1300mathformulas

CHAPTER 9. INTEGRAL CALCULUS 1152. fÑ=p=áë=ÖáîÉå=ÉñéäáÅáíäó=Äó=íÜÉ=ÑìåÅíáçå= ò(ñIó)I==íÜÉå==íÜÉ==ëìê- Ñ~ÅÉ=~êÉ~=áë== ∫∫^ = N +  ∂ò O +  ∂ò O ÇñÇó I== a(ñ Ió )  ∂ñ  ∂ó ïÜÉêÉ= a(ñIó)=áë=íÜÉ=éêçàÉÅíáçå=çÑ=íÜÉ=ëìêÑ~ÅÉ=p=çåíç=íÜÉ=ñó- éä~åÉK== = 1153. j~ëë=çÑ=~=pìêÑ~ÅÉ= ã = ∫∫µ(ñIóIò)Çp I== p ïÜÉêÉ= µ(ñI óIò) = áë= íÜÉ= ã~ëë= éÉê= ìåáí= ~êÉ~= = EÇÉåëáíó= ÑìåÅ- íáçåFK= = 1154. `ÉåíÉê=çÑ=j~ëë=çÑ=~=pÜÉää= ñ = jóò I= ó = jñò I= ò = jñó I== ããã ïÜÉêÉ== jóò = ∫∫ ñµ(ñIóIò)Çp I== p jñò = ∫∫ óµ(ñIóIò)Çp I== p jñó = ∫∫òµ(ñIóIò)Çp = p ~êÉ==íÜÉ==Ñáêëí==ãçãÉåíë==~Äçìí===íÜÉ=ÅççêÇáå~íÉ=éä~åÉë= ñ = M I= ó = M I= ò = M I==êÉëéÉÅíáîÉäóK== µ(ñI óIò) =áë=íÜÉ=ÇÉåëáíó=ÑìåÅíáçåK= = 1155. jçãÉåíë=çÑ=fåÉêíá~=~Äçìí=íÜÉ=ñó-éä~åÉ=Eçê= ò = M FI==óò-éä~åÉ== E ñ = M FI=~åÇ=ñò-éä~åÉ=E ó = M F= fñó = ∫∫òOµ(ñI óIò)Çp I== p fóò = ∫∫ ñOµ(ñIóIò)Çp I== p 291

CHAPTER 9. INTEGRAL CALCULUS fñò = ∫∫ óOµ(ñI óIò)Çp K= p = 1156. jçãÉåíë=çÑ=fåÉêíá~=~Äçìí=íÜÉ=ñ-~ñáëI=ó-~ñáëI=~åÇ=ò-~ñáë= fñ = ∫∫(óO + òO )µ(ñIóIò)Çp I== p fó = ∫∫(ñO + òO )µ(ñIóIò)Çp I== p fò = ∫∫(ñO + )óO µ(ñIóIò)Çp K== p = 1157. sçäìãÉ=çÑ=~=pçäáÇ=_çìåÇÉÇ=Äó=~=`äçëÉÇ=pìêÑ~ÅÉ= s = N ∫∫ ñÇóÇò + óÇñÇò + òÇñÇó == P p = 1158. dcr ê=~dîáãí~∫ípá∫çµå(~ñäI=cóçIòê)ÅêÉrêP= Çp I= ïÜÉêÉ=ã=áë=~=ã~ëë=~í=~=éçáåí= ñMIóMIòM =çìíëáÇÉ=íÜÉ=ëìêÑ~ÅÉI== rê = ñ − ñM I ó − óM Iò − òM I== µ(ñI óIò) =áë=íÜÉ=ÇÉåëáíó=ÑìåÅíáçåI== ~åÇ=d=áë=Öê~îáí~íáçå~ä=Åçåëí~åíK= = 1159. mcrê=Éë∫ë∫ìéê(Érê=c)ÇçprêÅI=É== p =rêéK=(rê ) ==~Åíë= = çå== íÜÉ= = ëìêÑ~ÅÉ==p==ÖáîÉå== Äó= ïÜÉêÉ=íÜÉ=éêÉëëìêÉ= íÜÉ=éçëáíáçå=îÉÅíçê= = 1160. cΦäì=áÇ∫=∫cîräì(rêñ)=⋅EÇ~ÅprêIç==ëë=íÜÉ=ëìêÑ~ÅÉ=pF= p 292

CHAPTER 9. INTEGRAL CALCULUS ïÜÉêÉ= îr(rê)=áë=íÜÉ=ÑäìáÇ=îÉäçÅáíóK= == 1161. jΦ ~=ëë∫=∫cρäìîrñ(rê=E)~⋅ÅÇêprçIë=ë==íÜÉ=ëìêÑ~ÅÉ=pF= p cr = ρîr =áë=íÜÉ=îÉÅíçê=ÑáÉäÇI= ρ =áë=íÜÉ=ÑäìáÇ=ÇÉåëáíóK= ïÜÉêÉ= = 1162. pìêÑ~ÅÉ=`Ü~êÖÉ= n = ∫∫σ(ñIó)Çp I== p ïÜÉêÉ= σ(ñI ó)=áë=íÜÉ=ëìêÑ~ÅÉ=ÅÜ~êÖÉ=ÇÉåëáíóK= = 1163. d~ìëë∞=i~ï= qÜÉ=ÉäÉÅíêáÅ=Ñäìñ=íÜêçìÖÜ=~åó=ÅäçëÉÇ=ëìêÑ~ÅÉ=áë=éêçéçêíáçå~ä= íç=íÜÉ=ÅÜ~êÖÉ=n=ÉåÅäçëÉÇ=Äó=íÜÉ=ëìêÑ~ÅÉ= ∫∫br ⋅Çpr Φ = = n I== p εM ïÜÉêÉ== Φbr ==ááëë==ííÜÜÉÉ==ãÉä~ÉÖÅíåêááíÅì=ÑÇäìÉ=ñçIÑ===íÜÉ=ÉäÉÅíêáÅ=ÑáÉäÇ=ëíêÉåÖíÜI= εM = UIUR × NM −NO c =áë=éÉêãáííáîáíó=çÑ=ÑêÉÉ=ëé~ÅÉK== ã = = 293

