Fourier Series

197 Pages·2005·8.11 MB·English

SCHAUM’S OUTLINE SERIES
THEORY AMD PROBLEMS OF

FOURIER
LNNTAVESLIS)

with applications to

BOUNDARY VALUE
PROBLEMS

MURRAY R. SPIEGEL

INCLUDING 205 SOLVED PROBLEMS.

SCHAUM’S OUTLINE SERIES

McGRAW-HILL BOOK COMPANY
SCHAIMS OUTLINE OF

THEORY AND PROBLEMS

FOURIER ANALYSI

with Applications to

Boundary Valuc Problems

by
MURRAY 2. SPIEGEL, Ph.D,

Former Frafasor vad Chairmen
of Mosbewntrs
Hassler Palyiesocie Ina
of Conneticut

SCHALOTS OUTLINE SERIES
‘MoGRAW-HILL DOOK COMPANY

New Fork, St Loui, San Francine, Dsceldey, obwrnetbarg, Bula Lampur, ladon. Weaeos
Montreal Nee Debi, Penna, io Paul, Sitaupare Sydney, and Poronts
Conyteht © 1974 dy Miu, Ine, AU vighte reread, Printed in fhe
Untal Staten of Amina No pave nt Si pohiotion tae be nepoted
Fserenit, mactaniee, phatonpying, Tending, oF adheres, thon
pros secben persian of the pl

Library of Congress Cadatog Cord

ping, Waray 1
Soherar's eve a theory and pL uf Honea

1 Tile, T, Ttle Teanga poles of Poti

oer a ee
Preface

Ta the ently years of the 19k coutury the Wrench mathematician J. 1k 4, Fonrier
fn hia rvseavches on heat eonductian was Tel (y Uke reuuirkatle discovery of veetaia
frigonwuetzie series whic wwe We his ounie. Since that time Fuarier series, and
seenerslizariing tu Fursiee incerale and avthogonal somes, huve become an essential
part of fev Wueksraund of anientigs, engcicers and custhematieiant Prom bolbs sn if
Plind an Unecretien” point of sles

‘The purpose uf this bak i to present che fondamoatsl camcepre snd applivations of
ouv'er series. Fourier integrals aad orthoxonal funetions ‘Bessel, Leyenda, Tere
sud Lacuerre tonetiong, ay well na others

The Dock is designed th'he need ether a8 4 Lestounk for a formal eomrse in Fancior
Analysis ur an a comprehensive supplement ‘a all ecrrene stander texts, I abval. be
Gf considerable calio ta those taking emiraea in enginoesiny, scieuee ur macher-atios In
‘whieh thece finporlunt exethads are frequently waed, IL soon alka prove naehut
‘book of ruferance to research wovkery crugloving Fourier mechngs ut tu Uhose inter
ste i the fel Fo at

ach rhupler bogies with a clear elatuinent of portisent datinitions, peiacivkes and
fhourencs. tigether with Hlushrative und other desex'ptive rater, The solved wrote
Jems serve to illnstrcce aad eaaplity the theory aaul lo provide the mepettion of basic
Principles o vila to efective learn, Numieeuus pecs of thentems and derivat ous
of Cormulus sre inlnded aamueye the sulved uroblemos. ‘The large aitiber uf saul
mentary prntvems with uussvesa aeree as A camplete review of "he material nf each
chapter

‘Considerably moro maverial hae beoa included bere than can he eovores ia meat
first ecutmes. This hina buen done to make the onal more flexible, to PrOviGe A mite
nsefil bou' of reference, and ca atimlnte Lurther interest in the tonics,

I wish tn take this upyuctanity Us Unank Hunry ttayden ané Davld Beckwith for
their aptendil eamurution,

