formulas ecuaciones diferenciales (by carrascal)

7/23/2019 Formulas Ecuaciones Diferenciales (by Carrascal)

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@jpajoeb`z nof`y`bjoah`z[@jpajoeb`z n` wyod`y eyn`bT [@jpajoeb`z n` z`lpbne eyn`bT [Ew`yaney NT [Zoz~`daz n` `jpajoeb`zT

[^èyäa n` ha `z~aiohonanT [Wyeih`da n` jeb~eybe n` Z~pyd $ Hoepvohh`T [@jpajoeb`z `b n`yovanaz wayjoah`zT [Fpbjoeb`z n` H`l`bny`T [Fpbjoeb`z n` I`zz`hT

Eyn`b n` pba `jpajoeb nof`y`bjoah Eyn`b n` ha n`yovana dásoda qp` away`j`

Ha zehpjoúb l`b`yah n` pba `jpajoúb nof`y`bjoah n` eyn`b b wez`` b jebz~ab~`z ayio~yayoaz

Lyane n` pba `jpajoúb nof`y`bjoah @sweb`b~` ah qp` `z~á h`vane ha n yovana n` eyn`b dásode

Ha zehpjoúb l`b`yah n` pba `jpajoúb nof`y`bjoah n` eyn`b b jeb~o`b` b jebz~ab~`z ayio~yayoaz # z` nì`yábawhojay b jebnojoeb`z waya ei~`b`y pba zehpjoúb way~ojphay/

@jpajoeb`z nof`y`bjoah`z eynobayoaz n` wyod`y eyn`b # lyane 1/=¾ 4 f #s( / zo`bne ¾4 n % ns

Vayoaih`z z wayanaz ¾ 4 f #s/ l #/

He qp` n`w`bn` n` s z` hh`va a pb do`diye he qp` n`w`bn` n` ah e~ye

Cedelïbàz ¾ 4 f # % s/

Jadioe p 4 % s

Z` hh`la a pba `jpajoúb `b vayoaih`z z`wayanaz

Y`npjoih`z a

cedelïbàz¾ 4 f # #a s + i + j/ % #n s + ` + f/ /

Z` jahjpha `h wpb~e n` ob~`yz`jjoúb `b~y` haz y`j~az=a s + i + j 4 7n s + ` + f 4 7

Zo z` jey~ab `b `h wpb~e W#se( e/

Jadioe=s 4 se + p

4 e + v

Z` hh`la a pba `jpajoúb cedelïbà

Zo y`zph~ab wayah`haz=Jadioe= } 4 as + i #ú } 4 n s + `/

Hobàh`z ¾ + w #s/ 4 q #s/

Z` dph~owhoja wey `h faj~ey ob~`lyab~` y #s/ ~ah qp`

y #s/ [ ¾ + w #s/ T 4 y #s/ q #s/ 4 n [ y #s/ T % ns



I`ybephho ¾ 4 a#s/ + i #s/ w

D`noab~` `h jadioe p 4 1$w( z` ei~o`b` pba `jpajoúb hob`ah

Yoja~~o ¾ 4 a#s/ + i #s/ 9 + j #s/

Jadioe= 4 1 + p( nebn` 1 `z pba zehpjoúb way~ojphay jebejona

@saj~azD #s( / + B #s( / ¾ 4 7 @z `saj~a zo z` v`yofoja qp` D #s ( / % 4 #s ( / % s

Ha zehpjoúb n` ha `jpajoúb `z F #s( / 4 j~`( nebn` F #s( / `z pba fpbjoúb ~ahqp`∉ F #s ( / % ∉ s 4 D #s( /∉ F #s ( / % ∉ 4 B #s( /

Be `saj~az" Faj~ey

ob~`lyab~`Dph~owhojadez ha jpajoúb D #s( / + B #s( / ¾ 4 7 wey ° #s( / n` feyda qp`ha jpajoúb y`zph~ab~` z`a nof`y`bjoah saj~a"

° #s/Jpabne #∉ D % ∉ $ ∉ Δ % ∉ s / % B `z fpbjoúb n` s° #s/ 4 `sw { ∫ [ # ∉ D % ∉ $ ∉ Δ % ∉ s / % B T ns t

