Unt-T Fourier Seríes
Qeiodie Fundion'
4funetíon to) Soid to be a priode fanebon
apositive
for all n, forT) f)
.
lonstant.
kost ke of T for ohich the Above pstion
pald Galled the peviod ot fo)
{x +2 sin(1+x)- sinx =fo),
1(xt 41) - Sin(a x)-fo) =Sinx
Sinx ia a periodie functin ith period,
Cosn a with period gT
periodi function oit
tonx i Periodi funetion with pero d T
Diichlet Conditions :
Said to divichlet Gndrtlons
the Gondtions are Satifed."
in
(:) ). Single Valued finide
(e,e12l)
G) g) Contiauous oo' pieiser tontinusus in
maxima ond
(G) 4) has & finite number f
hinima in (C, c4 3)
geiodic uncfion oih pmid
f o) is a
then it Gn be
Satsfies Dirichtet Gorditions,
ond ineoluig Sine and
an infnite Seris
Gusine
for the function foJ in
Fouríe Sries
Ihus the Fouvíer Seriea
n=
uhee
an=
Notes
founier ocffients (a
) The onstants Q,, ansbn
ti fulers Gnstant
The Formua we are Sed to fnd aan are
Galled Euler's fomula
) Dirichlet Conditíons not but onlg
Sufficient for the xisence of Fouvier Seiey.
the inteal
the
folnier Series igen by q0
is o,fu) or) (-1,TU)
hen
"basinn
2/
khere
27
On
bn fo)Sinnude
to 0.
5) he Fourier Sjes, of o)
IS 'poindt of Coatinity
ipoint adiscontint
the RHL LHL
Ahre Ha) fa) ae
the funchon to) ta
o at
ath
Q- hh
Root Meon Squane e
lue (RMs
he inteital a,bi
Ret )
lt be dfioad in
then MS lue
b-a
Parse val's heotem:
10)
the Fouríey Series
Conespondg
(G,cas8)
then
n
Bernoullr's Fomula
-u +u'V
kere , , , so on
Note' Bernozili'5. fermua s teplid hen one
of terms inside a and chose
Note
e Sinbvck (a Sinbx blosbn
SinnI =0 n is an
n s an
indyger.
ine -0s29
Cos = It(0s2
SinA osB- Sin(At6) + Sin(A-RB)
-6)
Cost (os& = os (A16) +os(A
2
Sint Sinb Cos(A-6)- asA 48)
obtaín the Fourier Series of thefundion,
f ) = -x in o,aT) intval
dex<i
FS for the given funion in,
o) ao
hl
bsinrx
where
u' -l V=SinNX-Gsn
Bentoult's dormuda
anm-2) Sinnx - Cas(rm)2
Jo
SinGnt)
91
bn
dussionvd
u': -I
- Cosank ,-Sinnr
n
Beonellis fomala.
(-)osnx Sinnx
n
Gs (onn) Si ont).
-(r-o)so Sino
n
20
b, n
9 h n
K)=e in (o, n)
Fourier
Garier Bevies fol he gven fn in
fo«) 4S 0nCosnx
Ahene Q, =
+{-te"e)
oith a=-,bn, e
t
ntJ
ba
(asinbx , beosbu)
Sinna - nlosn)
(-Sinetx -
n-ne
n't
X -21 -s)
20
4) 6) - in (o,n)
nt|
3
2Sinnx
an
dv cOSNK
)-o) u'=2x
an
n Va = -Sinn
ns
4) Sinnxdx
bo
20
u-x2r dvsin n
x sinnyd u'=X
0-(osnx
n
V, -Sinn
n
Cosnx 4 Sinnx x
1
V,CoSnx
n n2
n3
t0+2
x 4T3 n= n
) fa)xSinx
6) = Qo 4an Cosx 4 basinn
ISinsdx
+ Sinx
u'eV:
Vi-Soy
Sin Gina) Snn)
yo
9t x Sn(+nx t Sinl-ndk
nt|
V -Sn(4n)Sin-a)%
Gs(4n)
4 Sin(4n)*, Sinl-a)
- (-n
(4) (1:n)
-
7
-1)", (ts9 *P
psag
b, = A
ba Lf60 Stnnxdx
xSnx Sinnd
21
(os(-n)x - (os(4n) dx
2
u'|
VSina-n)x
, -Gsl-x Gslranlk
bn
Cosl-n
-2, an
9
9
the
Ffnd the fourler 3eriés a perhod A
inteval o,m)
Henee dedue that I-t;t 4
Doo fourier Seriy
fouier Seria f the given fuancion
Kroo that
(Gorntinuous pót) in O
Sin)
-Sin+.
4
FS in ,i1) henee
) ).-x(T -x)
deduce the Sum f
(0)
() 92
92
-|F
&T3)
3
3
( )Gsnl
u' V Sinh
u' -2 -Sinny
V,= -os n n
n
Apelying -bonoulls phmila
an
(EH-2)03nx + 2Shny)20
n
ne +o)-(o+M-9. h
an= - 20
-4
n2
,
(Coninuous point) in 9
9u9
(p-a)
d sinnK
V= -Cesnx
n n
nt
Aerirg Benols faamlo,
n n Jo
b=
n-I
(hsnx+D
3
4S!Cosnx
33
(Continuous point ) in 9
x-7
f ) - 32_ , Cosnx
2n92
34
2
(i) Rt X=0 (disontinues; point) in D.,
ne)
3
(
'o (A-o)
tot
k2,6)(o )
-2n)
9
-44
...2
A)Fnd the Fouriex
ene dedue 8 "
97
xdu
(M
+(
do'sosndk
V=Sinn u'
Ay Bin vulis piiple 2i
(er-)Sinnk.. (oshy
n n
n n
an=-4 n
odd
n
is even
fa) Sinnrdx
2u
(-)Sinnxdy
du = sinx dy
n
V, =-Snn
+ Snn
n no Sinna
4 Sinni
Sin n
n2
nn
n n
bo Ma
-4 n=eyen
h ,3 n'
f) 4 n=i, -4 Gsnx
n-3 n?
4
TT4 4
n
9
ool:
) -) e in (o. n)
Ahoc
Q= 1,b= h
on n
nt
eSindx
Sinbxda oyg(aSnbx- brasbu)
ax
al, bh
n@un
e ()
n
bn n -ae)
e n(
ttence
3 3
TI
6
n
u" 9 -Slan
,
24
An Jo
e')- ( (o h2
n2
ant
Jo) Sinnw dx
4
x)'Shnsdk
-(osnx
u (T-x) n
-Sinnk!
u'- 2(-) = -2(u-)
Cosn
n
Benoull kontm.
n
4
b,=0
ni
Cosnx
point)
x= 0 ( discontinuo
+3 tosnx
- It6)=
?MaM
4.
poit)
Casny
Cosnx
E
n
|2
