aem : fourier series of even and odd function
TRANSCRIPT
Active Learning Assignment ofAdvanced Engineering Mathematics
Topic: Fourier series of Even and Odd Function
* CONTENTS *Fourier Series.
Even and Odd Function.
Application of Fourier Series in Civil Engineering
FOURIER SERIES
JOSEPH FOURIER
(Founder of Fourier series)
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)
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.
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.
Even FunctionsWhen 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)=
Odd FunctionsDefinition: 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.
Odd FunctionsWhen 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==0an =bn=Therefore f(x)=
Note:
Application in Civil EngineeringThe most useful application is solving vibration problems.
Vibration are obviously important in many parts of civil
engineering.
We can find of frequency of earthquake.