Math 220, Section 10.3-4 Spring 2020, XT
10.3 Fourier Series & 10.4 Fourier Cosine and Sine Series
1. Compute the Fourier series for function f(x) = |x|, �⇡ < x < ⇡.
1
even function f IT11 n Azt ancoscnx 1 bn8Th n x
ao f txtdx foTxdx lot In IT
an f lxlcostnxdxEIfotxcos.mxdxx cosnnxL OT
Sinha sinks O
cosennIL05
n ED D
II 2 5 nnIodµ Notrequired
e
bn f txtsinCnx dx OTeen odd
1 1 n Iz t PE Ft KD 1 cos nx
pIf t EE 2k cos Gk Dx A odd 2k I
not required
2. Determine the ⇡-periodic extension f̃ , the odd 2⇡-periodic extension fo and the odd 2⇡-periodicextension fe for f(x) = ⇡ � x, 0 < x < ⇡.
2
Een
fix T X
fix IT X oak IT
fixes fix a periodic
toxin x
iwith foxtail fttreat
Feat
3. Determine the Fourier sine series and cosine series for f(x) = ⇡ � x, 0 < x < ⇡.
3
Fourier sine fix n bnsinlh bn Zfffcxys.innTXhdxTTxnEbnsinCnx3 FEiZnsinChx
Tybn fo x sincnxdx ffotksinlnxldx fottxsnlnxdx
ff o.hn oT fx nn sina.defotint o En
Fouriercosine fix anztnfzancos.IM an Zffcx7cosChII dx
Ti x aft ancostnx Ao fifth xdx tix lot's HEIL
tnEn tD4DcoscnxIT
An fo x7coslnxdxfotticosinxdx foxcoslnx dx
3 sina.usnnxPfcosnn yotyEEfosfnII ntD En.I titty
Formula you may need 1 but use WolframAlphais easier
fxsincnxdxIEI xcosf.NL Singh c
fx cosCnx dxEEP x sinffI asgn c
feaxoostbxdx atfbl.be sinlbxtaeaxooscbxDtCfeaXsincbxdx a Ebe scbx t aeaxsincbx t C