Chapter 10 Differential Equations = = = = cìåÅíáçåë=çÑ=çåÉ=î~êá~ÄäÉW=óI=éI=èI=ìI=ÖI=ÜI=dI=eI=êI=ò== ^êÖìãÉåíë=EáåÇÉéÉåÇÉåí=î~êá~ÄäÉëFW=ñI=ó= cìåÅíáçåë=çÑ=íïç=î~êá~ÄäÉëW= Ñ (ñIó) I= j(ñIó)I= k(ñIó) = cáêëí=çêÇÉê=ÇÉêáî~íáîÉW= ó′ I= ì′ I= ó& I= Çó I=£= Çí pÉÅçåÇ=çêÇÉê=ÇÉêáî~íáîÉëW= ó′′ I= &ó& I= ÇOf I=£= ÇíO m~êíá~ä=ÇÉêáî~íáîÉëW= ∂ì I= ∂Oì I=£= ∂í ∂ñ O k~íìê~ä=åìãÄÉêW=å= m~êíáÅìä~ê=ëçäìíáçåëW= óN I= óé = oÉ~ä=åìãÄÉêëW=âI=íI=`I= `NI= `O I=éI=èI= α I= β = oççíë=çÑ=íÜÉ=ÅÜ~ê~ÅíÉêáëíáÅ=Éèì~íáçåëW= λNI= λO = qáãÉW=í= qÉãéÉê~íìêÉW=qI=p= mçéìä~íáçå=ÑìåÅíáçåW= m(í) == j~ëë=çÑ=~å=çÄàÉÅíW=ã= píáÑÑåÉëë=çÑ=~=ëéêáåÖW=â= aáëéä~ÅÉãÉåí=çÑ=íÜÉ=ã~ëë=Ñêçã=ÉèìáäáÄêáìãW=ó= ^ãéäáíìÇÉ=çÑ=íÜÉ=Çáëéä~ÅÉãÉåíW=^= cêÉèìÉåÅóW= ω= a~ãéáåÖ=ÅçÉÑÑáÅáÉåíW= γ = mÜ~ëÉ=~åÖäÉ=çÑ=íÜÉ=Çáëéä~ÅÉãÉåíW= δ = ^åÖìä~ê=Çáëéä~ÅÉãÉåíW= θ = mÉåÇìäìã=äÉåÖíÜW=i= 294

CHAPTER 10. DIFFERENTIAL EQUATIONS ^ÅÅÉäÉê~íáçå=çÑ=Öê~îáíóW=Ö= `ìêêÉåíW=f= oÉëáëí~åÅÉW=o= fåÇìÅí~åÅÉW=i= `~é~Åáí~åÅÉW=`= = = 10.1 First Order Ordinary Differential Equations = 1164. iáåÉ~ê=bèì~íáçåë= Çó + é(ñ)ó = è(ñ)K== Çñ = qÜÉ=ÖÉåÉê~ä=ëçäìíáçå=áë== ó = ∫ ì(ñ )è(ñ )Çñ + ` I== ì(ñ) ïÜÉêÉ== ( )ì(ñ) = Éñé ∫ é(ñ)Çñ K= = 1165. pÉé~ê~ÄäÉ=bèì~íáçåë= Çó = Ñ (ñIó) = Ö(ñ)Ü(ó)= Çñ = qÜÉ=ÖÉåÉê~ä=ëçäìíáçå=áë=ÖáîÉå=Äó= ∫ Çó ) = ∫ Ö(ñ )Çñ + ` I== Ü(ó çê= e(ó) = d(ñ)+ ` K= = = = 295

CHAPTER 10. DIFFERENTIAL EQUATIONS 1166. eçãçÖÉåÉçìë=bèì~íáçåë= qÜÉ= ÇáÑÑÉêÉåíá~ä= Éèì~íáçå= Çó = Ñ (ñI ó) = áë= ÜçãçÖÉåÉçìëI= áÑ= Çñ íÜÉ=ÑìåÅíáçå= Ñ (ñI ó) =áë=ÜçãçÖÉåÉçìëI=íÜ~í=áë== Ñ (íñIíó) = Ñ (ñIó) K== = qÜÉ=ëìÄëíáíìíáçå= ò = ó =EíÜÉå= ó = òñ F=äÉ~Çë=íç=íÜÉ=ëÉé~ê~ÄäÉ= ñ Éèì~íáçå= ñ Çò + ò = Ñ (NIò)K== Çñ = 1167. _Éêåçìääá=bèì~íáçå= Çó + é(ñ)ó = è(ñ)óå K== Çñ = qÜÉ=ëìÄëíáíìíáçå= ò = óN−å =äÉ~Çë=íç=íÜÉ=äáåÉ~ê=Éèì~íáçå== Çò + (N − å)é(ñ ) ò = (N − å )è(ñ ) K== Çñ = 1168. oáÅÅ~íá=bèì~íáçå= Çó = é(ñ)+ è(ñ) ó + ê(ñ) óO = Çñ = fÑ=~=é~êíáÅìä~ê=ëçäìíáçå= óN =áë=âåçïåI=íÜÉå=íÜÉ=ÖÉåÉê~ä=ëçäì- íáçå=Å~å=ÄÉ=çÄí~áåÉÇ=ïáíÜ=íÜÉ=ÜÉäé=çÑ=ëìÄëíáíìíáçå= ò = N I=ïÜáÅÜ=äÉ~Çë=íç=íÜÉ=Ñáêëí=çêÇÉê=äáåÉ~ê=Éèì~íáçå== ó − óN Çò = −[è(ñ ) + OóNê(ñ )] ò − ê(ñ ) K== Çñ = = = 296

CHAPTER 10. DIFFERENTIAL EQUATIONS 1169. bñ~Åí=~åÇ=kçåÉñ~Åí=bèì~íáçåë= qÜÉ=Éèì~íáçå== j(ñIó)Çñ + k(ñIó)Çó = M == áë=Å~ääÉÇ=Éñ~Åí=áÑ== ∂j = ∂k I== ∂ó ∂ñ ~åÇ=åçåÉñ~Åí=çíÜÉêïáëÉK= = qÜÉ=ÖÉåÉê~ä=ëçäìíáçå=áë== ∫ j(ñIó)Çñ + ∫ k(ñIó)Çó = ` K== = 1170. o~Çáç~ÅíáîÉ=aÉÅ~ó= Çó = −âó I== Çí ïÜÉêÉ= ó(í) =áë=íÜÉ=~ãçìåí=çÑ=ê~Çáç~ÅíáîÉ=ÉäÉãÉåí=~í=íáãÉ=íI=â= áë=íÜÉ=ê~íÉ=çÑ=ÇÉÅ~óK== = qÜÉ=ëçäìíáçå=áë== ó(í) = óMÉ−âí I=ïÜÉêÉ= óM = ó(M) =áë=íÜÉ=áåáíá~ä=~ãçìåíK= = 1171. kÉïíçå∞ë=i~ï=çÑ=`ççäáåÖ= Çq = −â(q − p)I== Çí ïÜÉêÉ= q(í) =áë=íÜÉ=íÉãéÉê~íìêÉ=çÑ=~å=çÄàÉÅí=~í=íáãÉ=íI=p=áë=íÜÉ= íÉãéÉê~íìêÉ= çÑ= íÜÉ= ëìêêçìåÇáåÖ= ÉåîáêçåãÉåíI= â= áë= ~= éçëá- íáîÉ=Åçåëí~åíK== = qÜÉ=ëçäìíáçå=áë= q(í) = p + (qM − p) É−âí I== ïÜÉêÉ= qM = q(M) = áë= íÜÉ= áåáíá~ä= íÉãéÉê~íìêÉ= çÑ= íÜÉ= çÄàÉÅí= ~í= íáãÉ= í = M K== = = 297

