Summary
WTW238: Theme 4 (LU 4.1 - 4.2) Summary
- Institution
- University Of Pretoria (UP)
This document contains a summary of LU 4.1 - 4.2: Fourier Cosine and Sine functions, Even and Odd functions, Theorem of Convergence and definitions.
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. FOuRIER SERIES 39 191wpefini tiong
Inner produt ot 2 ta's fi and ft on Cab the nr
orthogonq| fn'S2 orthododl
fns Fi 4hd ta gre on Ca,bJ ik
F,fu." Fi(») fria)0(Inner product *o)
Example.
Fn fi(a) (ojlz,ft l) cos 3x dre orthogondl on , T )
ginte
Cos2.co[3x dz. 9 A
2 col (1x- in) ca (1a-3x)
N8
ox)
/2L gih5* Sin Lo
inner produch
pefini tion
ortho gonad setof veiluey- 4Setsk read vaued,fn'{f., Bi e
are ofrho iPm, tnl mion )om#n
Example n:h
T Jhow r t a vet sinainla... oftho on T tn
Jet. ard n ih (na) n,2,
Jhow thd o Pr(2) *o f nto
dm (a.( mtn
, Fourier serje
A31AUo
0 mfh->ortho
p if mh 1900
F M#h ortho
p t m=n
J-pSin m for al m,n. (orthogeral
P
.sin :o o
Suppase Phat f Fh detined on the intervd (Pp) and
n be written as
cdleuldte fo get Series
f(a)d4 Can cp bnsh
n
PJ-b
Ioen ify (
pn P Fim)sin D w l SPt Serjes in given
PJ- nier voJ.
Pietevise ont funtiong sd
fl)ad F(a) 2 ene-sidod limity
limflz4h) limFlz-h .
on Ch)
Piece vise jt
Ofinie nr. o f pojinti a, < X . . < Rn lh C a , b e
which f hes a finite dscontinuiy.
IP continuous on each $lep inlerval
2
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