9587Fourier Series Formulae List (Transforms & PDE Unit I Formulae List )
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Transcript of 9587Fourier Series Formulae List (Transforms & PDE Unit I Formulae List )
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Fourier series of f(x) in ,Fourier series of f(x) in 2,0
Unit I - Fourier series
Fourier series in an interval of length 2
Even Function Odd Function
Convergence of Fourier Series:
At a continuous point x = a, Fourier series converges to f(a)
At end point c or c+2l in (c, c+2l), Fourier series converges to2
)2c(f)c(f
At a discontinuous point x = a, Fourier series converges to2
)a(f)a(f
1n
nn0 xnsinb
xncosa
2
a)x(f
1n
nn
0 xnsinbxn
cosa2
a)x(f
2
0
0 dx)x(f1
a
2
0
n dx
xn
cos)x(f
1
a
2
0
n dxxn
sin)x(f1
b
1
0 sincos2
)(n
nn
xnb
xna
axf
dxxfa )(1
0
dx
xn
xfa n
cos)(
1
dxxn
xfbn
sin)(1
1n
n
0 xncosa
2
a)x(f
0
0 dx)x(f2
a
0
n dxxn
cos)x(f2
a
0b n
1n
n
xnsinb)x(f
0a 0
0na
0
n dxxn
sin)x(f2
b
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Fourier series in the Interval of length 2
Even Function Odd Function
1n
nn0 nxsinbnxcosa
2
a)x(f
1n
nn0 nxsinbnxcosa
2
a)x(f
2
0
0 dx)x(f1
a
2
0
n nxdxcos)x(f1a
2
0
n nxdxsin)x(f1
b
1n
nn0 nxsinbnxcosa
2
a)x(f
dx)x(f
1a 0
nxdxcos)x(f1a n
nxdxsin)x(f
1b n
1n
n
0 nxcosa2
a)x(f
0
0 dx)x(f2
a
0
n dxnxcos)x(f2
a
0b n
1n
n nxsinb)x(f
0a 0
0a n
0
n dxnxsin)x(f2
b
Fourier Series of f(x) in (0,2 ) Fourier Series of f(x) in (- , )
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Fourier Cosine Series Fourier Sine Series
Convergence of Fourier Cosine series:
At a continuous point x = a, Fourier cosine series converges to f(a). At end point 0 in(0,l), Fourier cosine series converges to f(0+)
At end point l in(0,l), Fourier cosine series converges to f(l-)
Convergence of Fourier Sine series:
At a continuous point x = a, Fourier Sine series converges to f(a). At both end points Fourier Sine series converges to 0.
Harmonic Analysis:
N
y2a 0
,
N
xncosy
2a n ,
N
xnsiny
2b n
Parsevals Theorem:
If
1n
nn0 xnsinb
xncosa
2
a)x(f
is the Fourier series of f(x) in (c, c+2l),
Then2
y
1n
22n
20 )nba(2
1
4
a(or)
1n
22n
20
2c
c
2)ba(
2
1
4
adx)]x(f[
2
1n
1n
n0 xncosa2
a)x(f
0
0 dx)x(f2
a
0
n dxxn
cos)x(f2
a
1n
n
xnsinb)x(f
0
n dxxn
sin)x(f2
b
Half Range Fourier series
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Root Mean Square Value:
2y is the effective value (or) Root Mean square (RMS) value of the function y = f(x), which is given b
2
dx)]x(f[
y
2c
c
2
Some Important Results:
1. Sin n =0 for all integer values of n
2. Cos n= (1)n for all integer values of n
3. Cos2n=1 for all integer values of n
4. Sin2n = 0 for all integer values of n
5. If f( x ) = f( x ) then f(x) is even and If f( x ) = f( x ) then f( x ) is odd.
),0()x(
)0,()x()x(f.6
2
1
is even if either )x()x( 21 or )x()x( 12
),0()x(
)0,()x()x(f.7
2
1
is odd if either )x()x( 21 or )x()x( 12
0,
0,.8
xx
xxx
bxsinbbxcosaba
ebxdxcose.922
axax
bxcosbbxsinaba
ebxdxsine.10
22
axax
..........11 321 vuvuuvudv Where
............dxvv,dxvv,dvv........,dx
udu,
dx
duu 231212
2