CHAPTER 10. DIFFERENTIAL EQUATIONS 1172. mçéìä~íáçå=aóå~ãáÅë=EiçÖáëíáÅ=jçÇÉäF= Çm = âmN − m  I== Çí  j  ïÜÉêÉ= m(í) =áë=éçéìä~íáçå=~í=íáãÉ=íI=â=áë=~=éçëáíáîÉ=Åçåëí~åíI= j=áë=~=äáãáíáåÖ=ëáòÉ=Ñçê=íÜÉ=éçéìä~íáçåK== = qÜÉ=ëçäìíáçå=çÑ=íÜÉ=ÇáÑÑÉêÉåíá~ä=Éèì~íáçå=áë== m(í) = mM jmM I=ïÜÉêÉ= mM = m(M)=áë=íÜÉ=áåáíá~ä=éçéì- + (j − mM )É−âí ä~íáçå=~í=íáãÉ= í = M K=== = = = 10.2 Second Order Ordinary Differential Equations = 1173. eçãçÖÉåÉçìë=iáåÉ~ê=bèì~íáçåë=ïáíÜ=`çåëí~åí=`çÉÑÑáÅáÉåíë== ó′′ + éó′ + èó = M K== qÜÉ=ÅÜ~ê~ÅíÉêáëíáÅ=Éèì~íáçå=áë== λO + éλ + è = M K== = fÑ= λN= ~åÇ= λO = ~êÉ= ÇáëíáåÅí= êÉ~ä= êççíë= çÑ= íÜÉ= ÅÜ~ê~ÅíÉêáëíáÅ= Éèì~íáçåI=íÜÉå=íÜÉ=ÖÉåÉê~ä=ëçäìíáçå=áë== ó = `NÉλNñ + `OÉλOñ I=ïÜÉêÉ== `N=~åÇ= `O =~êÉ=áåíÉÖê~íáçå=Åçåëí~åíëK== == fÑ= λN = λO = − é I=íÜÉå=íÜÉ=ÖÉåÉê~ä=ëçäìíáçå=áë== O ó = (`N + `Oñ )É− é ñ K== O = fÑ= λN=~åÇ= λO =~êÉ=ÅçãéäÉñ=åìãÄÉêëW= 298

CHAPTER 10. DIFFERENTIAL EQUATIONS λN = α + βá I= λO = α − βá I=ïÜÉêÉ== α = − é I= β = Qè − éO I== OO íÜÉå=íÜÉ=ÖÉåÉê~ä=ëçäìíáçå=áë== ó = Éαñ (`N Åçëβñ + `O ëáåβñ) K== = 1174. fåÜçãçÖÉåÉçìë=iáåÉ~ê=bèì~íáçåë=ïáíÜ=`çåëí~åí======================= `çÉÑÑáÅáÉåíë== ó′′ + éó′ + èó = Ñ (ñ)K== = qÜÉ=ÖÉåÉê~ä=ëçäìíáçå=áë=ÖáîÉå=Äó== ó = óé + óÜ I=ïÜÉêÉ== óé =áë==~=é~êíáÅìä~ê==ëçäìíáçå=çÑ==íÜÉ=áåÜçãçÖÉåÉçìë=Éèì~íáçå= ~åÇ= óÜ =áë=íÜÉ=ÖÉåÉê~ä=ëçäìíáçå=çÑ=íÜÉ=~ëëçÅá~íÉÇ=ÜçãçÖÉåÉ- çìë=Éèì~íáçå=EëÉÉ=íÜÉ=éêÉîáçìë=íçéáÅ=NNTPFK= = fÑ=íÜÉ=êáÖÜí=ëáÇÉ=Ü~ë=íÜÉ=Ñçêã== Ñ (ñ) = Éαñ (mN(ñ)Åçëβñ + mN(ñ)ëáåβñ)I== íÜÉå=íÜÉ==é~êíáÅìä~ê=ëçäìíáçå= óé =áë=ÖáîÉå=Äó== óé = ñ âÉαñ (oN(ñ)Åçëβñ + oO(ñ)ëáåβñ) I== ïÜÉêÉ= íÜÉ= éçäóåçãá~äë= oN(ñ)= ~åÇ= oO(ñ)= Ü~îÉ= íç= ÄÉ= ÑçìåÇ= Äó=ìëáåÖ=íÜÉ=ãÉíÜçÇ=çÑ=ìåÇÉíÉêãáåÉÇ=ÅçÉÑÑáÅáÉåíëK== • fÑ= α + βá =áë=åçí=~=êççí=çÑ=íÜÉ=ÅÜ~ê~ÅíÉêáëíáÅ=Éèì~íáçåI=íÜÉå= íÜÉ=éçïÉê= â = M I= • fÑ= α + βá =áë=~=ëáãéäÉ=êççíI=íÜÉå= â = NI= • fÑ= α + βá =áë=~=ÇçìÄäÉ=êççíI=íÜÉå= â = O K== = 1175. aáÑÑÉêÉåíá~ä=bèì~íáçåë=ïáíÜ=ó=jáëëáåÖ= ó′′ = Ñ (ñIó′)K== pÉí= ì = ó′ K=qÜÉå=íÜÉ=åÉï=Éèì~íáçå=ë~íáëÑáÉÇ=Äó=î=áë== ì′ = Ñ (ñIì)I== ïÜáÅÜ=áë=~=Ñáêëí=çêÇÉê=ÇáÑÑÉêÉåíá~ä=Éèì~íáçåK= 299

