Wedesday, May 5, 3 Erraa (Icludes criical correcios oly for he s & d repri) Advaced Egieerig Mahemaics, 7e Peer V O eil ISB: 978474 Page # Descripio 38 ie 4: chage "w v a v " "w v a v " 46 ie : chage "y y " "y y " 68 3rd lie from he bom: chage " bcos(ω ) " " bω cos(ω ) " 95 ie 7: chage 3 Theorem 36, lie 8: chage A A A A ( ) f τ δ( τ) dτ f( ) ( ) f τ δ( τ d ) τ f( ) 4 Eample 37, lie : chage "y() H( 3)e ( 3) si( 3)" "y() H( 3)e ( 3) si( 3)" 4 Figure 33 should loo lie his: 8 ie : chage m ( + ) + + f( ) ( ) m ( + ) + + f( )
Wedesday, May 5, 3 4 ie 4: chage " or " " or / " 4 ie : chage I order have lim I order have lim 5 ie 7: chage d d ( s Y ( s) sy ( ) y '( ) ) sy ( s) y ( ) ( sy ( s) y ( ) ) ds + + ds 5 7 h lie from he bom: chage 7 Eample 44, lies 4 & 9: chage 3 ies ad should read: d d ( sy ( s ) sy ( ) y '( ) ) + sy ( s ) y ( ) Y ( s ) ds ds a a 7 (7)(6) a 3 a (7)(6) 7 3 + r + r ( + r)( + r ) c + 5( + r) c + r + r ( + r)( + r ) c + 5( + r) c 4!! 3 5 4 3 d lie from he bom: chage "( + r)( + r )c + (+r ) " "( + r)( + r )c + ( + r ) " 34 5h ad 6h lies from he bom: chage!!( + )! ( ) * c+ ( ) c + +
Wedesday, May 5, 3!( + )! (! ) * * + + c ( + ) c 49 ie 4: chage F+ G a + a, b + b, c + c F+ G a + b, a + b, a + b 3 3 63 9h lie from he bom: chage F+ G F+ G F+ G F + G 7 3rd lie from he bom: chage c c V G V V 7 Corollary 6, lie 8: chage c c c c c V G V c c c V c + c + + c cj j X v v v v c + c + + c cj j X v v v v j 76 ie 4: chage 76 ie 6: chage V V X V V V 3 3 d, V3 V X3 V d V V hv V, V V d V V 3 V3 V V V
Wedesday, May 5, 3 X3 V d V V X3 V V 78 as lie: chage 8 Theorem 68, lie : chage 8 Theorem 68, lie : chage 8 Theorem 68, lie 4: chage u U u U u S S U U u u u v u S S u u S > u v > s S u v > u v s u v > u u 8 Eample 6, lie : chage R 5 R 6 8 ie, chage 84 ie : chage c c c c + + 3 + + c c + c + c + + 8 8 ( ) ( ) ( ) π si si 3 π d π 7π 8 8 ( ) ( ) ( ) π si si 3 π π 7π d 9 6 h lie from he bom: chage 8 6 6
Wedesday, May 5, 3 9 3 rd lie from he bom: chage 9 d ad 3 rd lies from he bom: chage A ad whose secod colums are I I ad whose secod colums are A 3 Eample 78, lie : chage 48 5 h lie from he bom: chage /5 6 /5 bbb + bbb bbb 33 3 3 3 3 bbb bbb bbb + 33 3 3 3 3 5 5h lie from he bom: chage "Bu by (3), A B A " "Bu by (3), A B A " 54 7h lie from he bom: chage "Add imes " "Add 5 imes " 56 Theorem 8, lie : chage < i < < < 56 Eample 84: chage 6 3 7 A 5 6 4 6 6 3 7 A 5 9 4 6 84 93 Orhogoal Marices, lie : chage 89 emma 9, lie 5: chage 89 emma 9, lie : chage 9 Eample 97, lie 4: chage A A A A If S is sew-hermiia, he ZHZ If S is sew-hermiia, he ZSZ ( ) Z HZ (Z HZ) Z H Z Z HZ ( Z ) HZ ZHZ (ZHZ) Z H Z
