Half-range series

HALF-RANGE SERIES

(a) HALF-RANGE SINE SERIES

(b) HALF-RANGE COSINE SERIES

Sine series: To expand f(x) as a sine series in (0,л) or (0,l), extend the function reflecting it in the origin, so f(-x) = −f (x).

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

Example 2.3.1: Write the formula for Fourier constants to expand f(x) as a sine series in (0, л).

Solution :

Example 2.3.2: Write the formula for Fourier constants to expand f(x) as a sine series in (0, l).

Solution :

Example 2.3.3: Write the formula for Fourier constants to expand f(x) as a cosine series in (0, π).

Solution :

Example 2.3.4: Write the formula for Fourier constants to expand f(x) as a cosine series in (0, l).

Solution :

(a) HALF-RANGE SINE SERIES

Problems based on Half-Range Sine Series

Example 2.3.a(1): Expand the function f(x) = x, 0 < x < π in Fourier sine series.

Solution: Given ƒ (x) = x in 0 < x < π

Example 2.3.a(2) : Find the Half range Fourier sine series for f(x) = x in (0, l).

Solution: Let the required Fourier series be

Example 2.3.a(3) : Find the half range Fourier sine series for sinh ax in 0 < x < π.

Solution: 

Example 2.3.a(4) : Find the half range sine series for the function f(x) = x- = x − x2, 0 < x < 1

Solution : 

Example 2.3.a(5) : Find the sine series of f (x) = ex in (0, π).

Solution: Let the required Fourier series be

Example 2.3.a(6) : Find the Fourier sine series of f(x) = l − x in (0, l)

Solution :

Example 2.3.a(7): Find the Half range sine series for f(x) = x (π − x) in (0, л). Deduce that

Solution: Let the half range sine series be

Example 2.3.a(8): Find the sine series of

Solution: Let the half range sine series be

Example 2.3.a(9) : Express f(x) as a Fourier sine series where

Solution : Let the required Fourier sine series be

Example 2.3.a(10): Obtain the sine series for the function

Solution: The sine series for the function f(x) in (0, l) is given by

Example 2.3.a(11): If ƒ (x) = k (lx − x2) in the range (0, l), show that the half range sine series for

Solution: -

2.3 (b) Half-range cosine series

Problems based on Half-Range cosine series

Example 2.3.b(1): Find the Fourier cosine series for f(x) = x2 in 0 < x < л

Solution: 

Example 2.3.b(2): Expand the function f(x) = sin x, 0 < x < л in Fourier cosine series.

Solution: Let the required Fourier cosine series be

Example 2.3.b(3) : Expand f(x) = as a cosine series.

Solution: 

Example 2.3.b(4) Find the Half-range cosine series for f(x) = (x − 1)2 in (0, 1). Hence show that

Solution: Here l = 1

⸫ The required Fourier cosine series be

Example 2.3.b(5): Find the cosine series of f (x) = ex in (0, 1).

Solution: Let the required Fourier cosine series be

Example 2.3.b(6) : Find the half range cosine series of f(x) = x in 0 < x < л.

Solution:

Let the required Half range cosine series be

Example 2.3.b(7): Find the half range cosine series of f(x) = (x − x2) in the interval (0,л). 

Solution:

Example 2.3.b(8): Find the half-range cosine series for the function f(x) = x(л  - x) in 0 < x < л.

Solution:

EXERCISE 2.3

Half range series

1. Express as a Fourier Sine Series

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

11. Find the half range cosine series for

19. Find the Fourier sine series for the function