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Fourier Periodic Function

Note, that the function,

is a periodic function in space, i.e. it repeats itself after space interval λ₁. Similarly, we find

Hence, since sin k₂ x repeats itself after distance λ₁/2, it certainly repeats itself after interval λ₁. In fact, every term on right hand side of eq. is a periodic function, with period λ₁. Hence ƒ(x) itself is a periodic function of x with period λ₁.

Now, there is a general theorem due to J. Fourier which states that a well behaved periodic function F (z) with period 2 π (i.e. F (z + 2 π) = F (z)), can be represented as a series of harmonic terms as

The above statement is known as Fourier’s theorem and the series on right hand side is called Fourier series.

In particular, if z = 2 π x/λ, then z + 2 π = (z + λ); hence, if a function ƒ(x) is periodic in x with period λ, then Fourier’s theorem states that,

As another example, if z = 2 π t/T, then z + 2 π = 2 π/T (t + T); hence, if a function ƒ(t) is periodic in t with period T, then according to Fourier’s theorem, we get

The coefficients A₀, A_n and B_n are called Fourier coefficients.