CHAPTER 10. DIFFERENTIAL EQUATIONS 1176. aáÑÑÉêÉåíá~ä=bèì~íáçåë=ïáíÜ=ñ=jáëëáåÖ= ó′′ = Ñ (óIó′)K== pÉí= ì = ó′ K=páåÅÉ== ó′′ = Çì = Çì Çó = ì Çì I== Çñ Çó Çñ Çó ïÉ=Ü~îÉ= ì Çì = Ñ (óIì) I== Çó ïÜáÅÜ=áë=~=Ñáêëí=çêÇÉê=ÇáÑÑÉêÉåíá~ä=Éèì~íáçåK== = 1177. cêÉÉ=råÇ~ãéÉÇ=sáÄê~íáçåë= qÜÉ=ãçíáçå=çÑ=~=j~ëë=çå=~=péêáåÖ=áë=ÇÉëÅêáÄÉÇ=Äó=íÜÉ=Éèì~- íáçå== ã&ó& + âó = M I== ïÜÉêÉ== ã=áë=íÜÉ=ã~ëë=çÑ=íÜÉ=çÄàÉÅíI= â=áë=íÜÉ=ëíáÑÑåÉëë=çÑ=íÜÉ=ëéêáåÖI= ó=áë=Çáëéä~ÅÉãÉåí=çÑ=íÜÉ=ã~ëë=Ñêçã=ÉèìáäáÄêáìãK= = qÜÉ=ÖÉåÉê~ä=ëçäìíáçå=áë== ó = ^Åçë(ωMí − δ)I== ïÜÉêÉ== ^=áë=íÜÉ=~ãéäáíìÇÉ=çÑ=íÜÉ=Çáëéä~ÅÉãÉåíI= ωM =áë=íÜÉ=ÑìåÇ~ãÉåí~ä=ÑêÉèìÉåÅóI=íÜÉ=éÉêáçÇ=áë= q = Oπ I= ωM δ =áë=éÜ~ëÉ=~åÖäÉ=çÑ=íÜÉ=Çáëéä~ÅÉãÉåíK= qÜáë=áë=~å=Éñ~ãéäÉ=çÑ=ëáãéäÉ=Ü~êãçåáÅ=ãçíáçåK== = 1178. cêÉÉ=a~ãéÉÇ=sáÄê~íáçåë= ã&ó& + γó& + âó = M I=ïÜÉêÉ== γ =áë=íÜÉ=Ç~ãéáåÖ=ÅçÉÑÑáÅáÉåíK== qÜÉêÉ=~êÉ=P=Å~ëÉë=Ñçê=íÜÉ=ÖÉåÉê~ä=ëçäìíáçåW= = 300

CHAPTER 10. DIFFERENTIAL EQUATIONS `~ëÉ=NK γO > Qâã =EçîÉêÇ~ãéÉÇF= ó(í) = ^ÉλNí + _ÉλOí I== ïÜÉêÉ== λN = − γ − γO − Qâã I= λO = − γ + γO − Qâã K== Oã Oã = `~ëÉ=OK= γO = Qâã EÅêáíáÅ~ääó=Ç~ãéÉÇF= ó(í) = (^ + _í)Éλí I== ïÜÉêÉ== λ = − γ K= Oã = `~ëÉ=PK= γO < Qâã =EìåÇÉêÇ~ãéÉÇF== ó (í) = É − γ í ^ Åçë(ωí − δ) I=ïÜÉêÉ== Oã ω = Qâã − γO K== = 1179. páãéäÉ=mÉåÇìäìã= ÇOθ + Ö θ = M I= ÇíO i ïÜÉêÉ= θ = áë= íÜÉ= ~åÖìä~ê= Çáëéä~ÅÉãÉåíI= i= áë= íÜÉ= éÉåÇìäìã= äÉåÖíÜI=Ö=áë=íÜÉ=~ÅÅÉäÉê~íáçå=çÑ=Öê~îáíóK= = qÜÉ=ÖÉåÉê~ä=ëçäìíáçå=Ñçê=ëã~ää=~åÖäÉë= θ =áë== θ(í) = θã~ñ ëáå Ö í I=íÜÉ=éÉêáçÇ=áë= q = Oπ i K== i Ö = 1180. oi`=`áêÅìáí= i ÇOf + o Çf + N f = s′(í) = ωbM Åçë(ωí) I= ÇíO Çí ` 301

CHAPTER 10. DIFFERENTIAL EQUATIONS ïÜÉêÉ= f= áë= íÜÉ= ÅìêêÉåí= áå= ~å= oi`= ÅáêÅìáí= ïáíÜ= ~å= ~Å= îçäí~ÖÉ= ëçìêÅÉ= s(í) = bM ëáå(ωí)K=== = qÜÉ=ÖÉåÉê~ä=ëçäìíáçå=áë== f(í) = `NÉêNí + `OÉêOí + ^ ëáå(ωí − ϕ)I= ïÜÉêÉ== − o ± oO − Qi ` I== êNI O = Oi ^ = ωbM I== O  iωO − N  + oOωO  ` ϕ = ~êÅí~å iω − N  I==  o o`ω  `NI= `O =~êÉ=Åçåëí~åíë=ÇÉéÉåÇáåÖ=çå=áåáíá~ä=ÅçåÇáíáçåëK= = = = 10.3. Some Partial Differential Equations = 1181. qÜÉ=i~éä~ÅÉ=bèì~íáçå= ∂Oì + ∂Oì = M = ∂ñ O ∂ó O ~ééäáÉë=íç=éçíÉåíá~ä=ÉåÉêÖó=ÑìåÅíáçå= ì(ñI ó)==Ñçê==~==ÅçåëÉê- î~íáîÉ= ÑçêÅÉ= ÑáÉäÇ= áå= íÜÉ= ñó-éä~åÉK= m~êíá~ä= ÇáÑÑÉêÉåíá~ä= Éèì~- íáçåë=çÑ=íÜáë=íóéÉ=~êÉ=Å~ääÉÇ=ÉääáéíáÅK== = 1182. qÜÉ=eÉ~í=bèì~íáçå= ∂Oì + ∂Oì = ∂ì = ∂ñ O ∂ó O ∂í 302

