n{\omega _0}t + {b_n}\sin n\;{\omega _0}t)\)
• A function is called even if f(x) = f(-x) and called odd if f(x) = -f(-x)