Download - ENGINEERING Math 4,Ch 10, Fourier Analysis
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10.1,10.2,10.3,10.4,10.5,10.9,10.10, & 10.11)
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2012
2010
Bio-medical-engineering
mathematics
By Mohammad Sikandar-khan-Lodhi
[FOURIER-ANALYSIS: CH-10]
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FOURIER-SERIES:(10.1)
PERIODIC-FUNCTIONS:Any function f(x) have some positive no (p) and its defined for all
real (x) .
[ * +, - ,
-.]
fimilar periodic function are sine & cosine functions.
(f=c=constant) => its also a periodic-function, because its
satisfied equation (A) for all (p).
GRAPH:
those function whose are not periodic are followed
Point
(a)
Point (b)
There is any
interval of
length (p) b/w
point (a) & (b)
f(x)
x-axes
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(i-e)
[ () .]
TRIGONOMETRIC-SYSTEM:Our problem in the first few sections of this chapter will be the
representation ofvarious functions of period ( ) in termsof the simple functions.
(i-e)
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And so on
Above figure show the cosine and sine function having the
period ( ) From (0 2[pi]).TRIGONOMETRIC-SERIES:There is a series which aries by above equation (D)
(i-e)
Where , a0,a1,a2,,an.; & b0,b1,b2,,bn .; all are real constant this
series is called as trigonometric-series we may also write
above series which was given in equation (E) as ,
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Also the an & bnare Coefficient of series where the set of
function in eq (D) above is called as Trigonometric-System.
Key point we see that each terms of series in equation (E)
above has the period ( 2) hence if the series of eq(E) convergesthen its sum will be a function of period (2);Key-point--> trigonometric series in equation (E) is also called
fourier-series of f .
_____________
(10.2) start here
FOURIER-SERIES:STATEMENTS;
The fourier series aries from the partical-task for representing a
periodic function [f(x)=f(x+p)] in terms of Cosine and Sine function.
The Fourier-series are often also called as trigonometric-
series(Eq-[E] in section 10.1), whose coefficient are find from f(x) by
using certain formula Euler-Formula (below).
Now, we first drive the Euler formulas which are most use full for
finding the coefficient of fourier or trigonometric series.
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EULER-FORMULAS-FOR-FOURIER-COEFFICIENTS:Let, f(x) is a periodic function of period (2), which can berepresented by trigonometric-series(or Fourier-Series).
(i-e)
We assume that this series converges & has f(x) as its Sum.
Key we want to find the coefficients *a0, an & bn of the above
fourier-series in equation (E).
Where * n=1,2,3,4,..,inifinity+.
FOR-a0 : (Eular-formula for a0):
Let , integrated on both sides in eq(E) from * ] we get,
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(above)(Eq-CE) Its the equation of a0(the coefficient of fourier-
series in equation[E]).
FOR an=m: FOR am :When (m=n),
[m= any fixed +ve No]
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Formulas:
__________________________________
We multiply eq(E) by cos(mx) & then integrated from ( ),where [m=any fixed +ve No].
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CONSIDER :
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Now placing the value of eq(Zi) & eq (Zii) in eq(G1)then
eq(G1)becomes;
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When (n=m):
FOR bn=m OR bm :
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*m=n=1,2,3,,+
_____________
Rough-work:
When (n=m):
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__________________
LIST OF EULAR-FORMULA OF FOURIER COEFFICIENT:
Where:
[ , -.]Where (an,bn & a0 ) are real-Integer-No,also these number(an,bn
& a0 ) are called fourier-Coefficients of f(x).
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And above eq(E) is called as fourier series or trigonometrical-
series.
_________________(FINISHED-HERE)__________________
EXAMPLE#1) SQUARE-WAVE:
Find the fourier coefficients of periodic-function f(x) in below
function which is given in eq (A) and sketch the graph of f(x)?
Solution:
REQUIRED:
A). Sketch the graph of f(x) of eq(A)=?;
B). find fourier coefficient of f(x) .A). Sketch the graph of f(x) in eq(A):
Graph :
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B). find fourier coefficient of f(x) .1)FOR a0 :By using Eular formula:
2
-2
+k
(amplitude)
-k
(Amplitude)
x-axis
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___________
Rough work:
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______________
FOR bn:By using Eular formula:
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Rough work:
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CONSIDER ():In general ()
FOR FOURIER SERIES-OF-f(x):
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FORMULA:
FOR PARTIAL SUM OF ABOVE FOURIER SERIES ARE:
GRAPHICAL-REPRESENTATION-OF-ABOVE-FOURIER-SERIES INEQUATION (V):
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k
-k
-
x
(2)=P=(-
For
./
k
-k
-
x
(2)=P=(-
For k
-k
-
x
(2)=P=(-
For
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Key these above figure show that fourier series is convergent
and has sum f(x) we notice that partial-sum cause to be
converge and has the sum f(x).
f(x)
k
f(x)
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In above example we noted that at (x=0) & (x=) & (x=)the f(x) is discontinutes & the value of partial sum is zero, this
means that the series of partial-sum is convergent and has the
sum is f(x) [as represented in above graph].
Let, f(x) is the sum of series then Assume we set ( ),
(10.3) Start-Here.
FUNCTION-OF-ANY-PERIOD[ ]:The function considered so far had period ( ) ,but in mostapplication the most periodic function will have other periods
rather than period( ),We can converted those function whose have ( ) in
the transition from functions of ( ) basically astretch of scale on the axis;
The periodic function f(x) of period (p=2L) has a fourierseries.
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With the fourier-coefficients of f(x) given by the euler-formulas.
That is,
[]=V
[]=V
-VV
V
f(x) (v) instead of(x);
[ ,-]
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PROOF (DERIVATION):Now, we want to drive the formula of
x=-L
x=L
x
V
f(x) is a
function of
(v) so, f(x) =
g(v)
* +
[]
[-]
P=
2L
x=-LP=
2L
x=L
x
f(x)
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Solution:
We want to set the scale, we set ,
Thus, regarded as a function of (v) which we call g(v);
Key-point) we derive this above formulas of f(x) , ao , an , & bn
from the section (10.2) formulas of f(x)1 , ao1 , an1 , & bn1 ;
Formula of section (10.2);
(i-e)
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FOR g(v):We know that, [:. g(v) = f(x)1 ].
So,
So, by replacing (v) insteated of (x) , so f(x)1 becomes as,
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STEP2:o FORf(x):
Where: [ g(v)=f(x) ]
Placing the above value in eq(G), then above eq (G) becomes,
FOR aO , an & bn :as we know that,
[ 0 1 0 1 * + ]Consider above eq (a) , (b) & (c) :
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[ 0 1 ,-
,- ,
01 ,-,- , ./ ,-
; ]
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____________________________________________________
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90
270
360=2
pi
1
-1
0
Cos(2x)
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