WR, Brunet

Sonuary 1974
hapee

CONTENTS

BOUNDARY VALCE PROELEMS
Wechemaisa Korean and Soft. ve Hays! Dookimne! Tote Dee
Caiteg ts Parl Kiertal Baus, Lcvvur Pare) eget Masa
Ua," Sime Tenor, Paci Mifesntal queues The Laslacien sn Ear
Al Curainac Hemeam elt af Boke Meudniy Vala Denon,

alee

FOURTRE SIiKIES AND APPLICATIONS
ist Dotririm of Watcier Series Dinh ee Wotton, dent ren
FFunahine, “TEnfeeuze ourer Sine ar fine Seren Turse Lents
Unitir Convertor Lueyeaton wn] Ditesoatatan of Pour hoties Con

20

Chapter

ORTHOGONAL FUNCTIONS
Definitions Fouling Golbegonal Ppcetha, Qribiarnnd eel Ongena
Srvee Arpuoimutiows "the Looe: Sgonren noe Fans Wo
DBrthtanemal Sesion Cenpltene Storie aycanm. Plgavclees cod

GAMMA, BETA AND OTHER SPECIAL PENCTIONS
patil Punetoe, The Gama Frist, Tae of Values aod Gragh of te
Ginna Pineusn apne Pattly tor Fin ieelencvan Rents base
se the Ganuua Suction The Bela Fanta, “Otter Spel, Pivesions
‘Aapngtoti Seen cx Bape

er

Chapter

FOURIER INTEGRALS AND APPLICATIONS.
‘The Seed ter Pecos negra Toye paver Tnteguet, wouonat Bohne of
Teanetrma,uracta' Teas fer Texvly'Iaaecele The Corvlution

0

BUSSEL FUNCTIONS AND APPLICATIONS

Besse Dita Kuat'n, ‘he Metbid of Halen Tess Punwons
‘ie Fire Rau Hewat Prnetice uf the Secon! Kid Conc Function fe
piss Because Temata Punetons Hailed fe rool Pawson, Rake
FPentica, Zevon of Hose Peete Delogansery of teal Pentre of te
hel Kink Seva af Bera oncom ofthe Piet Wigs DetNogonlty woe
Seren of Heel Pensions othe Seed Biel, Beane t Houndary Year
[roles Ung Bese Ponefone

oo
CONTENTS

Chayer 7 LEGENDRE FUNCTIONS AND APPLICATION: 80
Vordee's Diderentl Eqneron, Tayatate Yolymandls. Gooeraxing Punetca
{6c alin Payers Herma, ‘gends Pciaso e
Polynit, Asmacinte Covesdee Funvviors, Crogwealty of Assiate
Legends Pasctns, Hokrionsyis Bouaders. Value Brvtoewe Caing Laseadre

Chape § HERMITE, LAGUERRE
AND OLHRR ORTHOGONAL POLY NOMIATS st
Yor Heriniea Payrarialn | Tosorerare Roslvs fue. termite. Paleo,
GuiLojuualie of Termite Pagromisls. Herne u: Nermite Paynoutale La
uur Dideerncial Panation, Lavsezee Datmutials, Pane Injoctaal Pop

Appondie A Lainounees uf Ratarions 187
Swusbal Youriw Bavien 19
Special Pouiner rowormn 1
Tables of Veiuee tov Feit) avit dist 16

ANSWERS 70 SUPPLEMENTARY PHORLEDE

10

moe
Chapier 1

Boundary Valve Problems

MATHEMATICAL FORMULATION AND SOLUTION OF PHYSICAL PROBLEMS
Im solving problemas uf science and orginoetiny the following steps ave generally taken,

A, Mathematical formulation. To adhiove ouch Lormalolion we vavally adopt mathenvatica!
rmadete hich scrve lo approximate the real objecte under investigation.

