Understanding Even and Odd Functions in Mathematics

Q1a)
EVEN FUNCTION:
The following function defined as even, when the function has symmetrical to the y-axis. The
graph has reflected to the y –axis, therefore the following function has called as even function.
The mathematical expression for even function has shown in below.
Example for even function,
The following even function waveform has shown in figure 1-a.
ODD FUNCTION:
The following function defined as odd function, when the function has rotatinal sysmetrical to
the origin. The graph has refflected phase shift of the origin. Therefore the following
function has called as odd function.
The mathematical expression for even function has shown in below.
Example for even function,

The following odd function waveform has shown in figure 1-b.
Consider the fourier analysis,
If is odd or even function has shown in mathematical expression.
Consider the following even function,
Consider the following odd function.

2
a)
The following half wave symmetry has shown in figure 1-c.

b)
The following periodic waveform has shown in figure 1,
Consider the Fourier series at t=0.
EVEN SYMMETRY EXPANSION:
The Fourier series does not contain sinusoidal term.
ODD SYMMETRY
The following Fourier series expansion contains sin term.
Consider the half wave symmetry.
Consider, the time shifting property.
The above relationship has satisfied when n=odd integer.
Therefore
The Fourier series expansion of the following half wave symmetry signal contains only odd
harmonics.
EVEN HALF WAVE SYMMETRIC:
When, DC=0
The signal contains odd harmonics and cos terms.