Fourier Series: Half Range Cosine series

Half-range cosine series

Cosine series: To expand f(x) as a cosine series in (0, π) or (0,l), we extend the function reflecting it in the y‒axis, so that f (−x) = f(x).

 f(x) = g(x) in (0,π) or (0,l) is extended to f(x) = g(‒x) in (‒π, 0) or (‒1, 0).

Problems based on Half‒Range cosine series

Example 1: Find the Fourier cosine series for f(x) = x²  in 0<x<π.

Solution:

Example 2: Expand the function f(x) = sin x, 0 < x <π in Fourier cosine series.

Solution:

Let the required Fourier cosine series be

Example 3: Expand f(x) =  as a cosine series.

Solution: 

Example 4: Find the Half‒range cosine series for f(x) = (x − 1)² in (0, 1). Hence show that 1/1² + 1/2² + 1/3² + … = π²/6.

Solution: Here /= 1

The required Fourier cosine series be

Example 5: Find the cosine series of f(x) = e^x in (0,1).

Solution: Let the required Fourier cosine series be

Example 6: Find the half range cosine series of f(x) = x in 0 <x<π.

Solution: Let the required Half range cosine series be

Example 7: Find the half range cosine series of f(x) = (π − x²) in the interval (0,π).

Solution: Given f(x) = (π − x²) in (0,π)

Example 8: Find the half‒range cosine series for the function f(x) = x(π − x) in 0 < x < π.

Solution: Given f(x) = x (π‒x) in 0 < x <π

= πx ‒ x²

EXERCISE

Half range series

1. Express as a Fourier Sine Series

2. If f(x) = πx / 4, 0<x<π/2

f(x) = π/4 (π‒x), π/2<x<π

Express f(x) in a series of cosines only.

3. Find a Fourier sine series for f(x) = k in 0 < x <π

4. Find the Fourier cosine series, and Fourier sine series for f(x) =π‒x in (0, π)

5. Find the Fourier sine series for f(x) = ax + b in 0 < x < l

6. Expand 

7. Expand f (x) = cosx, 0 < x <π in half range sine series.

8. Expand x sinx as cosine series in 0 < x <π. Hence, show that

9. Find the half range sine series of f(x) = a in (0, l). Deduce the sum of .

10. Find the half range sine series of f(x) = sin ax in (0, l)