Wedesday, May 5, 3 98 4 h lie from he bom: chage 99 Theorem 3, lie : chage 7/ 7/ 7/ A 7/ X AX; X() O X AX; X( ) O X AX; X() E X AX; X( ) E ( j) ( j) 3 ies 3/4: chage c Ω C 3 3 e + ( ) e 3 3 e + e c 3 3 3 3 c( e ) c( ) e e ( ) e c 3 3 c 3 + 3 ce ce e e ( ) c Ω C e e c 3 3 e e 3 3 3 3 3 3 c( e ) + c( ) e e ( ) e c 3 3 c 3 + 3 ce + ce e e 34 Eample 7, lie : chage 5 4 4 X X 4 4 5 5 4 4 X X 4 4 5 35 6h lie from he bom: chage "The eigevalues of A are /, /5 wih correspodig eigevalues, " "The eigevalues of A are /, 3/5 wih correspodig eigevecrs, " 39 ie 6: chage
Wedesday, May 5, 3 3 ie : chage E E Φ () Ee + E e α (+ 3 α) / α (+ 3 α) / 3 + + Φ () Ee + E e 5 e 4 e 4 5 e + + 5 e 4 e 4 5 e 3 3 rd lie from he bom: chage E E AE E 3 E AE 3 3 3 ie : chage 5 5 5 E 3 4 5 5 5 5 E3 4 5 3 4h, 7h, 8h, 9h, ad h lies from he bom: chage "r " " r" 38 ie 3: chage 3 s 3 3s [( + ) s e e 4( s s) e e ] ds 3 s 3 3s [( s) e e + ( + s) e e ] ds 3 s 3 3s [( + ) s e e 4( s s) e e ] ds 3 s 3 3s [( s) e e + s( + s) e e ] ds
Wedesday, May 5, 3 348 4 h lie from he bom: chage 357 ie : chage G 3 3 3 3 ( s) cos s + si s +, i j G 3 3 3 3 ( s) si s + cos s +, i j ϕ i+ j y cos( yz) [ cos( yz) z si( yz)] y si( yz) ϕ i+ j 36 Eample 8, lie : chage 367 4 h lie from he bom: chage y cos( yz) [ cos( yz) yz si( yz)] y si( yz) ϕ (, y, z) z + y ϕ (, y, z) z + y ( ( ), y) ), z( )) ( ( ), y( ), z( )); ( ( ), y( ), z( )) ( ( ), y( ), z( )); 373 Eample 7, lies 5, 9, & 3: chage 4si 4cos 4si 4cos 373 3 rd lie from he bom: chage 3 6 3[ cos( )][ si( )] 4 si ( ) 4 cos 6 + d 3 π / 3[ cos( )][ si( )] 4 si ( ) 4 cos 6 + () d 38 4 Idepedece of Pah ad Poeial Theory, lie 9: chage ϕ ϕ ϕ CF dr C d+ dy+ dy y z ϕ ϕ ϕ CF dr C d+ dy + dz y z 39 Eample 6, lie : chage au cos v, y au si v, z u ( ) ( ) ( ) ( ) aucos v, y busi v, z u
Wedesday, May 5, 3 39 h lie from he bom: chage 394 ie 9: chage 394 Eample, chage 396 h lie from he bom: chage (, z ) (, z ) S/ S / y S, uv, ( y, ) y (, z ) (, z ) S/ S / y S, ( u, v) (, y) y π 5 cos( v)si( v)si( v) dv u u du + + π 5 cos( v)si( v)si( v) dv u u du D:, D:, y mass of Σ δ ( P ) ( P ) ΔuΔv j mass of Σ δ ( P ) ( P ) j j j j j ΔuΔv 397 as lie: chage 4 ie : chage 4 ie 4: chage ( V Δ) A Δ ( V Δ) A Δ j j VA V Δ j VA V A j j g f ( F ) y g f ( F) y d dy F Tds [ f (, y) i+ g (, y) j] i+ ds ds
