Convergence Theorem on Fourier Series

Convergence Theorem on Fourier Series:

Statement :

If f(x) is a periodic function with period 2π and f(x) and f ‘(x) are piecewise continuous on [‒π, π], then the Fourier series is convergent. The sum of the Fourier series is equal to f(x) at all points of x where f(x) is continuous. At the points of x where f(x) is discontinuous, the sum of the Fourier series is the average of the right and left limits, that is 1/2 [f(x+) + f(x−)]

Note :

• In the interval (0,2π) i.e., 0 < x < 2π

 x continuous at all points except 0 and 2π

• In the interval [0,2π] i.e., 0 ≤ x ≤ 2π

 x continuous at all points including 0 and 2π

• In the interval (‒π, π) i.е., ‒π < x < π

 x continuous at all points except at ‒π and π.

• In the interval [‒π, π] i.е., ‒π ≤ x ≤ π

 x continuous at all points including ‒π and π

• In the interval (‒π, 0) and (0,π) i.e., −π < x < 0 and 0 <x<π

 x continuous at all points except ‒π, π and 0.

Note: If f(x) is defined in the interval (0, 2π), then

 [the function f(x) may be continuous or discontinuous at the end points x = 0, x = 2π]

Example 10: Sum the Fourier series for f(x) = ½ (π ‒ x)

(i) in (0, 2π) at x= π/2

(ii) in (0, 2π) at x= π

Solution:

(i) Here f(x) is a continuous at x = π/2 in (0,2π),

so we substitute the value directly

 sum = ½ (π ‒π/2) = ½ (π/2) = π/4

(ii) Here f(x) is a continuous at x=π in (0, 2 π),

so we substitute the value directly

sum = 1/2 (π ‒ π) = 0

Example 11: Sum the Fourier series for f(x) = 

(i) at x = π/2  (ii) at x = π/2  (iii) at x= ‒ π (iv) at x=π (v) at x = 0

Solution:

(i) x = π /2 is a continuous point in (0, π)

f(x) = 0

(ii) x = ‒ π/2 is a continuous point in (‒π, 0)

(iii) x = ‒π is a discontinuous point in the extremum

(iv) x = π is a discontinuous point in the extremity.

(v) x=0 is a continuous point.

[f(0+) = f(0−) = f(0)]

Example 12: Sum the Fourier series for f(x) =  at x = 1

Solution:

 x = 1 is a finite point of discontinuity (in the middle) of(0, 2)

 [f (1−) ≠ f (1+)]

Example 13: Sum the Fourier series for

Solution:

x =0 is a point of discontinuity in the extreme of the given interval. [ f(0) ≠ f(2) ])

Sum = average of the extremes of the discontinuity

Sum = [ f(0) +f(2) ] / 2 = (0+2)/2 = 1

IMPORTANT FORMULAE

Note:

1. Bernoulli's formula: ʃuv dx = uv₁ ‒ u'v₂ + u"v₃ ‒, …. Where u and v are functions of x.

4. (‒1)ⁿ⁺¹ = (‒1)^n‒1

5. cos nπ = (−1)ⁿ, cos 2nπ = (−1)²ⁿ = [(−1)²]ⁿ = [1]ⁿ = 1

sin nπ = 0, sin 2nπ = 0 if n is an integer.