Mama 1
In the mattemstical formulation we use known piysiea! lawe to at op equations
dleece Phy tho problens, If Iho lows ave unkmeva we tay oten beled ta ae! up ezpere
vbente order 10 clnvover dhe

"a aging the matles of ¢ phe abl hee ws aie Began’ be to aston aw diferent
ruins sone the duces of te Planet tos the SH BL any Le

2. Mathematical solntion, Once w problem hus beer succesnfully formulated in terma at
‘equstions, we novd Lo anive them for the unknowns involved, pObject to the various
conditions which ar given ox implied in the phyzical pratlem, One imporiant con
sideration 1S whother such folutione actually sviat and, if Zhey do exine, ius Thay

Ip the aUlempl to tind achutions, the need far now kinds of mathematical analyeid—
to now Mulbemnstieal problema —may arise.

eal

Braap 5

ii. Poni stamping tr slew praem i owt ye whith be had “ivauatad in led
acti itera equations, ms ly tthe interne Sree of Sapte of ancien ee sti
[vciing since und exer Sor sri, son slag Peuior snr are af inet Et I pe af Ve
Mohali! eared im pao appicaten, a2 oul wt in Chagas 3

3. Phrsieal interpretation. After a solution hae uum obtained it in unelul to interpret
physically. Such ‘rfetyrele-ious muy be Uf vulue in suggesting other kinds of problems,
hick conld lead :n new kuowledie of & auuthematieal or physieal nature,
in thia book see shall be maialy wonearhed with the matkemuatical frmaiation uf physl-

cal problems in tara of partial iforentiol erations aad with the solstien of puch equalions

bby motheda commonly called Fourier metods,
WOENMARY WAIATE PROBLEMA conan

DEFINITIONS PERTAFNING TO PARTIAL DINVERENTIAL EQUATIONS
A pavtine diffevential equation ig an estation containing £m wnkmowa fometion of tev
or miore vasinbles and ils parti’ dovivativus sith espe. fa these arable.
‘The order of a pustial dlfoventiel equation fe che onder of the highest derivative
prevent

e+» few qui lceuia eatin of ure tay ob senicondey gate! Aieentn

A zolusion of purl dUferential equation is uny funelive which wulislice the eyuation
Merten.

The generut wuintiox is 4 golvtion whieh contains » qumber of axbilrury Indepeudeal
Tuustious eguat Le he order of Une ezualipa

A verticular sohtion is one whith can be obtained frum the xenorol zololion by putiou-
lac duo of Ihe wcbilruey fumetions,

Mose ena, ¢° 9K fe FeCl acento etn
Thin poeliclae Pir tssasy igi Sy, ee aewin the partccar seis

~ fort — Bene — a6

A super aoiation ie one which camor he obtained from the yeuerad solution by par:
Aeolus hie of the arbitrary functions,

ws at (M
ooh) gt de 2 ane pbtivie, Phe the general eaatoeinslvr gw algny
Tuscon ft "The sovond, whlen canal be vealed tron Li gencenlroaton Ry may Ch ot gh

A boundery ewlue probleo: involving: a puta! eifferential equation sucks oll olutZone
fof the equation which zatisty voulizoms eelled haundary ooaditiegs,.‘Phourems relating to
the oxislence and uniqueweny of such eoutiona sue ealled eindsvee wad wdiueneas theorema

here 9 ik a fetlon of © ane a, we ame by subaiation Eu tat

UINEAR FAUTIAL DIFFERENTIAL EQUATIONS,

‘The venecat Knorr porcie? diderenNel eqwatica. uf ueder two im wa indepeslent vaei-
ablea hus the form

Beg te. ge, yaw
Aan” Pasay ~ Sap} Poe = 6 a
sphere AVF, 6 may depend on and y but ner ov eA souomdhonder aquttion will

Independent sziublan

Ay 6 ~0 ienivally the otualion is eed homogeneous, while it (#0 JL called xen
Jenngmenas, Generaliatios to higher order wysationa ave easily made

anil y which does not have fhe “ure (21 i called woatheoat

‘Becaune of the ualure of the solutuns of (1), tho eauliom iz often oladsied ae citie,
Jorwerbitix, or paratot srcnvdins we E4408 lege Ubeny eer shan, Ov oa
respect
cman ROTNTARY VALUE PROBLEMS

