MATH241

LECTURE 7 Fourice Series ( Sect .3 .1-3 .3 )

" Last day. Method of separation of variedles Chaplace eguation)

Exercise: Solve toplace equation inside a ae secta da

circolan annales acrob, or oa subject to

e(,0)= 0 " 6,0)=

c o

u(r., J= 0 aCs,o) =glos

aarab

BC :

Separatia &vanades alr,g)=( ( )Z (O).

a (e.o )= 0 = W (+)Z (0 ) - Z (0)= 0

- G6 ,3 )= 0 = w (a)Z (7 ) ~ Z ( 7/2 )=0

a(a,0)= 0 = 6 (0)2 (o) w (a)= 0

ODES: 2(8)12( W'(.))+1-2W(c)2"(0)=0

ien tent(read(as)= 20 =1

• Bup(eigenvalues) fa Z(O):

<"(0)+ 12(0)=07 1*=47 )=45,n=1,2,...

8 ( )=0 J 12(o)= sin(ano)

Z ( o)= 0

For t= 4m,n= 1,2,.... ODE fa W ( ):

a (aw(r))= theare (r)+xW"(x)=4nW()=0

Let u (e)=rmo rpretrap(p-1) 2 421P=0

ap= tan

Therefore, wer)= c.22n test

B.C. for W (s): ( (a)= 0 scan teater-co

aca = - c.an . So

W ( )= c.ro-c.a c.(12. th )

• Product solution :

al ,g)= c.(mn_el )sin(208)

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nohh

Superposition principle:

a(,0)= [ Ba( . alha )sin(200),

with the nonhomore on BC

u(6,5)=f(0)= [ Ba(8 al )sin(2nd).

Thus, using the outlogoality propertyof sin(200) over

[0,1 ],we obtain the values of Bri

But the grossuCeneddo.

o do

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[Chapter3 ]Fourier Server

Definition: The Fourie Series of a function f(x) over the

stewel hexsh is

To

C

Fego(a)= a. £ ..ces (M )+ E bensin( ha ),

som toestand,heteen

e , an =

We want to answer the following questions:

1) Are the coefficients well-defined ?

- 3 Is the series"well- defined" (ie, converges)?

3) If the series converges, does it converge to S(+)? Thatis,

is f(-) equal to its Fouria secies?

Responk . In the previous problems we assumed all these

pcrats were true.

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ounica slnics .

Theorem. Convergence of the Fourier services

If f(t) is piecewise smooth a hexsh ,then the

Foncia series F(f(x) converges to

1) the periodic extension I J(x) where the periodic extension

is continuous

2) to the average of the two sided limits, thatis,

- CA( +)+ f(x)).

where the percidic entesion has a jump discontinuity

Remark : Ifwe limit ourselves to -Lexon,we have then

+ (S(++)+J(=)) = +(88(4), and in particular,where

f(x) is continuans we have f(x)= f(f)(x).

LE3

Remark: Notice that,assuming the convergence, the definition of

the coefficients is simply a caseguace of theothogaality

property for six (mux),cos(max ) over -Lexsh .

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Exemple: Sketch St) and the Fourier series of JG )

over -Lexeh . Find the Fourien series coefficiels .

a) f(x)= x

To sketch the Fourier series F(JJ(t)weuse the previous

theorem: o sketeh ft) over Lex&h.

@ Sketch its periodic extension.

on the jump discontinuitics, F.(5)(+)is the average.

Here .

e as IF($)(=)

L x -sh -26 -L 22 32 a

Since the periodic extension iscontinuous everywhere,then

JG)= F(JJE ) for all alexsh.

Coefficerede acrefut de= ffades a

an= tfaca(min)alt, usa duaand

du= cos(ml+) dx ou= ha sia(mult ) J

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2x o

-

an en sie( 2)de+thetrain 1 =

u=2- ode=ade

toasimma)dan (n){

-- - -- 2of Joe)-

or(22cos(12)+2Lcos(a) =(7 )] =

LL cos

e

1 +1 ur nodd.

baztesin ( 2)dx = 0 !!

but I all of

even odd

We see that its

Remack : f(x)=x is an even function

Fourie series only has cosines .

à f(x)= 1 +x . . L = 1 .

sketch of g(x) and FCG):

- F(DG)

go y 1.1X / /

1+x in -1< x < 1

the we have that F.(B)(=)= 1 fa x =-bl

Coefficiente,

de= (2+0)=1.

an= f'altacon(ra)dx=2/co najde=2th a sucasas1- 0

1(1+x)sin (aux)dx= (x sin (atx de=alxsin (aut)dx =

3 u =xode = dx

3du= sin (ona)dx osho ?

- - 2 +colano)] 12 4 cos(haw)de=

- - 12 ha costam )+2 singlanad].=3 colum)=1-2 here

. nodd .

neve

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ourie

• Definition: The Fourie Sine Series of (o) over the

interval ostet is

sig G = I ben sin(928),with ba= 2 ga sn (m )se.

Remark: Netice the relationship with the (full)Fourier secies:

Given S(x) over osxah .we can define its old extension

even bexsh as follows

S !9(+) osast

Lode(t)= q1-S(-+) -braco

Then,the (full) Fourien series of food(W was

FlSoma(a)=at a cos(a ) bensin(at) with

as= some woon.ac.

ans & L Sand(0)cos(24)dx=0,

ba=ts Code(a)si(ang)dx= 325 g )sin( Jolt.

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IA

Thatis, the Tamir Sine series of a functia SC ) over

Osxet is simply the Fourien series of the odd extension

f g ) in Lexsh.

Example Sketch , and find the coefficiets, of the Fourier

sime series ever osxs ū of f(t) = 10 .

Solution : I food(a)

S

Tc

I

s

5(1)(G)= f ( Jode ( ) .

Coefruesis: sega)= busin(na),

ba=2 se silno)dhe 2 sahan de

= 2 (200(na)-1)={ to do ishold the con cons] =

o ifn even .

Remort: Notice that

S(4 = 10 .x ) in orxate but it is not

a true at the boundary pecats.

Example: Find the coefficients and sketch the fie Fourica

series oa oxxen f f(x) = cos(x).

h Job(A)= codyofasa

1) Tasaco

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An

th S(g)(A)=F(Jade(t)).

SCOJ(G)= 8busin(aa), with

ba=2 ( cas(+)sia(Ax)dx to!! (cos( 2),sn(mar)

I are rethogonal

over -Lsash

11

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