Wedesday, May 5, 3 d dy F Tds [ f (, y) i+ g (, y) j] i + j ds ds ds 4 ie 5: chage 4 ie 5: chage π [ 3 si( )( 3 cos( )) + 3 cos( )(3 cos( ))] d π [ 3 si( )( 3 s i ( )) + 3 cos( )(3 cos( ) )] d π 3 [ cos ( ) si ( )] r θ r θ rdrd θ π 3 [ cos ( ) si ( ) + ] r θ r θ rdrd θ 44 ie : chage "q q (, y, z) " "q q (, y, z) " 48 ie : chage 49 8 h lie from he bom: chage 43 Equaio (36): chage ( ) π 4 b ( π )si ( ) d 3 π π ( ) π 4 b ( π )si ( ) d 3 π π ( ) a π a ( ) a a a f( )cos( π / ) d a f( )cos( π / ) d 43 5h & 6h lies from he bom: chage "[, ]" [, ]" 43 Figure 3 should loo lie his:
Wedesday, May 5, 3 439 5h lie from he bom: chage "P[, ]" "PC[, ]" 44 Theorem 3, lie : chage "[, ] "(, ) 444 Theorem 33, lie : chage "[, ] "(, )" 446 as lie: chage ( ) + (cos( ) ( ) ) ( ) (cos( ) ( ) ) 447 ies -4: chage π A F()cos d π π F () si si F() d π π π f () a si d π π π f () si d a si d π + π b π
Wedesday, May 5, 3 π A F()cos d π π F () si si F() d π π π f () a si d π π π f () si d a si d π + π b π 447 4 h lie from he bom: chage F ( ) a b cos( π ) π F( ) a 447 3 rd lie from he bom: 448 Eample 33, lie : chage 4 6 + 3 π + ( ) 4 6 + ( ) 3 π cos( π / ) cos( π / ) 449 ie 3: chage 3 If eiss, he he Fourier coefficies of f() 3 If eiss, he he Fourier coefficies of g() 449 ie 4: chage (g()) (f()) 45 ie 4: chage b ( f( )) d
Wedesday, May 5, 3 b ( g( )) d 45 ie 6: chage b ( f( )) d b ( g( )) d 458 6 h lie from he bom: chage 466 ie : chage π + 468 8 h lie from he bom: chage π + d a f( ) d d a f( ) d ( f ( ξ) cos( ωξ) dξ) cos( ω) ( ( ξ ) si ( ωξ ) dξ ) si ( ω ) dω ( f ( ξ) cos( ωξ) dξ) cos( ω) ( f ( ) ( ) d ) ( ) d ξ si ωξ ξ si ω ω Bω fe ( ξ) cos( ωξ) dξ π Bω fe ( ξ) s i ( ωξ) dξ π 47 5 h lie from he bom: chage ˆ 5 iω 5 iω ( 5 iω ) f ω H e e d e e d e + d ( ) ( ) ( 5+ iω ) e 5+ iω 5+ iω ˆ 5 iω 5 iω ( 5 iω ) f ω H e e d e e d e + d ( ) ( ) ( 5+ iω ) e 5+ iω 5+ iω
