Determination of Fourier Coefficients: (Euler's Formulae)

DETERMINATION OF FOURIER COEFFICIENTS: (Euler's Formulae)

Let f(x) be represented in the interval (c, c + 2π) by the Fourier Series

To find the coefficients of a₀, a_n and b_n.

We assume that the series (1) can be integrated term by term x = c to x = c + 2π

To find a₀:

Integrate both sides of equation (1) from

 x = c to x = c + 2π. Then,

To find a_n:

Multiply both sides of (1) by cos nx and integrate from

 x = c to x = c +2π. Then,

To find b_n

Multiply both sides of (1) by sinnx and integrate from

x = c to x = c +2π. Then,

CHANGE OF INTERVAL

In practice, we often require to find a Fourier series for an interval which is not of length 2π.

In many problems, the period of the function to be expanded is not 2π, but some other interval say 2l.

Suppose f(x) is defined in the interval (‒l, l).

Let z = πx / l, Hence x = lz/π

Also,

when x = ‒l we have z = ‒π and

when x = l we have z = π and

Hence, the function F(z) = f(lz/π) is defined in the interval (‒π, π)

The Fourier series of f(z) is given by

Example 1: State the Euler's formulae when f(x) is expanded as a Fourier series in c<< x < c + 2 π.

Solution:

The Fourier Series for f(x) in the c< x < c + 2π

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

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

Solution:

 Let the Fourier Series for f(x) in (0, 2π) be

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

Example 3: Write the formula for finding Euler's constant of a Fourier series in (‒π, π).

Solution:

Let the Fourier Series for f(x) in (‒π, π) be

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

Example 4: Write the formula for Fourier Constants for f(x) in (c, c + 21).

Solution:

The Fourier expansion for f(x) in the interval

 c<x<c+2l is given by

Example 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 < 2l is given by

Example 6: Write the formulas for Fourier Constants for f(x) in (‒l, l).

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