CHAPTER 10. DIFFERENTIAL EQUATIONS ~ééäáÉë= íç= íÜÉ= íÉãéÉê~íìêÉ= ÇáëíêáÄìíáçå= ì(ñI ó)= áå= íÜÉ= ñó- éä~åÉ=ïÜÉå=ÜÉ~í=áë=~ääçïÉÇ=íç=Ñäçï=Ñêçã=ï~êã=~êÉ~ë=íç=Åççä= çåÉëK=qÜÉ=Éèì~íáçåë=çÑ=íÜáë=íóéÉ=~êÉ=Å~ääÉÇ=é~ê~ÄçäáÅK== = 1183. qÜÉ=t~îÉ=bèì~íáçå= ∂Oì + ∂Oì = ∂Oì = ∂ñ O ∂ó O ∂íO ~ééäáÉë=íç=íÜÉ=Çáëéä~ÅÉãÉåí= ì(ñIó)=çÑ=îáÄê~íáåÖ=ãÉãÄê~åÉë= ~åÇ= çíÜÉê= ï~îÉ= ÑìåÅíáçåëK= qÜÉ= Éèì~íáçåë= çÑ= íÜáë= íóéÉ= ~êÉ= Å~ääÉÇ=ÜóéÉêÄçäáÅK== = = 303

Chapter 11 Series = = = = 11.1 Arithmetic Series = fåáíá~ä=íÉêãW= ~N = kíÜ=íÉêãW= ~å = aáÑÑÉêÉåÅÉ=ÄÉíïÉÉå=ëìÅÅÉëëáîÉ=íÉêãëW=Ç= kìãÄÉê=çÑ=íÉêãë=áå=íÜÉ=ëÉêáÉëW=å= pìã=çÑ=íÜÉ=Ñáêëí=å=íÉêãëW= på = = = 1184. ~å = ~å−N + Ç = ~å−O + OÇ = K = ~N + (å −N)Ç = = 1185. ~N + ~å = ~O + ~å−N = K = ~á + ~å+N−á = = 1186. ~á = ~ á −N + ~á+N = O = (å − N)Ç 1187. på = ~N + ~å ⋅å = O~N + O ⋅å = O = = = = = 304

CHAPTER 11. SERIES 11.2 Geometric Series = fåáíá~ä=íÉêãW= ~N = kíÜ=íÉêãW= ~å = `çããçå=ê~íáçW=è= kìãÄÉê=çÑ=íÉêãë=áå=íÜÉ=ëÉêáÉëW=å= pìã=çÑ=íÜÉ=Ñáêëí=å=íÉêãëW= på = pìã=íç=áåÑáåáíóW=p= = = 1188. ~å = è~å−N = ~Nèå−N = = 1189. ~N ⋅ ~å = ~O ⋅ ~å−N = K = ~á ⋅ ~å+N−á = = 1190. ~á = ~á−N ⋅ ~á+N = = ( )1191. på = ~åè − ~N = ~N èå −N = è −N è −N = 1192. p = äáã på = ~N = N− è å→∞ cçê= è < NI=íÜÉ=ëìã=p=ÅçåîÉêÖÉë=~ë= å → ∞ K= = = = 11.3 Some Finite Series = kìãÄÉê=çÑ=íÉêãë=áå=íÜÉ=ëÉêáÉëW=å= = = 305

CHAPTER 11. SERIES 1193. N+ O + P + K + å = å(å + N) = O = 1194. O + Q + S +K+ Oå = å(å + N)= = 1195. N+ P + R +K+ (Oå −N) = åO = = å(Oâ + å −N) 1196. â + (â +N)+ (â + O)+K+ (â + å −N) = O = = å(å N)(Oå N) 1197. NO + OO + PO +K + åO = + S + = = 1198. NP + OP + PP +K+ åP = å(å + N)O =  O  = ( )1199. NO + PO + RO +K+ (Oå −N)O = å QåO −N = P = ( )1200. NP + PP + RP +K+ (Oå −N)P = åO OåO −N = = 1201. N+ N + N + N +K+ N +K= O= O Q U Oå = 1202. N + N + N +K+ N +K=N= N⋅ O O⋅P P⋅Q å(å +N) = 1203. N+ N + N + N + K+ (å N N)> + K = É = N> O> P> − = = = 306

CHAPTER 11. SERIES 11.4 Infinite Series = pÉèìÉåÅÉW= {~ å }= cáêëí=íÉêãW= ~N = kíÜ=íÉêãW= ~å = = = 1204. fåÑáåáíÉ=pÉêáÉë= ∞ ∑~å = ~N + ~O +K+ ~å +K= å=N = 1205. kíÜ=m~êíá~ä=pìã= å ∑på = ~å = ~N + ~O +K + ~å = å=N = 1206. `çåîÉêÖÉåÅÉ=çÑ=fåÑáåáíÉ=pÉêáÉë= ∞ ∑~å = i I=áÑ= äáã på = i = å=N å→∞ = 1207. kíÜ=qÉêã=qÉëí= ∞ ∑• fÑ=íÜÉ=ëÉêáÉë= ~å áë=ÅçåîÉêÖÉåíI=íÜÉå= äáã ~å = M K== å=N å→∞ • fÑ= äáã ~ å ≠ M I=íÜÉå=íÜÉ=ëÉêáÉë=áë=ÇáîÉêÖÉåíK= å→∞ = = = 11.5 Properties of Convergent Series = ∞∞ `çåîÉêÖÉåí=pÉêáÉëW= ∑~å = ^ I= ∑Äå = _ = å=N å=N oÉ~ä=åìãÄÉêW=Å= 307

CHAPTER 11. SERIES ∞ ∞∞ 1208. ∑ (~å + Äå ) = ∑ ~å + ∑ Äå = ^ + _ = å=N å=N å=N = ∞∞ 1209. ∑ Å~å = Å∑ ~å = Å^ K== å=N å=N = = = 11.6 Convergence Tests = 1210. qÜÉ=`çãé~êáëçå=qÉëí= ∞∞ ∑ ∑iÉí= ~å =~åÇ= Äå =ÄÉ=ëÉêáÉë=ëìÅÜ=íÜ~í= M < ~å ≤ Äå =Ñçê=~ää=åK== å=N å=N ∞∞ ∑ ∑• fÑ= Äå áë=ÅçåîÉêÖÉåí=íÜÉå= ~å áë=~äëç=ÅçåîÉêÖÉåíK== å=N å=N ∞∞ ∑ ∑• fÑ= ~å áë=ÇáîÉêÖÉåí=íÜÉå= Äå áë=~äëç=ÇáîÉêÖÉåíK= å=N å=N = 1211. qÜÉ=iáãáí=`çãé~êáëçå=qÉëí= ∞∞ ∑ ∑iÉí= ~å =~åÇ= Äå =ÄÉ=ëÉêáÉë=ëìÅÜ=íÜ~í= ~å =~åÇ= Äå =~êÉ=éçëá- å=N å=N íáîÉ=Ñçê=~ää=åK== ∑ ∑• M< äáã ~å ∞ ∞ fÑ= Äå < ∞ = íÜÉå= = ~åÇ= ~êÉ= ÉáíÜÉê= ÄçíÜ= å→∞ ~å Äå å=N å=N ÅçåîÉêÖÉåí=çê=ÄçíÜ=ÇáîÉêÖÉåíK= ∑ ∑• ~å ∞ ∞ fÑ= äáã Äå = M =íÜÉå= ÅçåîÉêÖÉåí=áãéäáÉë=íÜ~í= =áë= å→∞ Äå ~å å=N å=N ~äëç=ÅçåîÉêÖÉåíK= 308