Wedesday, May 5, 3 476 Proof: chage F ω iω iω [ e f()]( ω) e f() e d ˆ( ω ω ) i( ω ω ) e d f iω iω iω F [ e f( )] ( ω) e f( ) e d ˆ( ω ω ) i( ω ω ) e f( ) d f 479 8 h lie from he bom: chage fˆ( ω ) F [ g ( )]( ω ) iω[ g( )]( ω) iωf f ( τ) dτ( ω) fˆ( ω ) F [ g ( )]( ω ) iωf [ g( )]( ω) iω f ( τ) dτ( ω) 48 43 Filerig ad he Dirac Dela Fucio, lie 6: chage H( ) δ( ) 48 ie : chage a iω F [ H ( + a)] H ( a)] e d e a iω iω ( aω ) iaω iaω si ( e e ) iω ω a iω F [ H ( + a)] H ( a)] e d e a iω iω a ( aω ) iaω iaω si ( e e ) iω ω a a 486 ie : chage 487 ie 8: chage π π f f e e d π πω i / iω () ω π π f f e e d πω i / iω () ω π () ˆ i f f ) e ω ω π ω ω
Wedesday, May 5, 3 () ˆ ) i f f ( e ω ω π ω ω 493 d lie from he bom: chage π U e e e e i j i π U e e e e i i ija ij / ija πij / ija ij / ija πij / j j 494 ie : chage 494 ie 9: chage 494 ie 4: chage U U ia πij / ia πij / ( e ) ( e ) i e i e ia πij / ia πij / ia πi / ia πi / ( e ) ( e ) i e i e ia πi / ia πi / U U U U e e i e i e 5i 5 i πi/5 i πi/5 5i 5i e e i πi/5 i πi/ 5 i e i e 5i e i e i e i 5i i πi/5 πi/5 5i 5i e e i πi/5 πi/5 i e i e i 495 h lie from he bom: chage r j π( r j) r j ( W ) e ad W πi( r j)/ e ( ) ( W r j ) e r j ad W r j ( )/ e πi 496 ie 9: chage ( j + ) 4 j 4, for j,,, ( j + ) 4 j 4, for j,,,
Wedesday, May 5, 3 496 h lie from he bom: chage u u j j 4 j si 4 j si 497 ies 6 ad 8: chage 4 4 i i e e U + ( 4 i / πi / ) i[ + ( 4 i/ πi/ )] 4i 4i e e 4 π π + + + π 4i 4i 4i 4i [4 πie ( e ) ( e e )] 4 4 4 4 i i e e U i + ( 4 i / πi / ) i[ + ( 4 i/ πi/ )] 4i 4i e e 4 π π + + π e e e + e 4 π 4 498 46 Sampled Fourier Series, lie 7: chage 4i 4i 4i 4i [4 ( ) ( ) M () i/ p M S d e π M M () i / p M S d e π ] 498 as lie: chage + M M πij / πij / Ue Ue M + M πij / πij / Ue Ue M 5 ie 5: chage U 7 j 64 j ij/64 e π
Wedesday, May 5, 3 U 7 j 64 j ij e π /64 5 ie 3: chage + + + + πi/ 7 ( 4735 ie ) ( 355 ) + ( + 3557 ie ) + ( + 53 ie ) 3 πi/ πi + ( + 878 ie ) + ( + 6745 ie ) 5 πi/ 3πi + ( + 5763 ie ) + ( + 573 ie ) 7 πi/ 4πi + ( + 4453 ie ) + ( + 399 ie ) 9 πi/ 5πi πi + ( 399i) e + ( 4453 i) e ie πi 8 / 9 πi/ + ( 573 ie ) + ( 5763 ie ) πi/ π / + ( 6745 ie ) + ( 878 ie ) πi/ 3 πi/ + ( 53 ie ) + ( 3557 ie ) 4 πi/ 5 πi/ + ( 355 ie ) + ( 3735 ie ) 6483 6 πi/ 7πi/ + + + + πi/ 7 ( 4735 ie ) ( 355 ) + ( + 3557 ie ) + ( + 53 ie ) 3 πi/ πi + ( + 878 ie ) + ( + 6745 ie ) 5 πi/ 3πi + ( + 5763 ie ) + ( + 573 ie ) 7 πi/ 4πi + ( + 4453 ie ) + ( + 399 ie ) 9 πi/ 5πi 8 πi/ + ( 399i) e + ( 4453 i) e + ( 573 ie ) + ( 5763 ie ) ie πi 9πi/ πi/ π/ + ( 6745 ie ) + ( 878 ie ) πi/ 3 πi/ + ( 53 ie ) + ( 3557 ie ) 4 πi/ 5 πi/ 6 πi/ + ( 355 ie ) + ( 4 735 ie ) 6483 7πi/ 5 6 h lie from he bom: chage 5 3 rd lie from he bom: chage c c ( ) π π 6+ 6+ 3 4 4π ( ) 4π π 6+ 6+ 3 4 4π
