Mathematical Methods in the Physical Sciences

MATHEMATI CAL METHODS I N THE PHYSI CAL SCI ENCES Thi r d Edi t i on MARY L . BOAS DePaul Uni ver si t y CONTENTS 1 I NFI NI TE SERI ES, POW ER SERI ES 7 1. The Geomet r i c Ser i es 2. 3. Def i ni t i ons and Not at i on 4 Appl i cat i ons of Ser i es 6 Conver gent and Di ver gent Ser i es 6 Test i ng Ser i es f or Conver gence ; t he Pr el i mi nar y Test 9 Conver gence Test s f or Ser i es of Posi t i ve Ter ms : Absol ut e Conver gence A. The Compar i son Test 10 4. 5. 6. B. The I nt egr al Test C. The Rat i o Test 7. 8. 1 11 13 D. A Speci al Compar i son Test Al t er nat i ng Ser i es 17 15 9. 10 . Condi t i onal l y Conver gent Ser i es 18 Usef ul Fact s About Ser i es 19 Power Ser i es ; I nt er val of Conver gence 11 . 12 . Theor ems About Power Ser i es 23 Expandi ng Funct i ons i n Power Ser i es 13 . 20 23 Techni ques f or Obt ai ni ng Power Ser i es Expansi ons 25 A. Mul t i pl yi ng a Ser i es by a Pol ynomi al or by Anot her Ser i es B . Di vi si on of Two Ser i es or of a Ser i es by a Pol ynomi al 27 26 10 xi v Cont ent s 28 C. Bi nomi al Ser i es D. Subst i t ut i on of a Pol ynomi al or a Ser i es f or t he Var i abl e i n Anot her Ser i es 29 E. Combi nat i on of Met hods 30 F. Tayl or Ser i es Usi ng t he Basi c Macl aur i n Ser i es G. Usi ng a Comput er 14. 15 . Accur acy of Ser i es Appr oxi mat i ons Some Uses of Ser i es 36 16 . Mi scel l aneous Pr obl ems 2 30 31 33 44 COMPLEX NUMBERS 1. 2. 3. 4. 5. 46 I nt r oduct i on 46 Real and I magi nar y Par t s of a Compl ex Number The Compl ex Pl ane 47 Ter mi nol ogy and Not at i on Compl ex Al gebr a 51 47 49 A. Si mpl i f yi ng t o x+i y f or m 51 B. Compl ex Conj ugat e of a Compl ex Expr essi on C. Fi ndi ng t he Absol ut e Val ue of z 53 D. Compl ex Equat i ons 54 6. E. Gr aphs 54 F. Physi cal Appl i cat i ons Compl ex I nf i ni t e Ser i es 7. 8. Compl ex Power Ser i es ; Di sk of Conver gence El ement ar y Funct i ons of Compl ex Number s 9. 10 . 11 . 12 . 13 . 14. 15 . 16 . 17 . 3 1. 2. 3. 55 56 Eul er ' s For mul a 61 Power s and Root s of Compl ex Number s 64 The Exponent i al and Tr i gonomet r i c Funct i ons Hyper bol i c Funct i ons 70 Logar i t hms 52 58 60 67 72 Compl ex Root s and Power s 73 I nver se Tr i gonomet r i c and Hyper bol i c Funct i ons Some Appl i cat i ons 76 Mi scel l aneous Pr obl ems 80 74 LI NEAR ALGEBRA I nt r oduct i on 82 82 4. 5. Mat r i ces; Row Reduct i on 83 Det er mi nant s ; Cr amer ' s Rul e 89 Vect or s 96 Li nes and Pl anes 106 6. Mat r i x Oper at i ons 7. 8. 9. 10 . Li near Combi nat i ons, Li near Funct i ons, Li near Oper at or s Li near Dependence and I ndependence 132 Speci al Mat r i ces and For mul as 137 Li near Vect or Spaces 142 11 . 12 . Ei genval ues and Ei genvect or s ; Di agonal i zi ng Mat r i ces Appl i cat i ons of Di agonal i zat i on 162 114 124 148 Cont ent s <, x v 13 . 14. A Br i ef I nt r oduct i on t o Gr oups Gener al Vect or Spaces 179 15 . Mi scel l aneous Pr obl ems 4 184 PARTI AL DI FFERENTI ATI ON 1. 2. 3. 4. 188 I nt r oduct i on and Not at i on 188 Power Ser i es i n Two Var i abl es 191 Tot al Di f f er ent i al s 193 Appr oxi mat i ons usi ng Di f f er ent i al s 196 5. 6. Chai n Rul e or Di f f er ent i at i ng a Funct i on of a Funct i on I mpl i ci t Di f f er ent i at i on 202 7. 