# Fourier Series Examples #10 Half Range Fourier Cosine Series Examples and Solutions in Hindi|PTU|GTU

**EVEN FUNCTION**

A function y = f(x) is said to be even, if f(-x) = f(x). The graph of the even function is always symmetrical about the y-axis.

**ODD FUNCTION**

A function y=f(x) is said to be odd, if f(-x) = – f(x). The graph of the odd function is always symmetrical about the origin. For example, the function f(x) = in [-1,1] is even as f(-x) = = f(x) and the function f(x) = x in [-1,1] is odd as f(-x) = -x = -f(x). The graphs of these functions are shown below : 1. If f(x) is even and g(x) is odd, then

• h(x) = f(x) x g(x) is odd

• h(x) = f(x) x f(x) is even

• h(x) = g(x) x g(x) is even

For example,

1. h(x) = x2 cosx is even, since both x2 and cosx are even functions

2. h(x) = xsinx is even, since x and sinx are odd functions

3. h(x) = x2 sinx is odd, since x2 is even and sinx is odd.

2. If f(x) is even, then bn=0 This is FOURIER COSINE SERIES ao and an is present

3. If f(x) is odd, then ao=an=0 This is FOURIER SINE SERIES only bn is present