Wedesday, May 5, 3 + + ( ) 6+ 6+ π 3π 4π ( ) 6+ 6+ 4π 3π 4π cos ( ) cos ( ) 57 ie 6: chage f cφ f cφ f cmφm f cφ f cφ f cmφm m 5 ies & : chage 3 3 3 4 5 3 3 + + + + 8 5 4 5 5 5 5 5 6 35 3 5 5 6 + + + + 4 8 8 6 4 6 3 3 3 3 4 5 3 3 + + + + + 8 5 5 5 5 35 4 4 35 5 + + + + 4 8 8 6 8 4 6 539 ie 8: chage 4 5 5 6 3 5 6 y ( ) c J( b) + cj( b) a c a c ν ν y ( ) c J( b) + c Y ( b ) a c a c ν ν 539 8 h lie from he bom: chage b 6 539 Eample 56, lie 6: chage "The b e," "The b 7 e," 545 Equaio (53): chage 55 Theorem 5, lie 8: chage ce I( ) ( + 8 ce I( ) ( + 8 J ν + ν Jν( ) Jν ( )
Wedesday, May 5, 3 55 ie 6: chage 554 ie 8: chage 554 ie 8: chage J ν ( ) + ν Jν( ) Jν ( ) α+ π α α+ π α ( uv vu ) d ( uv vu ) d ( v ( )) J d ( v ( )) J j d ( ( )) ( ( )) j Jv J j Jv j d ( ( )) ( ) ( ) j Jv J j Jv j d 58 3 rd lie from he bom: chage y y iω F ( ω) (, ) e d iω y(, ) e d fˆ ( ω, ) y y iω F ( ω) (, ) e d iω y (, e ) d yˆ ( ω, ) 58 as lie: chage 583 ie : chage ˆ ω ω ˆ y(, ) c y(, ) ˆ ˆ y( ω, ) c ω y( ω, ) ˆ ( ω ) ω ˆ y, + c y(, )
Wedesday, May 5, 3 583 ie 9: chage ˆ ( ) ˆ y ω, + c ω y( ω, ) yˆ y ( ω, ) cωbω F (, ) ( ω,) Fg [ ( )], ( ω) gˆ ( ω) yˆ y ( ω, ) cωbω F (, ) ( ω) Fg [ ( )], ( ω) gˆ ( ω) 59 ie : chage "Y (s, )" "Y (, s)" 59 Case, lie 3: chage 59 ies 7 & 8: chage K Fs () K s K Fs () K [ ] s ck ( ) E ck E ( ) e s ck ( ) E ck E e s (( (+ ) )/ c) s (( (+ ) + )/ c) s e s e s ((( + ) )/ c) s (((+ ) + )/ c) s 59 ie 3: chage 593 ie : chage () g () g for / c, 4 / c for / c 4 / c for / c, + 4 / c for / c 4 / c
Wedesday, May 5, 3 596 6 h lie from he bom: chage 6 ie : chage + + ( e e ) + si(4 ) cos(8 ) 8 + + ( e e ) + cos(4 ) si(8 ) 8 M + M + M ud cud ucd c d c du c + M M + u M ud cud ucd c d c du c 6 ie 6: chage F(, y) da Δ F(, y) da Δ + c c + c c ( ) g w dw ( ) g w dw 6 3 rd lie from he bom: chage + 5 w e dw X T d X dt 5 Δ + + + 5 w e dw X T dx dt 5 Δ + + 64 Figure 6: chage J () y J () 65 as lie: chage T T + c T + λc T 66 ies & 5: chage 66 3 h lie from he bom: chage 67 ies 5 & 6: chage F + ( / r) F F rf + rf F () + () + ( ) () ; ( ) rf r rf r r Fr FR () + () + ( λ ) () ; ( ) rf r rf r r Fr FR