CHAPTER 11. SERIES ∑ ∑• äáã ~å ∞ ∞ fÑ= Äå = ∞ =íÜÉå= ÇáîÉêÖÉåí=áãéäáÉë=íÜ~í= =áë= å→∞ Äå ~å å=N å=N ~äëç=ÇáîÉêÖÉåíK= = 1212. é-ëÉêáÉë= ∑é-ëÉêáÉë= ∞ N =ÅçåîÉêÖÉë=Ñçê= é > N=~åÇ=ÇáîÉêÖÉë=Ñçê= å=N åé M < é ≤ NK= = 1213. qÜÉ=fåíÉÖê~ä=qÉëí= iÉí= Ñ (ñ)= ÄÉ= ~= ÑìåÅíáçå= ïÜáÅÜ= áë= ÅçåíáåìçìëI= éçëáíáîÉI= ~åÇ= ÇÉÅêÉ~ëáåÖ=Ñçê=~ää= ñ ≥ NK=qÜÉ=ëÉêáÉë== ∞ ∑Ñ (å) = Ñ (N)+ Ñ (O)+ Ñ (P)+K+ Ñ (å)+K= å=N ∞ ÅçåîÉêÖÉë=áÑ= ∫ Ñ (ñ)Çñ ÅçåîÉêÖÉëI=~åÇ=ÇáîÉêÖÉë=áÑ= N å ∫ Ñ (ñ)Çñ → ∞ =~ë= å → ∞ K= N = 1214. qÜÉ=o~íáç=qÉëí= ∞ iÉí= ∑~å =ÄÉ=~=ëÉêáÉë=ïáíÜ=éçëáíáîÉ=íÉêãëK= å=N ∑• fÑ= ~ å +N ∞ äáã ~å < N =íÜÉå= å=N ~å áë=ÅçåîÉêÖÉåíK= å→∞ ∑• ~ å +N ∞ fÑ= äáã ~å > N=íÜÉå= å=N ~å =áë=ÇáîÉêÖÉåíK= å→∞ ∑• fÑ= äáã ~ å +N ∞ ~å = N= íÜÉå= = ã~ó= ÅçåîÉêÖÉ= çê= ÇáîÉêÖÉ= ~åÇ= å→∞ ~å å=N íÜÉ= ê~íáç= íÉëí= áë= áåÅçåÅäìëáîÉX= ëçãÉ= çíÜÉê= íÉëíë= ãìëí= ÄÉ= ìëÉÇK== = 309

CHAPTER 11. SERIES 1215. qÜÉ=oççí=qÉëí= ∞ iÉí= ∑~å =ÄÉ=~=ëÉêáÉë=ïáíÜ=éçëáíáîÉ=íÉêãëK= å=N ∞ ∑• fÑ= äáã å ~å < N=íÜÉå= ~å =áë=ÅçåîÉêÖÉåíK= å→∞ å=N ∞ ∑• fÑ= äáã å ~å > N=íÜÉå= ~å =áë=ÇáîÉêÖÉåíK= å→∞ å=N ∞ ∑• fÑ= äáã å ~å = N=íÜÉå= ~å = ã~ó= ÅçåîÉêÖÉ= çê= ÇáîÉêÖÉI= Äìí= å→∞ å=N åç=ÅçåÅäìëáçå=Å~å=ÄÉ=Çê~ïå=Ñêçã=íÜáë=íÉëíK= = = = 11.7 Alternating Series = 1216. qÜÉ=^äíÉêå~íáåÖ=pÉêáÉë=qÉëí=EiÉáÄåáò∞ë=qÜÉçêÉãF= = iÉí= {~ å }=ÄÉ=~=ëÉèìÉåÅÉ=çÑ=éçëáíáîÉ=åìãÄÉêë=ëìÅÜ=íÜ~í= ~å+N < ~å =Ñçê=~ää=åK= äáã ~ å = M K== å→∞ ∞∞ (−N)å ~å = ~åÇ= (− )N~å−N ∑ ∑qÜÉå= íÜÉ= ~äíÉêå~íáåÖ= ëÉêáÉë= å = å=N å=N ÄçíÜ=ÅçåîÉêÖÉK==== = 1217. ^ÄëçäìíÉ=`çåîÉêÖÉåÅÉ= ∞ ∑• ^= ëÉêáÉë= ~å = áë= ~ÄëçäìíÉäó= ÅçåîÉêÖÉåí= áÑ= íÜÉ= ëÉêáÉë= å=N ∞ ∑ ~å =áë=ÅçåîÉêÖÉåíK== å=N 310

CHAPTER 11. SERIES ∞ ∑• fÑ=íÜÉ=ëÉêáÉë= ~å áë=~ÄëçäìíÉäó=ÅçåîÉêÖÉåí=íÜÉå=áí=áë=Åçå- å=N îÉêÖÉåíK= = 1218. `çåÇáíáçå~ä=`çåîÉêÖÉåÅÉ= ∞ ^= ëÉêáÉë= ∑~å áë= ÅçåÇáíáçå~ääó= ÅçåîÉêÖÉåí= áÑ= íÜÉ= ëÉêáÉë= áë= å=N ÅçåîÉêÖÉåí=Äìí=áë=åçí=~ÄëçäìíÉäó=ÅçåîÉêÖÉåíK= = = = 11.8 Power Series = oÉ~ä=åìãÄÉêëW=ñI= ñM = ∞∞ ∑ ∑mçïÉê=ëÉêáÉëW= ~åñå I= ~å (ñ − ñM )å = å=M å=M tÜçäÉ=åìãÄÉêW=å= o~Çáìë=çÑ=`çåîÉêÖÉåÅÉW=o= = 1219. mçïÉê=pÉêáÉë=áå=ñ= ∞ ∑~åñå = ~M + ~Nñ + ~OñO +K+ ~åñå +K= å=M = 1220. mçïÉê=pÉêáÉë=áå= (ñ − ñM ) = ∞ ∑ ~å (ñ − ñ M )å = ~M + ~N (ñ − ñ M ) + ~O (ñ − ñ M )O + K + ~å (ñ − ñ M )å + K å=M = 1221. fåíÉêî~ä=çÑ=`çåîÉêÖÉåÅÉ=== qÜÉ=ëÉí=çÑ=íÜçëÉ=î~äìÉë=çÑ=ñ=Ñçê=ïÜáÅÜ=íÜÉ=ÑìåÅíáçå= ∞ Ñ (ñ) = ∑~å (ñ − ñM )å =áë=ÅçåîÉêÖÉåí=áë=Å~ääÉÇ==íÜÉ==áåíÉêî~ä=çÑ= å=M ÅçåîÉêÖÉåÅÉK= 311

