: Fourier series of Even and Odd Function

1. Active Learning Assignment of
Advanced Engineering Mathematics
Topic: Fourier series of Even and Odd Function

2. * CONTENTS *
Fourier Series.
Even and Odd Function.
Application of Fourier Series in Civil Engineering

3. FOURIER SERIES
JOSEPH FOURIER
(Founder of Fourier series)

4. Fourier Series
Fourier series make use of the orthogonality relationships of
the sine and cosine functions.
FOURIER SERIES can be generally written as,
Where,
……… (1.1)
……… (1.2)
……… (1.3)

5. BASIS FORMULAE OF FOURIER SERIES
The Fourier series of a periodic function ƒ(x) with
period 2п is defined as the trigonometric series with
the coefficient a0, an and bn, known as FOURIER
COEFFICIENTS, determined by formulae (1.1), (1.2)
and (1.3).
The individual terms in Fourier Series are known as
HARMONICS.

6. Even Functions
Definition: A function f(x) is said to be even if
f(-x)=f(x).
e.g. cosx are even function
Graphically, an even function is symmetrical
about y-axis.


7. Even Functions
When function is even:
When f(x) is an even function then f(x)sinx is an odd
function.
Thus an =
a0=
an =
bn=
Therefore f(x)=


10. Odd Functions
Definition: A function f(x) is said to be even if
f(-x)=-f(x).
e.g. sin x, are odd functions.
Graphically, an even function is symmetrical about
the origion.


11. Odd Functions
When function is odd:
When f(x) is an odd function then f(x)cosnxis an odd
function and f(x)sinx is an even function.
a0==0
an =
bn=
Therefore f(x)=


14. Note:


15. Application in Civil Engineering
The most useful application is solving vibration problems.
Vibration are obviously important in many parts of civil
engineering.
We can find of frequency of earthquake.