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