SOD IMPORTANT PARTIAT, DIFPERERTIAL

‘This oatatiun JS applicable to tke amat
rnsnyverts vockUloms Uf § tae, Hee acing,
sas a viois stung, iia Hated the
Seuis eat ae: itn mcion ‘seg Fig. tj, The
Tunelior yf) is the dieplacorent of any
dicks: 9 of the strings al Law f. The conetane eat
Guns whut + iy the foxuntant tens in
fy Avie wl 9 by cha femostant masa per
Ut our of the string. a ast ed that
fe external furese eet an the ering nd Gal Fut
IU vibrates only dua ta its elasccity
The eynation ean easily be penoralized Wo luher dimensions, 23 for exammle thy
Sioretinns uf 9 mombeune or drarakead in twa dimensions, "wlio dimen, the

2 Meat eondnetioneeuation | SF = gt

ere aia. 6.9142 the temparatare al position (e-y.2) int so8d at Hime ¢. The oon
Saal call coe fiery, iy enual ty King, where she Phased contustiity #, the
‘usc heat » and the deusity eases nee unit volume) , nee aasomned renslan:, We call
‘Vin che Lainie of vy it ia given in. taven-cimensional vecramgulae vous
(nay hy

EQUATIONS

ve eB

Laplace's equation | wer

Tis equation oceuea in many Salle, Te te theoey of henl eondaction. for example,
wie tho steedyfate tempevuters, jan Ube semaperalice war a long thre bas elapsed,
vrloso equation fs abiaiael by pulliye @u/bi ~ 0. in the heae condaction equation above.
In Une theory af gravilation or oleekrisity x represenes the greaitativan of electri
ovtentik espectively, For Uiis reason Ihe epration te atten called the polenta wowelia.

‘The ponblm sf evans Wie O inside a aeyion = wher x is une piven Lunelion
ba the boeriary af ia often ealled a Dirichlet proba

4. Lonsiludinal vitations of a bean ae 8 ate

This e1vatiun doseribes the mation a a beam (Fig. 1-2, pow 4h whieh cua vibeate
Tenyittdindly (2. tn the Airey: ina Cae yculiuns oie savanna avail. "The vacinble
wis fia the longitu! dcoplecement from the eynilfariee position of the erase section
The umlual «2 — Bjs, where £ ia the modulus of caatiity istrens divided
Uy strain: ane deporte on the prupurties of Ihe beam, «it the cenaity ‘nase per nil
vyokam

‘ote thar this equation is ke suis we Ul Lor a vibrating sbring.
4 ‘BOUNDARY Vain: benwiiee HALL

0

~ pees
E }

, my
& Transom vation wizam | 3
‘ts euutton deserts Ge motion of beam fatally Kale om he sath te
sin 18) she eli: eee ft porpendale tthe eto arene
fil bration In haem ae) rngrerge pica or deen ant
{ine to any ait, the comtant: <a were Ec te rman of ety,
Fg tho ment of face ot any eons tse ube ay 4 te a ne
{acon ad Ue capo tong ate an estar Snamervo nes Fs

SS tient right-side a he coat is Fplced by E19 St

Learns ‘base

Meta pars

‘THE LAPLACIAN IN DIFFERENT COORDINATE SYSTEMS

‘The Laplisian Chu often seiies jn partial ditfesentiol equations af science’ and eagi-
necring: Depending on the type-of problem tnvolves, the’caoice of coazdinaze system may
be ixportunt im obisining eolotions. Far example, #€ thé problem involves a aylinaen, #
ill uftnne convenient ' ase eytivatical coordéuatea; chile it H-vaives a spheve, St el
‘We eonvanient to uze suferial coordinates.

‘The Laptuciun in eylindsical coordinates ip. ,2) (908 Fig. 1a ie geen bir
a lan 1d, oe P
aot" dy mae | ant ®,

‘The Lrausformatici options between rectangular and éylinavicel coqrdinaten are

ve

peeg, Hm pang, 2

whére
‘The Laplacian in Spheviesl epordinates (7,2, 4) fee Fis. 1-8) is glen by

Chg et, execs,