CHAPTER 11. SERIES 1222. o~Çáìë=çÑ=`çåîÉêÖÉåÅÉ= fÑ=íÜÉ=áåíÉêî~ä=çÑ=ÅçåîÉêÖÉåÅÉ=áë== (ñM − oI ñM + o) ==Ñçê==ëçãÉ= o ≥ M I=íÜÉ=o=áë=Å~ääÉÇ==íÜÉ=ê~Çáìë=çÑ=ÅçåîÉêÖÉåÅÉK==fí=áë=ÖáîÉå= ~ë= o = äáã N =çê= o = äáã ~å K== ~å→∞ å ~å→∞ å å+N = = = 11.9 Differentiation and Integration of Power Series = `çåíáåìçìë=ÑìåÅíáçåW= Ñ (ñ)= ∞ ∑mçïÉê=ëÉêáÉëW= ~åñå = å=M tÜçäÉ=åìãÄÉêW=å= o~Çáìë=çÑ=`çåîÉêÖÉåÅÉW=o= = = 1223. aáÑÑÉêÉåíá~íáçå=çÑ=mçïÉê=pÉêáÉë= ∞ ∑iÉí= Ñ (ñ) = ~åñå = ~M + ~Nñ + ~OñO +K=Ñçê= ñ < o K== å=M qÜÉåI= = Ñçê= ñ < o I= Ñ (ñ)= áë= ÅçåíáåìçìëI= íÜÉ= ÇÉêáî~íáîÉ= Ñ′(ñ)= Éñáëíë=~åÇ= Ñ ′(ñ ) = Ç ~M + Ç ~Nñ + Ç ~Oñ O + K= Çñ Çñ Çñ ∞ ∑= ~N + O~Oñ + P~PñO + K = å~åñå−N K= å=N = = = 312

CHAPTER 11. SERIES 1224. fåíÉÖê~íáçå=çÑ=mçïÉê=pÉêáÉë= ∞ ∑iÉí= Ñ (ñ) = ~åñå = ~M + ~Nñ + ~OñO +K=Ñçê= ñ < o K== å=M qÜÉåI==Ñçê= ñ < o I=íÜÉ=áåÇÉÑáåáíÉ=áåíÉÖê~ä= ∫ Ñ (ñ)Çñ Éñáëíë=~åÇ== ∫ Ñ (ñ)Çñ = ∫ ~MÇñ + ∫ ~NñÇñ + ∫ ~OñOÇñ +K = ∑= ~Mñ ñO ñP ∞ ñ å+N + ` K= + ~N O + ~O P +K= å+N ~å å=M = = = 11.10 Taylor and Maclaurin Series = tÜçäÉ=åìãÄÉêW=å= aáÑÑÉêÉåíá~ÄäÉ=ÑìåÅíáçåW= Ñ (ñ)= oÉã~áåÇÉê=íÉêãW= oå = = = 1225. q~óäçê=pÉêáÉë= Ñ(ñ) = ∞ Ñ (å ) (~ ) (ñ − ~)å = Ñ (~)+ Ñ ′(~)(ñ − ~)+ Ñ ′′(~)(ñ − ~)O +K ∑ å> O> å=M == + Ñ (å)(~)(ñ − ~)å + oå K== å> = 1226. qÜÉ=oÉã~áåÇÉê=^ÑíÉê=åHN=qÉêãë=áë=ÖáîÉå=Äó== oå = Ñ (å+N)(ξ)(ñ − )~ å+N I=== ~ < ξ < ñ K= (å +N)> = 1227. j~Åä~ìêáå=pÉêáÉë= 313

CHAPTER 11. SERIES ∑Ñ(ñ) = ∞ Ñ (å) (M) ñ å = Ñ (M)+ Ñ ′(M)ñ + Ñ ′′(M)ñO +K+ Ñ (å)(M)ñ å + oå å=M å> O> å> = = = = 11.11 Power Series Expansions for Some Functions = tÜçäÉ=åìãÄÉêW=å= oÉ~ä=åìãÄÉêW=ñ= = = 1228. Éñ =N+ ñ + ñO + ñP +K+ ñå +K= O> P> å> = 1229. ~ñ =N+ ñ äå ~ + (ñ äå ~)O + (ñ äå~)P +K+ (ñ äå ~)å +K= N> O> P> å> = 1230. äå(N+ ñ)= ñ − ñO + ñP − ñQ +K+ (− N)å ñå+N ±KI= −N< ñ ≤ NK= O P Q å+N = 1231. äå N+ ñ = O ñ + ñP + ñR + ñT + K I= ñ < N K= N− ñ P R T = 1232. äå ñ =  ñ − N + N  ñ − N P + N  ñ − N R  I= ñ > M K= O ñ + N P  ñ + N  R  ñ + N  K   = 1233. Åçë ñ =N− ñO + ñQ − ñS +K+ (− N)å ñOå ±K= O> Q> S> (Oå)> = 314