° #/Jpabne #∉ Δ % ∉ s $ ∉ Θ % ∉ / % D `z fpbjoúb n` ° #s/ 4 `sw { ∫ [ #∉ Δ % ∉ s $ ∉ Θ % ∉ / % D T n t

° #s " /Jpabne #∉ Δ % ∉ s $ ∉ Θ % ∉ / % #s D $ B/ `z fpbjoúb n` #s " /° #s " / 4 `sw { ∫ [ #∉ Δ % ∉ s $ ∉ Θ % ∉ / % #s D $ B/ T n #s "/ t

° #s + /Jpabne #∉ Δ % ∉ s $ ∉ Θ % ∉ / % #D $ B/ `z fpbjoúb n` #s + /° #s + / 4 `sw { ∫ [ #∉ Δ % ∉ s $ ∉ Θ % ∉ / % #D $ B/ T n #s + / t

° #sd " b/

Jpabne #∉ Δ % ∉ s $ ∉ Θ % ∉ / % #b sd b$1 D $ d sd$1 b B/ `z fpbjoúb n` #sd

" b/° #sd " b/ 4 `sw { ∫ [ #∉ Δ % ∉ s $ ∉ Θ % ∉ / % #b sd b$1 D $ d sd$1 b B/ T n#sd " b/ t

@jpajoeb`z nof`y`bjoah`z eynobayoaz n` wyod`y eyn`b n` lyane daey qp` 1

Hhadadez ¾ 4 w

@jpajoeb`z y`zehpih`z `b

Jpabne wen`dez n`zw`may ha `b fpbjoúb n` s n` ¾" 4 f #s( w/

Z` n`yova ha `jpajoúb y`zw`j~e n` s


s

Jpabne wen`dez n`zw`may ha s `b fpbjoúb n` n` ¾"s 4 f #( w/



Z` n`yova ha `jpajoúb y`zw`j~e n`


wJpabne ~`b`dez pb wehobedoe n` lyane b `b w"

@jpajoúb n` Halyabl` $ N

¾Ah`di`y~ 4 s f #w/ + l #w/

Z` n`yova ha `jpajoúb y`zw`j~e n` s" Y`zph~a pba `jpajoúb hobàh nebn` havayoaih` n`w`bno`b~` `z ha w s ha obn`w`bno`b~`"

@jpajoeb z n` Jhaoyap~ 4 s w + l #w/

Zehpjoúb l`b`yah= 4 s J + l #J/( nebn` J `z pba jebz~ab~`"Zehpjoúb zoblphay= s 4 $ l¾#w/

Awhojajoeb`z n` haz `jpajoeb`z nof`y`bjoah`z n` wyod`y eyn`b

^ya`j~eyoazey~elebah`z

Zeb haz jpyvaz qp` z` ob~`yz`jab feydabne pb áblphe y`j~e"Zo ha fadohoa n` jpyvaz ~o`b` wey `jpajoúb F #s( ,( / 4 7( ha `jpajoúb nof`y`bjoah n`haz ~ya`j~eyoaz ey~elebah`z a `hha `z e~ya fadohoa n` ha feyda F #s( $ 1% , ( / 4 7

Da~`yoah yanoaj~ovePb da~`yoah yanoaj~ove z` n`zob~`lya a pba ya}úb wyeweyjoebah a ha jab~onan

wy`z`b~`"n B % n~ 4 $ k B 442 B #~/ 4 Be `sw #$ k ~/

Jph~ove n` iaj~`yoazZo ha yawon`} n` dph~owhojajoúb `z wyeweyjoebah ah býd`ye n` iaj~`yoaz wy`z`b~`z `bjana obz~ab~`"n B % n~ 4 k B 442 B #~/ 4 Be `sw #k ~/

H` n` `bfyoado`b~en` B`|~eb

Ha v`hejonan a ha qp` z` `bfyäa pba zpz~abjoa ah aoy` hoiy` `z wyeweyjoebah a hanof`y`bjoa n` ~`dw`ya~pyaz n` ha zpz~abjoa n`h aoy`"n ^ % n~ 4 $ k #^ $ â/ 442 ^ #~/ 4 â + #ê $ â/ `sw #$ k ~/nebn` ê `z ha ~`dw`ya~pya obojoah â ha ~`dw`ya~pya n`h aoy` qp` h` yenà"

