Fourier series : infinite series that re
present period ic function in terms of sines and cosines Fourier Integral : Problems that invdve function that are non periode and are of
interest on the Whale × -
axis
Period ic function : f- IN for AN real ✗ ( except possible at some Points )
de fired > Consider
any period ic function f , (x) of period 2L that can be representeert by
and if there is a positive number p called the period Of f-IN ,
a Fourier series :
for which f- ( ✗ tp ) f- ( x )
=
; or : f- ( ✗ tnp ) =
f-( x ) •
Lo£1 ) s
( cost Wnx ) tbncoslwnx )) with Wn × =
ao +
„=,
an =
Fourier series for Zr period ic function s : f- IN lancoslnxltbnsinlnx)) with :
'
aotn
- = :
'
What happens if L
= , we let -
> as : the series turn into an
Integral
}
r as
ao =
In /
-
r
flxldx >
f- ( x ) =
/[ AIW ) COSIWX ) t Btw ) SINIWX ) dw ] with Fourier
integral coefficients
r
Fourier coefficients 0 as
an , yg„ „ „ „ „ „ „„
given by the AIW ) =
In / f ( v) COSIWV) dv
Euler f- ormulas das
-
„
£ / f- (
"
=L / f- ( sinlnxldx Btw ) v) sinlwv) dv
=
bn x)
- as
-
r
If f- ( x ) is
piecewise continuous in
every finite interval and has a right and
-
left hand der Native at every point and if the a ntegraiexists then f- ( x ) can Derepresented
Trigonometrie System Function zr ( sininx ) , cosinxl )
,
: with a
period of
↳ The
trigonometrie System is
orthogonalon.rs/srlalsoosxszr) ;
by a Fourier
integral .
↳
ftp.ixildx
- as
of any functions in the trig System over that
that is : the Integral two .
At a
poinowhere f- ( x ) is dis continuous , the value of the Fourier integrale qu als the
interval is zero , so that for any integers hand m :
hand limits of
^
"M average of the left -
and right f- ( x ) at that point
{
-
.
°
-
|
r
COSINX ) COSIMX ) DX
= 0 for ( n =/ m) * Kronecker delta QMT
:
Smr 1 AM
Fourier BIW ) -01 integraal of Btw ) dd)
Integral I ff has Fourier
^
cosine : a
Integral representation and is ever, then
{
,
sincnx ) sinlmxld ✗ = 0 for in # m )
then the Fourier integral reduces to a Fourier cosine
as
Integral :
/ AIW { { f- (
i.| interval 0)
f- ( X ) =
) COSIWX ) dw with AIW ) = V) COSIWV ) dv
m)
sinlnx) COSIMX ) DX for in # morn ( Odd Integral symmetrie
= =
= 0 on
.
Fourier integral iff AIW ) even )
sine : has a Fourier
integral representation and is odd , the = 0 / integraal of AIW ) -
_
reducestoa Fourier sine integral :
then the jtourier integral as
Representation by a Fourier series : Jf f- ( x ) is
zr-periodicandpiecewi.se continuous in rsxsr -
f- ( X ) =
| Btw ) sincwx ) dw with BIW ) =
{ { f- In sinlwvldv
and we let f- (x ) have a left and right handed derivative
- -
at each a
Points of that interval , then the Fourier series ( has limit →
approaches real number)
converges a a as
|
coslwx ) kx
f- ( x )
21g e-
Its sum is except at points where f- ( x ) is dis continuous → there the sum of the series
is the
average of the left
-
and
right hand limits of f- ( x ) at ✗o .
Laplace integra / s :
k
'
t wz
dw =
☐
as
f.
wsinlwx )
Fourier series for period 2L &
p
:
dw
7- e-
-
-
=
pz + wz
We set ✓= Î × ,
so dv = Ê dx ,
this
givesus :
Integral trans form
as
f- ( x ) =
aot an cos /Ê × + bnsin 7- × ; with :
: a transformation in the form of an Integral that producers from given function s
function s dep ending on a different variable
a a ,
New .
a. = te / f- klok > can be used in DE 's ,
P DE 's and Integral equations
an
.
_
:
Êfflxicos ( Ex)
L
"
Fourier cosinetransform : concerns even function and is obtained from the Fourier cosine
Integral
bn -
-
E.{ f- als in / 7- ) × "×
Set AIW ) = % Êclw) where c
suggests cosine
VÉ}
If f- ( x ) is oneven function its Fourier series reduces to a Fourier cosine series :
h f- IN =
ÊCIW ) cos ( wxldx } inverse Fourier cosinetransform :
§ } { f- ÊIW)
as
f- ( X ) =
aotn =,
ancos (% ×
with coefficients ao =L flx ) DX ; an = IN cos
/7- ) DX ×
gives bach fl ✗ I from
Odd function s : f- 1- × ) = -
f- ( x ) Fourier sine trans form : concerns Odd function s and is obtained from the Fourier sine integral
Jf f- ( x ) is an Odd function its Fourier series reducestoa Fourier sine series : set BIW ) =
VÉFÌIW ) where s suggests sine
L
} / f- (7- ) DX Fc / f) Êclw )
as
f- ( x ) =
n= 1
bnsin % × with coefficients bn = IN Sin ×
Writing v. ×
as
gives : Other rotation s : =
0
ÊSIW )
ÊSIW ) VK.f.fi/1siniwxIdx } Fourier sinetransform Fs ( f )
a a =
| hlxldx = 0 for Odd h
gives bach
fl ✗ I from ÊSIW )
* The Fourier transforms
operations : -51 aftbg ) AF / f) Fig )
[
are linear t b
-
=
Half range expansions Spitting the Fourier series
-
: in a Fourier sine series and Fourier cosine series as a
%/ | f- In eiw
" "
-
of the Fourier series :(Best approximation off trigonometrie pdynomial of the same degree N) complex form of the Fourier Integral f/✗ ) dvdw
The Nthpartial by : =
sum a
↳ error is as smalt as
possible cs cs
- -
N
"
Bnsinlnx ))
f-( x )
Aotn (An COSINX ) + Euler for make : e cost ✗ It Isin ( X )
= =
as
= ,
]
*
[ /anztbn )
'
with E- [
*
E ? E-
*
E:-[
*
BN Other rotation s : 5- If ) f F- (
b n r
If Aóao
only if
=
r > 0 and so and =
;
-
- -
, ,
. . .
.
=, as
r
f IÊIWIÎCIW
-
as
Bessel 's in
equality :
Zaoztn /anztbn )
'
s f / ✗ Îdx Total
energy of a
System
:
,
-
r - as
5- { f- ( x )}
'
For function f Parsevaistheoremholds that
such a is Bessel 's in
equality Fourier transform of the derivative : in 5- { Fix ) }
=
,
as
hdds the equality sign so that it becomes Parseval 's
a
Identity :
=/ / f x-p glpldp
,
n
Control ution f- * g hix ) f- * g) IN flplg / x-p ) dp
te / f- ( ✗ Îdx
•
: = ( =
/ )
Zaoz +
'
lanztbn ) =
- cs -
cs
na r
2nF / f)
5-( f #
g)
=
as
Fcg)
If # g) 1×1=9 ÊIW )
g iw) eindW
ij ij
- as
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