Wedesday, May 5, 3 j f ( r, θ ) a J r R j j + a J r cos( θ ) b J r si ( θ ) R + R j f ( r, θ ) a J r R j j + a J r cos( θ ) + b J r si ( θ ) R R 67 8 h lie from he bom: chage j π a J r f(, r θ) dθ α(), r R π π j π a J r f(, r θ) dθ α(), r R π π 65 Eample 7, lie : chage ( ) π ( ) π / A A u(, ) + cos e π + ( ) ( ) π ( ) π / A A u(, ) + cos e π 66 ie 3: chage 69 ie 7: chage 69 ies 5 & 6: chage X ( ) + AX( ) c+ c X ( ) + AX( ) c+ Ac π U(, ) c si e π U(, ) c si e π / π / 3 πξ c si d ξ πξ si dξ
Wedesday, May 5, 3 c 3 πξ si d ξ ξ πξ ξ si dξ 6 Equaio (76): chage si si 69 3 rd lie from he bom: chage π πω si ( πω) b si d 633 ie 5: chage 635 ie : chage c c 635 ie 3: chage b ω ω ( ) ( ) π ξ ωξ ξ π ω ω ( πω) π πω si ( π ξ ) si ( ωξ) dξ π π ω deomiar by deomiar by s/ e s / e s/ (, ) ce + U s U s 635 ies 4, 8, 9 & : chage c c 635 ie 6: chage 636 5 h lie from he bom: chage : : A s A s s / (, ) ce +,, T + λt ad F + F + λf r T + λt ad F + F + λf r
Wedesday, May 5, 3 637 ie 7: chage () + () + ω () rf r rf r Fr () + () + ω () rf r rf r r Fr 637 5 h lie from he bom: chage ( ξ) ( ξ) f r aj j ( Rξ) ( ξ) f aj j 637 d lie from he bom: chage 638 3 h lie from he bom: chage 638 4 h lie from he bom: chage 644 Eample 8, lie : chage X + λx, Y + μy, T + ( λ+ μ) T, X + λx, Y + μy, T + ( λ+ μ) T, T + + m π T ( ) T + + π ( m ) T π 4 ( π ξ)si( ξ) dξ ( ) 3 π π π 4 ( π ξ)si( ξ) dξ ( ) 3 π ξ π 645 4 h lie from he bom: chage π π a f ( ) cos( ) d ad b f ( ) si( ) d R ξ ξ ξ ξ ξ ξ π R π π π a f ( ξ) cos( ξ) d ad b f ( ) si( ) d π R ξ ξ ξ ξ π π R π 645 d lie from he bom: chage
Wedesday, May 5, 3 647 ie : chage 8 ( ) ( ) π 4 π π 648 6 h lie from he bom: chage π 8 cos ξ si ξ dξ π, π ( ) ( ) 8 cos ξ si ξ dξ π, 4 8 z + r cos( θ ) Re + z Re + z z + r cos( ζ ) Re + z Re + z 65 85 The Upper Half-Plae, lie 7: chage 65 85 The Upper Half-Plae, lie 8: chage u (,) XT ()() u (,) X( ) Y( y) X + λx, T λt X + λx, Y λy 65 h lie from he bom: chage 655 ie : chage C C m m AC sih AC 4 ( β B) m sih ( β B) m 657 ie : chage d [ Φ ( ϕ)si( ϕ) dϕ
Wedesday, May 5, 3 659 Proof of emma 8, lie : chage 659 d lie from he bom: chage 66 Eample 87, lie : chage 66 Eample 87, lie : chage: 66 ie 9: chage d [ Φ ( ϕ)si( ϕ) ] dϕ g g ds ( g f) ds ( g f) da C C D f g ds ( g f) ds ( g f) da C C D g + + + f f g f g f y y g + + + f f g f g f y y y u < < < y< for,, u for < < < y< C C,, u ds y dy 3 u ds y dy 3 X () d, X () c, 663 ie 8: chage π b f( ξ) si( ξ) dξ π π π π π b R f ( ξ) si( ξ) dξ 68 3 rd lie from he bom: chage u y v,