@jpajoeb`z nof`y`bjoah`z eynobayoaz hobàh`z n` z`lpbne eyn`b=

a #s/ ¾¾ + i #s/ ¾ + j #s/ 4 y #s/

Ha zehpjoúb l`b`yah n` ha `jpajoúb jedwh`~a `z ha zpda n` ha zehpjoúb l`b`yah n` ha `jpajoúb cedelïbà[ a #s/ ¾¾ + i #s/ ¾ + j #s/ 4 7 T dáz pba zehpjoúb way~ojphay n` ha `jpajoúb jedwh`~a [ a #s/ ¾¾ + i #s/¾ + j #s/ 4 y #s/ T

4 c

+ w

Ha zehpjoúb l`b`yah n` ha cedelïbà `z ha qp` jeb~o`b` haz jebz~ab~`z ayio~yayoaz"

c

@z pba jediobajoúb hobàh n` zehpjoeb`z hobàhd`b~` obn`w`bno`b~`z #1 ` 9/ n`

ha jpajoúb cedelïbà

N`w`bn`bjoa òbn`w`bn`bjoa hobàh= `h|yebzqpoabe

@h |yebzqpoabe `z pb n`~`ydobab~` nebn` `b ha wyod`ya foha jehejadez hazfpbjoeb`z 1 ` 9 `b ha z`lpbna foha zpz n`yovanaz 1, ` 9,"

\ #1( 9/ 4 n`~ [ [1( 9T( [1,( 9,T T 4 1" 9, $ 1," 9



Zo haz zehpjoeb`z zeb hobàhd`b~` obn`w`bno`b~`z nì`yá z`y noz~ob~e n` j`ye"

Zehpjoúb n` ha way~` cedelïbà

@jpajoeb`z nof`y`bjoah`z eynobayoaz hobàh`z n` z`lpbne eyn`b n` je`fojo`b~`z jebz~ab~`z=

a ¾¾ + i ¾ + j 4 7

Zpweb`dez ha zehpjoúbn` ha feyda=

4 `ds

Hh`ladez a= a d9 + i d + j 4 7

i9 + > a j 2 7

Nez zehpjoeb`z yàh`z noz~ob~az #d1 d9/

Zehpjoúb= 4 J1 `sw #d1s/ + J9 `sw #d9s/

i9 + > a j 4 7

Zehpjoúb yàh neih` #d/Zehpjoúb= 4 J1 `ds + J9 s `ds

Zo cpio`ya zone ~yowh`= Zehpjoúb= 4 J1

`ds + J9

s `ds + J:

s9 `ds

i9 + > a j 8 7

Zehpjoeb`z jedwh`maz jebmplanaz= d 4 α ± ι o

Zehpjoúb= 4 `α s [ J1 jez #ι s/ + J9 z`b #ι s/ T

@jpajoeb`z nof`y`bjoah`z eynobayoaz hobàh`z n` z`lpbne eyn`b n` je`fojo`b~`z vayoaih`z" @jpajoeb`z

n` @ph`y=

a s9 ¾¾ + i s ¾ + j 4 7

Zpweb`dez ha zehpjoúb

n` ha feyda=

4 sd

Zehpjoeb`z yàh`z= d ≧d¾

4 J1 sd + J9 sd¾

Zehpjoúb neih`= d 4 J1 sd + J9 #hb s/ sd

Yaäj`z jedwh`mazjebmplanaz= d 4 α ± ι

o

4 sα [ J1 jez #ι hb s/ + J9 z`b #ι hb s/ T

Zehpjoúb n` ha way~` be cedelïbà

A emeWen`dez ei~`b`y pba zehpjoúb way~ojphay n` ha way~` be cedelïbà zob dáz qp`

wyeiay a )eme) zehpjoeb`z jedwyeiay zo v`yofojab e be ha jpajoúb nof`y`bjoah"

Dï~ene n` hezje`fojo`b~`z

obn`~`ydobanez

N`w`bn` n` ha feyda qp` ~`bla y #s/

Wehobedoe n` lyane b= y #s/ 4 Wb #s/

w #s/ 4 sz [ ae + a1 s + a9 s9 + """ + ab sbT

nebn` z `z `h däbode `b~`ye #n`zn` z 4 7/ qp` nì` ~eday waya qp` be jeobjonajeb haz zehpjoeb`z n` ha cedelïbà"

y #s/ 4 `α s Wb #s/ w #s/ 4 sz `α s [ ae + a1 s + a9 s9 + """ + ab sbT



y #s/ 4 `α s Wb #s/ jez

#ι s/

w #s/ 4 sz `α s [ #ae + a1 s + a9 s9 + """ + ab sb/ z`b #ι s/ + #ie + i1 s + i9 s9

