Examples

Example 2.1.1. State the Euler's formulae when f(x) is expanded as a Fourier series in c < x < c + 2 л.

Solution: The Fourier Series for f (x) in the c < x < c + 2л

Formulas (1), (2) and (3) are known as the Euler formulas.

Example 2.1.2. Write the formula for finding Euler's constant of a Fourier series in (0, 2л).

Solution: Let the Fourier Series for f (x) in (0, 2л) be

Formulas (1), (2) and (3) are known as the Euler formulas.

Example 2.1.3. Write the formula for finding Euler's constant of a Fourier series in (-л, л).

Solution: Let the Fourier Series for f (x) in (-л, л) bе

Formulas (1), (2) and (3) are known as the Euler formulas

Example 2.1.4. Write the formula for Fourier Constants for f(x) in (c, c + 2l).

Solution: The Fourier expansion for f(x) in the interval

Example 2.1.5. Write the formula for Fourier Constants for f(x) in (0, 2l).

Solution: The Fourier expansion for f(x) in the interval 0 < x < 21 is given by

Example 2.1.6. Write the formulas for Fourier Constants for f(x) in (−l, l).

Solution: The Fourier expansion for f(x) in the interval

§ CONDITIONS FOR A FOURIER EXPANSION :

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

Note 4: cosecx cannot be expanded as a Fourier series, since one of the Dirichlet's conditions is not satisfied. (ZAR)

Example 2.1.7: The function f(x) expanded as a Fourier series. Explain why?

Solution :Given f (x) 

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 2.1.8: 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

Example 2.1.9: Examine whether the function sin 1/x can be expanded in a Fourier series in -π ≤ X ≤ π

 (i.e.,) sin attains its maximum value '+1' and minimum value '-1' for the above values of θ. Hence the function f(x) = sin1/ x attains its maximum and minimum values when

For large values of n as n→ ∞o, 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 -л≤x≤л in which the point x 0 is included.

§ Convergence Theorem on Fourier Series :

Statement:

If f(x) is a periodic function with period 2л and f(x) and f'(x) are piecewise continuous on [-л, л], 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 [f(x+) + f (x−)]

Note :

n

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

Example 2.1.10 Sum the Fourier series for f (x) =1/2 (л − x)

Solution :

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

so we substitute the value directly

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

so we substitute the value directly

sum= ½   (л - л) = 0

Example 2.1.12: Sum the Fourier series for

Solution: x = 0 is a point of discontinuity in the extreme of the given interval. [': ƒ (0) ƒ (2) ]

Sum = average of the extremes of the discontinuity

§ IMPORTANT FORMULAE

Note :

1. Bernoulli's formula : Suv dx = uv₁-u'v₂+u”' V₃ —,…..where u and v are functions of x.