8. Mor e Chai n Rul e 203 Appl i cat i on of Par t i al Di f f er ent i at i on t o Max i mum and Mi ni mum Pr obl ems 211 199 9. 10. Max i mumand Mi ni mum Pr obl ems wi t h Const r ai nt s ; Lagr ange Mul t i pl i er s Endpoi nt or Boundar y Poi nt Pr obl ems 223 11 . 12. Change of Var i abl es 228 Di f f er ent i at i on of I nt egr al s ; Lei bni z' Rul e 13 . Mi scel l aneous pr obl ems 5 6 233 MULTI PLE I NTEGRALS 241 I nt r oduct i on 241 Doubl e and Tr i pl e I nt egr al s 3. 4. Appl i cat i ons of I nt egr at i on ; Si ngl e and Mul t i pl e I nt egr al s 258 Change of Var i abl es i n I nt egr al s ; Jacobi ans 5. 6. Sur f ace I nt egr al s 242 6. 7. 8. 9. 249 270 Mi scel l aneous Pr obl ems 273 VECTOR ANALYSI S 1 . I nt r oduct i on 276 2 . Appl i cat i ons of Vect or Mul t i pl i cat i on 4. 5. Tr i pl e Pr oduct s 278 Di f f er ent i at i on of Vect or s 276 276 285 Fi el ds 289 290 Di r ect i onal Der i vat i ve ; Gr adi ent 296 Some Ot her Expr essi ons I nvol vi ng V 10 . 11 . Li ne I nt egr al s 299 309 Gr een' s Theor em i n t he Pl ane The Di ver gence and t he Di ver gence Theor em 324 The Cur l and St okes' Theor em 12 . Mi scel l aneous Pr obl ems 314 336 FOURI ER SERI ES AND TRANSFORMS I nt r oduct i on 340 Si mpl e Har moni c Mot i on and Wave Mot i on ; Per i odi c Funct i ons 345 3 . Appl i cat i ons of Four i er Ser i es 4 . Aver age Val ue of a Funct i on 347 1. 2. 214 238 1. 2. 3. 7 172 340 340 xvi C> Cont ent s 5. 6. Four i er Coef f i ci ent s Di r i chl et Condi t i ons 7. 8. Compl ex For m of Four i er Ser i es Ot her I nt er val s 360 Even and Odd Funct i ons 364 9. 350 355 10 . 11 . An Appl i cat i on t o Sound Par seval ' s Theor em 375 12 . Four i er Tr ansf or ms 13 . Mi scel l aneous Pr obl ems 8 358 372 378 386 ORDI NARY DI FFERENTI AL EQUATI ONS 1. 2. I nt r oduct i on 390 390 3. Separ abl e Equat i ons 395 Li near Fi r st - Or der Equat i ons 4. Ot her Met hods f or Fi r st - Or der Equat i ons 5. Second- Or der Li near Equat i ons wi t h Const ant Coef f i ci ent s and Zer o Ri ght - Hand Si de 408 6. Second- Or der Li near Equat i ons wi t h Const ant Coef f i ci ent s and Ri ght - Hand Si de Not Zer o 417 Ot her Second- Or der Equat i ons 430 7. 8. 9. 10 . 11 . 12 . 13 . 9 401 404 The Lapl ace Tr ansf or m 437 Sol ut i on of Di f f er ent i al Equat i ons by Lapl ace Tr ansf or ms Convol ut i on 444 The Di r ac Del t a Funct i on 449 A Br i ef I nt r oduct i on t o Gr een Funct i ons Mi scel l aneous Pr obl ems 466 440 461 CALCULUS OF VARI ATI ONS 472 1. I nt r oduct i on 2. 3. The Eul er Equat i on 474 Usi ng t he Eul er Equat i on 478 The Br achi st ochr one Pr obl em; Cycl oi ds 482 Sever al Dependent Var i abl es ; Lagr ange' s Equat i ons I soper i met r i c Pr obl ems 491 4. 5. 6. 7. 8. 10 1. 2. 3. 4. 5. 6. 7. 8. 9. 472 Var i at i onal Not at i on Mi scel l aneous Pr obl ems 485 493 494 TENSOR ANALYSI S 496 I nt r oduct i on 496 Car t esi an Tensor s 498 Tensor Not at i on and Oper at i ons 502 I ner t i a Tensor 505 Kr onecker Del t a and Levi - Ci vi t a Symbol Pseudovect or s and Pseudot ensor s 514 Mor e About Appl i cat i ons 518 Cur vi l i near Coor di nat es 508 521 Vect or Oper at or s i n Or t hogonal Cur vi l i near Coor di nat es 525 Cont ent s zyxwvutsrqponmlkjihgfedcbaZYXWVUT " ' xvi i 10 . 11 . 11 1. 2. 3. 4. 5. 6. 7. 8. 