+ """ + ib sb/ jez #ι s/ T

y #s/ 4 `α s

Wb #s/ z`b#ι s/

w #s/ 4 sz `α s [ #ae + a1 s + a9 s9 + """ + ab sb/ z`b #ι s/ + #ie + i1 s + i9 s9

+ """ + ib sb/ jez #ι s/ T

Dï~ene n` vayoajoúb

n` hez wayád`~yez w 4 1 p + 9 v

nebn` 1 ` 9 zeb nez zehpjoeb`z n` ha cedelïbà hobàhd`b~` obn`w`bno`b~`z

#Ha zehpjoúb l`b`yah n` ha way~` cedelïbà `z c #~/ 4 J1 1 #~/ + J9 9 #~/( wey

`h dï~ene n` vayoajoúb n` hez wayád`~yez zpz~o~podez J1 wey J1#~/ 4 p( J9 wey

J9#~/ 4 v( azä qp` zpweb`dez ha zehpjoúb n` ha feyda #~/ 4 J1 #~/ 1 #~/ + J9 #~/9 #~/" Wen`dez jahjphay hez je`fojo`b~`z J1#~/( J9#~/( """ zpz~o~p`bne `b ha

`jpajoúb eyolobah/

p 4 ∫ [ $ y #s/ 9 % \ #1( 9/ T ns

v 4 ∫ [ y #s/ 1 % \ #1( 9/ T ns

Eyn`b daey #e olpah/

a ~y`z/

w 4 p1 1 + p9 9 + p: : ; po 4 ∫ [ y#s/ \o #s/ % \ #s/ T ns ; \o zpz~o~podez ha

jehpdba o n` \ wey 7( 7 """ 1

Y`npjjoúb n` eyn`b n` pba `jpajoúb nof`y`bjoah n` z`lpbne eyn`b cedelïbà

,, + w #s/ , + q #s/ 47

Y`npjodez `h eyn`b n` ha `jpajoúb ab~`yoey d`noab~` `h jadioe 4 1 p #s/

nebn` 1 `z pba zehpjoúb way~ojphay qp` v`yofoja ha `jpajoúb"

Ew`yaney N

Zo N y`wy`z`b~a `h ew`yaney n`yovana( N s 4 s , 4 n s % n ~ ( 1 % N z` n`fob` jede `h ew`yaney obv`yze n`

N= #1 % N/ f #s/ 4 ∫ f #s/ n s

1" @sweb`bjoah" Pba zehpjoúb way~ojphay n` ha `jpajoúb f #N/ 4 A às `z~á nana wey w 4 [A % f #a/T às

#zo f #a/ ≧ 7/Zo f #a/ 4 7( ~`bny`dez `b jp`b~a qp` [1 % #N$a/dT às 4 [sd às T % d!

9" Wehobedoe" Pba zehpjoúb way~ojphay n` ha `jpajoúb f #N/ 4 A sd `z~á nana wey w 4 [A % abT [1 + l

#N/T$1 4 [A % abT {1 $ l #N/ + [l #N/T9 $ [l #N/T: + """ t sb

Ahlpbez n`zayyehhez b z`yo` ob~`y`zab~`z=1 % #1 + s/9 4 1 $ 9s + :s9 $ >s: + 0s> $ 6s0 + """1 % #1 $ s/9 4 1 + 9s + :s9 + >s: + 0s> + 6s0 + """



1 % #1 + s/ 4 1 $ s + s9 $ s: + s> $ s0 + """1 % #1 $ s/ 4 1 + s + s9 + s: + s> + s0 + """

:" Wehobedoe wey `sweb`bjoah= [1 % f #N/T f #s/ às 4 às [1 % f #N+a/T f #s/

>" Fpbjoúb ~yolebedï~yoja= Waya ipzjay pba zehpjoúb way~ojphay n` ha `jpajoúb f #N/ 4 A jez #a s/ ú

