Conditions for a Fourier Expansion (Dirichlet's Conditions)

CONDITIONS FOR A FOURIER EXPANSION: [Dirichlet's Conditions]

Any function f(x) can be developed as a Fourier series a0/2 + an cos nx + bn sin nx where a0, an, bn are constants, provided.

(i) f(x) is periodic, single‒valued and finite

(ii) f(x) has a finite number of finite discontinuities in any one period and has no infinite discontinuity.

(iii) f(x) has at the most a finite number of maxima and minima.

Note 1: Dirichlet's conditions are not necessary but only sufficient for the existence of Fourier series.

Note 2: Peter Gustav Lejenune Dirichlet (1805 - 1859), Great German Mathematician is known for his contributions to Fourier Series and Number Theory.

Note 3: tanx cannot be expanded as a Fourier series, since tanx has infinite number of infinite discontinuties, so Dirichlet's conditions are not satisfied.

Note 4: cosec x cannot be expanded as a Fourier series, since one of the Dirichlet's conditions is not satisfied.

Example 7: The function f(x) = 1/ x‒2 (0 ≤ x ≤ 3) cannot be expanded as a Fourier series. Explain why?

Solution: Given f(x) = (1 / x‒2) (0 ≤ x ≤3)

At x = 2, f(x) becomes infinity, it has an infinite discontinuity at x = 2.

So it does not satisfy one of the Dirichlet's conditions.

Hence it cannot be expanded as a Fourier series.

Example 8: Can you expand f(x)=(1 − x2 )/(1 + x2) as a Fourier series in any interval.

Solution:

Let f(x) = (1 − x2 )/(1 + x2)

This function is well defined in any finite interval in the range (‒∞, ∞) it has no discontinuities in the interval

Differentiate f '(x) = 

 f(x) is maximum or minimum when f '(x) = 0

 (i.e.,) when 4x = 0, (i.e.,) x = 0

So it has only one extreme value, and has

 (i.e.,) a finite number of maxima and minima in the interval (‒∞, ∞)

Since it satisfies all the Dirichlet's conditions, it can be expanded in a Fourier Series in a specified interval in the range (‒∞, ∞).

Example 9: Examine whether the function sin(1/x) can be expanded in a Fourier series in ‒π ≤x≤ π.

Solution:

Consider the function f(x) = sin(1/x)

We know that sin θ = ±1

when θ = ± π/2, ± 3 π/2, ± 5 π/2, ….

 (i.e.,) when θ = (2n − 1) π/2 when n is zero (or) any integer.

 (i.e.,) sin θ attains its maximum value '+1' and minimum value '‒1' for the above values of θ. Hence the function f(x) = sin(1/x) attains its maximum and minimum values when

where n is zero or an integer.

For large values of n as n→∞, the values of x as given by (2) tend to become indefinitely small and to be crowded near to the value of x = 0.

Hence, the function (1) has an infinite number of maxima and minima near x = 0, so it does not satisfy one of the Dirichlet's conditions. It cannot be expanded in a Fourier series in the range ‒π ≤ x ≤ π in which the point x = 0 is included.