9. Non- Car t esi an Tensor s Mi scel l aneous Pr obl ems 529 535 SPECI AL FUNCTI ONS I nt r oduct i on 537 537 The Fact or i al Funct i on 538 Def i ni t i on of t he Gamma Funct i on ; Recur si on Rel at i on The Gamma Funct i on of Negat i ve Number s 540 Some I mpor t ant For mul as I nvol vi ng Gamma Funct i ons Bet a Funct i ons Bet a Funct i ons i n Ter ms of Gamma Funct i ons The Si mpl e Pendul um 545 Er r or The Funct i on 547 Asympt ot i c Ser i es 11 . 12 . St i r l i ng' s For mul a 552 El l i pt i c I nt egr al s and Funct i ons 13 . Mi scel l aneous Pr obl ems 543 549 554 560 SERI ES SOLUTI ONS OF DI FFERENTI AL EQUATI ONS; LEGENDRE, BESSEL, FUNCTI ONS HERMI TE, AND LAGUERRE 562 1. 2. I nt r oduct i on 562 Legendr e' s Equat i on 3. 4. Lei bni z' Rul e f or Di f f er ent i at i ng Pr oduct s Rodr i gues' For mul a 568 5. 6. Gener at i ng Funct i on f or Legendr e Pol ynomi al s 575 Compl et e Set s of Or t hogonal Funct i ons 7. Or t hogonal i t y of t he Legendr e Pol ynomi al s Nor mal i zat i on of t he Legendr e Pol ynomi al s Legendr e Ser i es 580 8. 9. 564 567 569 577 578 10 . 11 . 12 . The Associ at ed Legendr e Funct i ons 583 Gener al i zed Power Ser i es or t he Met hod of Fr obeni us 13 . 590 591 Gr aphs and Zer os of Bessel Funct i ons Recur si on Rel at i ons 592 Di f f er ent i al Equat i ons wi t h Bessel Funct i on Sol ut i ons 14 . 15 . 16 . 17 . 18 . 19 . 20 . 21 . 22 . 23 . 541 542 10 . 12 538 585 Bessel ' s Equat i on 587 The Second Sol ut i on of Bessel ' s Equat i on 593 Ot her Ki nds of Bessel Funct i ons 595 Pendul um 598 The Lengt heni ng Or t hogonal i t y of Bessel Funct i ons 601 Appr oxi mat e For mul as f or Bessel Funct i ons 604 605 Ser i es Sol ut i ons ; Fuchs' s Theor em Her mi t e Funct i ons ; Laguer r e Funct i ons ; Ladder Oper at or s Mi scel l aneous Pr obl ems 615 607 xvi i i 13 Cont ent s PARTI AL DI FFERENTI AL EQUATI ONS 619 1. 2. I nt r oduct i on 619 Lapl ace' s Equat i on ; St eady- St at e Temper at ur e i n a Rect angul ar Pl at e 3. The Di f f usi on or Heat Fl ow Equat i on ; t he Schr 6di nger Equat i on The Wave Equat i on ; t he Vi br at i ng St r i ng 633 4. 5. St eady- st at e Temper at ur e i n a Cyl i nder 638 6. 7. Vi br at i on of a Ci r cul ar Membr ane 644 St eady- st at e Temper at ur e i n a Spher e 647 8. 9. Poi sson' s Equat i on 652 I nt egr al Tr ansf or m Sol ut i ons of Par t i al Di f f er ent i al Equat i ons 10 . 14 Mi scel l aneous Pr obl ems FUNCTI ONS OF A COMPLEX VARI ABLE I nt r oduct i on 2. 3. Anal yt i c Funct i ons Cont our I nt egr al s 4. 5. Laur ent Ser i es 678 The Resi due Theor em 6. 7. Met hods of Fi ndi ng Resi dues 683 Eval uat i on of Def i ni t e I nt egr al s by Use of t he Resi due Theor em 8. 9. The Poi nt at I nf i ni t y ; Resi dues at I nf i ni t y Mappi ng 705 Some Appl i cat i ons of Conf or mal Mappi ng Mi scel l aneous Pr obl ems 718 15 2. Sampl e Space Pr obabi l i t y Theor ems 7. 8. 9. 10. 11 . 667 674 682 687 702 710 PROBABI LI TY AND STATI STI CS I nt r oduct i on 5. 6. 666 666 1. 3. 4. 659 663 1. 10 . 11 . 621 628 722 722 724 Met hods of Count i ng 729 736 Random Var i abl es 744 Cont i nuous Di st r i but i ons 750 Bi nomi al Di st r i but i on 756 The Nor mal or Gaussi an Di st r i but i on 761 The Poi sson Di st r i but i on 767 St at i st i cs and Exper i ment al Measur ement s Mi scel l aneous Pr obl ems 776 770 REFERENCES 779 ANSW ERS TO SELECTED PROBLEMS 781 I NDEX al l