A z`b #a s/; zo f #N/ `z pb wehobedoe( caj`dez `h jadioe N

9

4 $ a

9

y`npjodez f #N/ a pb ew`yaney hobàh `b N n` ha feyda α N + ι

Zoz~`daz n` `jpajoeb`z nof`y`bjoah`z

Jebzon`y`dez h zoz~`da hobàh n` jpajoeb`z nof`y`bjoah`z be cedelïbè= s , #~/ 4 W #~/ s #~/ + i #~/"(nebn` s , #~/( s #~/ i #~/ zeb v`j~ey`z jehpdba W #~/ `z pba da~yo}"Zo i #~/ 4 7( `h zoz~`da `z cedelïbè"Zo an`dáz W #~/ 4 A #da~yo} n` je`fojo`b~`z jebz~ab~`z/ `h zoz~`da qp`na( s , #~/ 4 A s #~/" Ei~`b`dez hezvahey`z λo v`j~ey`z wyewoez o n` ha da~yo} A"

Zehpjoúb n` ha way~` cedelïbà= s , #~/ 4 A s #~/

1" Jaze noalebaho}aih`" Zo waya λ 4 λ1 ei~`b`dez `h v`j~ey wyewoe 1 #v`j~ey jehpdba/; waya λ 4 λ9

ei~`b`dez h v`j~ey 9 """

Wen`dez jebz~ypoy zehpjoeb`z n` ha cedelïbà= s1 #~/ 4 1 `sw #λ1~/ ; s9 #~/ 4 9 `sw #λ9~/ """

ha zehpjoúb ha `zjyoiodez jede jediobajoúb hobàh= s #~/ 4 J1 s1 #~/ + J9 s9 #~/ + """

9" Jaze be noalebaho}aih "

$ Zpwebladez qp` ei~`b`dez pb vahey wyewoe #neih`/ λ1ah qp` zúhe z` azejoa pb v`j~ey wyewoe 1= s1 #~/4 1 `sw #λ1~/

pba z`lpbna zehpjoúb ha jebz~ypodez jede= s9 #~/ 4 ~ 1 `sw #λ1~/ + `sw #λ1~/ nebn` wen`dez ei~`b`y

a way~oy n`= #A $ λ1O/ 4 1

Ha zehpjoúb l`b`yah ha jebz~ypodez jede jediobajoúb hobàh= s #~/ 4 J1 s1 #~/ + J9 s9 #~/ + """

$ Zpwebladez qp` ei~`b`dez pb vahey wyewoe #~yowh`/ λ1ah qp` zúhe z` azejoa pb v`j~ey wyewoe 1= s1 #~/ 4

1 `sw #λ1~/

pba z`lpbna zehpjoúb ha jebz~ypodez jede ab~`z= s9 #~/ 4 ~ 1 `sw #λ1~/ + `sw #λ1~/ nebn` wen`dez

ei~`b`y a way~oy n`= #A $ λ1O/ 4 1 ha ~`yj`ya zehpjoúb= s: #~/ 4 #~9 % 9!/ 1 `sw #λ1~/ + ~ `sw #λ1~/ + `sw #λ1~/ nebn` `h dozde qp`

ab~`z he ei~`b`dez a way~oy n`= #A $ λ1O/ 4

Ha zehpjoúb l`b`yah ha jebz~ypodez jede jediobajoúb hobàh= s #~/ 4 J1 s1 #~/ + J9 s9 #~/ + J: s: #~/ + """

:" Yaäj`z jedwh`maz #jebmplanaz/= Zàb λ 4 α + ι o; 4 a + i o ( hez vahey`z v`j~ey`z wyewoezy`zw`j~ovad`b~`"s #~/ 4 J1 s1 #~/ + J9s9 #~/

nebn` s

1

#~/ 4 `sw #α ~/ [a jez #ι ~/ $ i z`b #ι ~/T

s9 #~/ 4 `sw #α ~/ [a z`b #ι ~/ + i jez #ι ~/T

Zehpjoúb n` ha way~` be cedelïbà= s , #~/ 4 W #~/ s #~/ + i #~/



Dï~ene n` vayoajoúb n` hez wayád`~yez= Y`zehv`dez ha way~` cedelïbà( ei~`bo`bne ha zehpjoúbjede= s #~/ 4 J1 s1 #~/ + J9 s9 #~/ + """( wey `h dï~ene n` vayoajoúb n` hez wayád`~yez zpz~o~podez J1 wey