Wedesday, May 5, 3 u y v, y 686 5 h lie from he bom: chage 689 Eample 93, lie : chage iz iz z iz cos( z) ( e + e ) ad si( z) ( e e ) i iz iz cos( ) ( ) ad si( ) ( iz iz z e + e z e e ) i z e π π ( /4 ) i + i z e π π ( /4 ) i + 689 Eample 93, lie : chage π log( + iz) l + i + π i 4 π log( + i) l + i + π 4 69 Eample 95, lie 3: chage /8 πi/6 /8 i( π/4+ π) /8 i( π/4+ 4 π)/4 /8 i( π/4+ 6 π)/4 e, e, e, ad e /8 πi/6 /8 i( π/4+ π)/4 /8 i( π/4+ 4 π)/4 /8 i( π/4+ 6 π)/4 e, e, e, ad e 76 Eample 7, lie 4: chage < z < < z < 78 ef-had side colum, lie 8: chage f ( w) f( w) f ( w) dw dw πi + Γ w z πi Γ w z f ( w) f( w) f ( z) dw dw i + Γ w z i Γ w z 78 8 h lie from he bom: chage f ( w) f( w) f ( z) dz dw πi γ w z πi γ w z f ( w) f( w) f ( z) d dw i w γ w z iγ w z π π 736 Figure : chage Γ γ (chage uppercase Gamma lowercase Gamma) 738 ie 8: chage π π
Wedesday, May 5, 3 γ f ( zdz ) πi + + cos( i) + cos( i) πi( + cos( i)) γ f ( zdz ) πi + + cos( i) + cos( i) πi( + cos( i)) 738 Eample 5, lie 8: chage si( z) + Res( f, i) lim z i z ( z i )( z i ) si( z) + Res( f, i) lim z i z ( z i ) 744 h lie from he bom: chage 3 3 75 Eample 3, lie 7: replace a wih obai cos ad si 75 as lie: chage 753 ies & : chage u e si( b) ad v e cos( b) u e cos() b ad v e si () b e si( b) + ie cos( b) e cos( b) + ie si( b) 754 Figure 35, lef side: chage lefmos π/ π/ 754 Blue bo, #: chage All C ( ) All C ( ) 758 Eample 34, lie : chage horizoally by Re(z) ad verically by Im(z) horizoally by Re(b) ad verically by Im(b) 758 Eample 34, lie 3: chage Tz () i Tz () z + i 763 Theorem 34, lie 5: chage ( w w)( z z )( z z) ( z z)( z z )( w w ) 3 ( w w)( z z )( z z) ( z z)( z z )( w w ) 3 3 767 Figure 3 capio: chage z o w > 3 z < o w > 3
Wedesday, May 5, 3 773 8 h lie from he bom: chage gz ( ) a( ξ ) ( ξ ) ( ξ ) gz ( ) a( z ) ( z ) ( z ) α α α α α α 775 Righ-had colum, las lie: chage 9< arg w < π /3 ( ) ( ) < arg w < π / 3 86 d lie from he bom: chage Y() y( ) Y() y(e ) 86 Problem 9, lie : chage a,,c, a,,a, 88 Problem 9, lie 5: chage (EA) sj ( row s of E) colum j of A) (EA) sj (row s of E) (colum j of A) 84 Problem, lie 4: chage S S () () ( ) 4 π si π ( ) 4 π si π 84 Problem 5, lie 4: chage S () cos ( π / ) ( ) si π S () cos ( π / ) ( ) si π π ( ) ( π ) ( ) ( ) 864 Problem 9, lie : chage ( ( w u+ iv y + yi u y v y 4, so ad 4 ( ) ( ) w u+ iv y + 4 yi, so u y ad v 4 y