It is often necessary to obtain a Fourier expansion of a function for the range (0, p) which is half the period of the Fourier series, the Fourier expansion of such a function consists a cosine or sine terms only.
(i) Half Range Cosine Series
(ii) Half Range Sine Series

**HALF RANGE SERIES**

It is often necessary to obtain a Fourier expansion of a function for the range (0, p) which is half the period of the Fourier series, the Fourier expansion of such a function consists a cosine or sine terms only.

**(i) Half Range Cosine Series**

The Fourier cosine series for f(x) in the interval (0,p) is given by

**(ii) ****Half Range Sine Series **

The Fourier sine series for f(x) in the interval (0,p) is given by

**Example 10**

If c is the constant in ( 0 < x < p) then show that

c = (4c / p) { sinx + (sin3x /3) + sin5x / 5) + ... ... ... }

Given f(x) = c in (0,p).

**Example 11**

Find the Fourier Half Range Sine Series and Cosine Series for f(x) = x in the interval (0,p).

**Sine Series**

**Example 12**

Find the sine and cosine half-range series for the function function . f(x) = x , 0 < x £π/2

= π-x, £x< pπ/2

**Sine series**

Exercises

1.Obtain cosine and sine series for f(x) = x in the interval 0< x < p. Hence show that 1/12

2.Find the half range cosine and sine series for f(x) = x2 in the range 0 __<__ x __<__ p

3.Obtain the half-range cosine series for the function f(x) = xsinx in (0,p)..

4.Obtain cosine and sine series for f(x) = x (p-x) in 0< x < p 5.Find the half-range cosine series for the function

6.f(x) = (px) / 4 , 0<x< (p/2)

= (p/4)(p-x), p/2 < x < p.

7.Find half range sine series and cosine series for

f(x) = x in 0<x< (p/2)

= 0 in p/2 < x < p.

8.Find half range sine series and cosine series for the function f(x) == p- x in the interval 0 < x < p.

9.Find the half range sine series of f(x) = x cosx in (0,p)

10.Obtain cosine series for

f(x) = cos x , 0<x< (p/2)

= 0, p/2 < x < p.

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