J1#~/( J9 wey J9#~/( """( azä qp` zpweb`dez ha zehpjoúb n` ha feyda s #~/ 4 J1 #~/ s1 #~/ + J9 #~/ s9 #~/ + """

Wen`dez jahjphay hez je`fojo`b~`z J1#~/( J9#~/( """ zpz~o~p`bne `b `h zoz~`da eyolobah= s , #~/ 4 A s #~/ + i

#~/

Zo W # ~ / 4 A #noalebaho}aih`/" Zo `b `h zoz~`da s , #~/ 4 A s #~/ + i #~/ caj`dez `h jadioe s #~/ 4 ^ #~/nebn` ^ `z pba da~yo} y`lphay jpaz jehpdbaz zeb hez v`j~ey`z wyewoez n` A( wp`n` ~yabzfeydayz` `b ,

#~/ 4 N #~/ + ^ $1i #~/ nebn` N `z ha da~yo} noalebah #jpez `h`d`b~ez zeb hez vahey`z wyewoez(jehejanez `b `h dozde eyn`b `b qp` z` zo~payeb hez v`j~ey`z wyewoez `b ha da~yo} ^/"

âdioïb wen`dez p~oho}ay `h dï~ene n` hez je`fojo`b~`z obn`~`ydobanez"

@sweb`bjoah n` pba da~yo} `sw #A ~/ 4 À~ 4 O + A ~ + A9 ~9 % 9! + A: ~: % :! + """

^èyäa n` ha `z~aiohonan

Nane `h zoz~`da hobàhcedelïbè

n s % n ~ 4 a s + i n % n ~ 4 j s + n jeb a n $ i j ≧ 7

`h wpb~e s 4 7 ` 4 7 z` hhada wpb~e n` y`weze n`h zoz~`da e wpb~e zoblphay"Jahjphay`dez hez vahey`z wyewoez #λ1 λ9/ azejoanez a ha da~yo} n`h zoz~`da ab~`yoey"

Ba~pyah`}a n` hez wpb~ez

O" λ1 ≧λ9 #yàh`z/

λ1 8 7( λ9 8 7 wpb~e n` y`weze n` `z~aiohonan azob~ú~oja 42 Bene odwyewoe azob~ú~ojad`b~``z~aih`

λ1 2 7( λ9 2 7 wpb~e n` y`weze ob`z~aih` 42 Bene odwyewoe ob`z~aih`

λ1 2 7( λ9 8 7 wpb~e n` y`weze ob`z~aih` 42 Wpb~e n` `bzohhanpya ob`z~aih`

OO" λ1 4 λ9 #yàh`z/

λ1 4 λ9 8 7 wpb~e n` y`weze n` `z~aiohonan azob~ú~oja 42 bene wyewoe `z~aih`

λ1 4 λ9 2 7 wpb~e n` y`weze ob`z~aih` 42 bene wyewoe ob`z~aih`

OO" λ 4 α ± ι o #jedwh`mazjebmplanaz/

α 8 7( ι ≧ 7 wpb~e n` y`weze n` `z~aiohonan azob~ú~oja 42 Feje #wpb~e `zwoyah/azob~ú~ojad`b~` z~aih`



α 2 7( ι ≧ 7 wpb~e n` y`weze ob`z~aih` 42 Feje #wpb~e `zwoyah/ ob`z~aih`

α 4 7( ι ≧ 7 wpb~e n` y`weze z~aih` 42 Wpb~e j`b~ye

E~ya dab`ya n` jhazofojay hez wpb~ez #zob hh`lay a jahjphay hez vahey`z wyewoez/

A `z ha da~yo} n` hez je`fojo`b~`z n`h zoz~`da hobàh ab~`yoey; ~y A 4 a + n( ha ~ya}a n` ha da~yo} A; n`~ A 4a n $ i j( `h n`~`ydobab~` n` A

∆ 4 #~y A/9 $ > n`~ A

∆ 2 7

n`~ A 2 7" ~y A 2 7 42 Bene odwyewoe ob`z~aih`

n`~ A 2 7" ~y A 8 7 42 Bene odwyewoe azob~ú~ojad`b~` `z~aih`

n`~ A 8 7 42 Wpb~e n` `bzohhanpya ob`z~aih`

∆ 4 7

~y A 2 7 42 dabab~oah #bene wyewoe ob`z~aih`/

~y A 8 7 42 zpdon`ye #bene wyewoe azob~ú~ojad`b~` `z~aih`/

∆ 8 7

~y A 4 7 42 wpb~e j`b~ye #`z~aih`/

~y A ≧ 7 ( ~y A 2 7 42 Feje #wpb~e `zwoyah/ ob`z~aih`

~y A ≧ 7 ( ~y A 8 7 42 Feje #wpb~e `zwoyah/ azob~ú~ojad`b~` `z~aih`

@z~aiohonan

Nane `h zoz~`da hobàh

cedelïbè ab~`yoey

$ Zo haz way~`z yàh`z n` ~enaz haz yaäj`z n` ha `jpajoúb jayaj~`yäz~oja n`hzoz~`da zeb b`la~ovaz( `h wpb~e n` y`weze n`h zoz~`da #s 4 4 7/ `zazob~ú~ojad`b~` `z~aih`"

$ Zo ah d`bez pba yaä} ~o`b` ha way~` yàh wezo~ova( `h wpb~e n` y`weze `zob`z~aih`"

$ Zo haz yaäj`z zeb jedwh`maz jeb way~` yàh bpha( `z `z~aih`"

Nane `h zoz~`da be hobàh n` `jpajoeb`z nof`y`bjoah`z= n so % n ~ 4 f o #s1( s9( """/ # o 4 1( 9( """ ( b/( `h wpb~e

so 4 7 `z pb wpb~e n` y`weze n`h zoz~`da f o #7( 7( """/ 4 7" N`zayyehhabne `b z`yo` n` we~`bjoaz

qp`nábnebez jeb ~ïydobez hobàh`z=1" Zo haz way~`z yàh`z n` ~enaz haz yaäj`z n` ha `jpajoúb jayaj~`yäz~oja zeb b`la~ovaz( haz zehpjoeb`z bphazzeb azob~ú~ojad`b~` `z~aih`z"9" Zo ha way~` yàh n` ah d`bez pba yaä} `z wezo~ova( ha zehpjoúb bpha n`h zoz~`da `z ob`z~aih`":" Zo haz way~`z yàh`z n` ~enaz haz yaäj`z n` ha `jpajoúb jayaj~`yäz~oja be zeb wezo~ovaz #zo`bne olpah a j`yeha way~` yàh n` ah d`bez pba yaä}/( `h `z~pnoe n` ha `z~aiohonan z`lýb ha wyod`ya awyesodajoúb `z( `bl`b`yah( odwezoih` #obfhp`b ~ïydobez be hobàh`z/"

Zoz~`daz ap~úbedez

s , 4 F #s( / , 4 L #s( /

Zeb aqp`hhez qp` )z` leio`ybab a zä dozdez); haz fpbjoeb`z F L ben`w`bn`b n` ~( zúhe n` s ` "Hez wpb~ez n` `qpohoiyoe n`h zoz~`da zeb haz zehpjoeb`z n`h zoz~`da F #s( / 4



7 L #s( / 4 7

Awyesodajoúb hobàh

s , 4 a s + i , 4 j s + n nebn` a 4 ∉ F % ∉ s ; i 4 ∉ F % ∉ ; j 4 ∉ L % ∉ s ; n 4 ∉ L % ∉ #zpz~o~ponez `bhez wpb~ez n` `qpohoiyoe/

@z~aiohonan z`lýb Hoawpbev

Zà `h zoz~`da s , 4 F #s( / ; , 4 L #s( / #1/

Pba fpbjoúb @ #s( / n`fobona wezo~ova ~ah qp` n @ % n~ 4 #∉ @ % ∉s / F + #∉ @ % ∉ / L `z z`don`fobona

b`la~ova y`joi` `h bediy` n` fpbjoúb n` Hawpbev n`h zoz~`da #1/" Wey `m`dwhe( pba fpbjoúb n`Hawpbev dp `dwhàna `z @ #s( / 4 a s9 + j 9 jeb a( j 2 7"

$ Zo `soz~` pba fpbjoúb n` Hawpbes @ #s( / waya `h zoz~`da #1/ `b~ebj`z `h Wj #7(7/ `z `z~aih`"

$ _ zo n @ % n~ 4 #∉ @ % ∉s / F + #∉ @ % ∉ / L `z n`fobona b`la~ova( `b~ebj`z `h Wj #7(7/ `z azob~ú~ojad`b~`

`z~aih`"$ Zo waya `h zoz~`da #1/ @ #s( / 2 7 n @ % n~ 2 7( `h Wj #7(7/ `z ob`z~aih`"

Jáhjphe vayoajoebah

qp`y`dez caj`y `s~y`dah ha ob~`lyah #`b~y` s1 s9/ O [#s/T 4 ∫ f #s( ( ¾/ ns

Fúydpha n` @ph`y $ Halyabl` ∉ f % ∉ $ n % ns [ ∉ f % ∉ ¾T 4 7

Zo f 4 f #s( ¾/ ∉ f % ∉ ¾ 4 j~`

Zo f 4 f #s( / ∉ f % ∉ 4 7

Zo f 4 f #s( ( ¾/ f $ ¾ [∉ f % ∉ ¾T 4 7

@jpajoeb`z `b n`yovanaz wayjoah`z

@j" jebnpjjoúb n`h jahey= ∉ p % ∉ ~ 4 k ∉9 p % ∉ s9 @j" ebnaz= ∉9 p % ∉ s9 4 α9 [∉9 p % ∉~9T

@j" Hawhaj`= ∌9 p 47

Hawajoabe b wehay`z=

Wyeih`da n` jeb~eybe n` Z~pyd $ Hoepvohh`

,, + λ 4 7 jeb #7/ 4 #j/ 4 7Zehpjoúb #nof`y`b~` n` ha ~yovoah/= 4 J z`b #b χ s %j/ nebn` λ 4 b9

χ9 %j9

Fpbjoeb`z n` H`l`bny`

@jpajoúb nof`y`bjoah #1 $ s9/ ,, $ 9 s , + b #b + 1/ 4 7



Zehpjoúb= #s/ 4 J1 Wb #s/ + J9 Qb #s/

nebn`

Wb #s/ zeb hez wehobedoez n` H`l`bny` W7 #s/ 4 1 ; W1 #s/ 4 s ; W9 #s/ 4 #: s9 $1/ % 9

; W: #s/ 4 #0 s: $ :s/ % 9( `~j"

Qb #s/ zeb haz fpbjoeb`z n` H`l`bny` n` z`lpbna jhaz`

Fúydpha n`Yenyälp`z=

Fúydpha n`y`jpyy`bjoa

#b+1/ Wb+1 #s/ ‚ #9b+1/ s Wb #s/ + b Wb$1 #s/ 4 7

Z`yo`z n` wehobedoezn` H`l`bny`

Fpbjoeb`z n` I`zz`h

@jpajoúb n` I`zz`h n` eyn`b b≠ 7 s9 ,, + s , + #λ9 s9 $ b9/ 4 7

Zehpjoúb= #s/ 4 J1 Mb #λ s/ + J9 _b #λ s/

nebn`

Mb #s/ `z ha fpbjoúb n` I`zz`h n` wyod`ya jhaz` n` eyn`b b

_b #s/ `z ha fpbjoúb n` I`zz`h n` z`lpbna jhaz` n` eyn`b b #fpbjoúb n`

B`pdabb/

Wáloba obojoah #ab~`z `b c~~w=%%`z"lèjo~o`z"jed%fozojaz/= c~~w=%%|||"jayyazjah"b`~>6"b`~%fozojaz%

@z~a z`jjoúb #n`zn` ha wáloba obojoah/= FÚYDPHAZ % DA^@DÁÔJAZ % @JPAJOEB@ZNOF@Y@BJOAH@Z

¯ Hez ap~ey`z= Dayo Wa} Cey~`habe Lúd`} ` Oñako Jayyazjah De}e ¯

Jaz~yohhe n` Neb Mpab" Wah`bjoa" #@zwaña/

Jeyyè h`j~yúboje= fozojazGacee"`z

@b ha y`n n`zn` `h 10%7:%1??3 $ Ýh~odaz denofojajoeb`z= 71%76%977: $ 9<%76%977: $ 99%73%977> $

16%7?%977> $ 1>%7>%9711 $ 96%70%9711

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