CHAPTER 11. SERIES 1234. ëáå ñ = ñ − ñP + ñR − ñT +K+ ( )− N å ñOå+N ±K= P> R> T> (Oå +N)> = 1235. í~å ñ = ñ + ñP + Oñ R + NT ñ T + SOñ V + K I= ñ < π K= P NR PNR OUPR O = 1236. Åçí ñ = N −  ñ + ñP + Oñ R + Oñ T +K I= ñ < π K= ñ P QR VQR QTOR = 1237. ~êÅëáå ñ = ñ + ñP + N⋅Pñ R +K+ N⋅ P ⋅ RK(Oå − N)ñ Oå+N + K I= O⋅P O⋅Q⋅R O ⋅ Q ⋅ SK(Oå)(Oå + N) ñ < N K= = 1238. ~êÅÅçë ñ = π −  ñ + ñP + N⋅ Pñ R +K+ N⋅ P ⋅ RK(Oå − N)ñ Oå+N + KI O O⋅P O⋅Q⋅R O ⋅ Q ⋅ SK(Oå)(Oå + N) ñ < N K= = 1239. ~êÅí~å ñ = ñ − ñP + ñR − ñT +K+ ( )− N å ñ Oå+N ±K I= ñ ≤ N K= P R T Oå + N = 1240. ÅçëÜ ñ =N+ ñO + ñQ + ñS +K+ ñ Oå +K= O> Q> S> (Oå)> = 1241. ëáåÜ ñ =ñ + ñP + ñR + ñT +K+ ñ Oå+N +K= P> R> T> (Oå + N)> = = = = = 315

CHAPTER 11. SERIES 11.12 Binomial Series = tÜçäÉ=åìãÄÉêëW=åI=ã= oÉ~ä=åìãÄÉêW=ñ= `çãÄáå~íáçåëW= å `ã = = = 1242. (N+ ñ)å = N+ å`Nñ + å`OñO + K + ã`åñã + K + ñå = = å(å −N)K[å − (ã −N)] I= 1243. å `ã = ã> ñ < N K= = 1244. N =N− ñ + ñO − ñP + K I= ñ < N K= N+ ñ = 1245. N =N+ ñ + ñO + ñP + K I= ñ < N K= N− ñ = 1246. N+ ñ =N+ ñ − ñO + N⋅ Pñ P − N⋅ P ⋅ Rñ Q + K I= ñ ≤ N K= O O⋅Q O⋅Q⋅S O⋅Q⋅S⋅U = 1247. P N+ ñ =N+ ñ − N⋅ OñO + N⋅ O⋅RñP − N⋅ O⋅R⋅UñQ +K I= ñ ≤ N K= P P⋅S P⋅S⋅V P⋅ S ⋅ V ⋅NO = = = 11.13 Fourier Series = fåíÉÖê~ÄäÉ=ÑìåÅíáçåW= Ñ (ñ)= cçìêáÉê=ÅçÉÑÑáÅáÉåíëW= ~M I= ~å I= Äå = tÜçäÉ=åìãÄÉêW=å== 316

CHAPTER 11. SERIES Ñ(ñ) = ~M ∞ ëáå åñ ) = ∑1248. O + Åçë åñ + Äå (~å å=N = ∫1249.~å= N π Çñ == π Ñ (ñ)Åçëåñ −π = ∫1250.Äå= N π Çñ == π Ñ (ñ)ëáååñ −π = = 317

Chapter 12 Probability = = = = 12.1 Permutations and Combinations = mÉêãìí~íáçåëW= å mã = `çãÄáå~íáçåëW= å `ã = tÜçäÉ=åìãÄÉêëW=åI=ã= = = 1251. c~Åíçêá~ä= å>= N⋅O⋅PK(å − O)(å −N)å = M>= N= = 1252. å må = å> = = 1253. å mã = (å å> = − ã)> = 1254. _áåçãá~ä=`çÉÑÑáÅáÉåí= å `ã = ãå  = å> ã)> = ã>(å − = 1255. å `ã =å `å−ã = = 1256. å `ã +å `ã+N =å+N `ã+N = = 318

CHAPTER 12. PROBABILITY 1257. å `M +å `N +å `O + K +å `å = Oå = = 1258. m~ëÅ~ä∞ë=qêá~åÖäÉ= = oçï=M= = = = = = = N= = = = = = = oçï=N= = = = = = N= = N= = = = = = oçï=O= = = = = N= = O= = N= = = = = oçï=P= = = = N= = P= = P= = N= = = = oçï=Q= = = N= = Q= = S= = Q= = N= = = oçï=R= = N= = R= = NM= = NM= = R= = N= = oçï=S= N= = S= = NR= = OM= = NR= = S= = N= = = = 12.2 Probability Formulas = bîÉåíëW=^I=_= mêçÄ~ÄáäáíóW=m= o~åÇçã=î~êá~ÄäÉëW=uI=vI=w= s~äìÉë=çÑ=ê~åÇçã=î~êá~ÄäÉëW=ñI=óI=ò= bñéÉÅíÉÇ=î~äìÉ=çÑ=uW= µ = ^åó=éçëáíáîÉ=êÉ~ä=åìãÄÉêW= ε == pí~åÇ~êÇ=ÇÉîá~íáçåW= σ = s~êá~åÅÉW= σO = aÉåëáíó=ÑìåÅíáçåëW= Ñ (ñ)I= Ñ (í)= = = 1259. mêçÄ~Äáäáíó=çÑ=~å=bîÉåí= m(^) = ã I== å ïÜÉêÉ== ã=áë=íÜÉ=åìãÄÉê=çÑ=éçëëáÄäÉ=éçëáíáîÉ=çìíÅçãÉëI== å=áë=íÜÉ=íçí~ä=åìãÄÉê=çÑ=éçëëáÄäÉ=çìíÅçãÉëK= = 319

CHAPTER 12. PROBABILITY 1260. o~åÖÉ=çÑ=mêçÄ~Äáäáíó=s~äìÉë= M ≤ m(^) ≤ N= = 1261. `Éêí~áå=bîÉåí= m(^) = N= = 1262. fãéçëëáÄäÉ=bîÉåí= m(^) = M = = 1263. `çãéäÉãÉåí= m(^)= N− m(^) = = 1264. fåÇÉéÉåÇÉåí=bîÉåíë= m(^L_) = m(^)I== m(_L ^) = m(_) = = 1265. ^ÇÇáíáçå=oìäÉ=Ñçê=fåÇÉéÉåÇÉåí=bîÉåíë= m(^ ∪ _) = m(^) + m(_)= = 1266. jìäíáéäáÅ~íáçå=oìäÉ=Ñçê=fåÇÉéÉåÇÉåí=bîÉåíë= m(^ ∩ _) = m(^)⋅ m(_)= = 1267. dÉåÉê~ä=^ÇÇáíáçå=oìäÉ= m(^ ∪ _) = m(^)+ m(_)− m(^ ∩ _)I== ïÜÉêÉ== ^ ∪ _ =áë=íÜÉ=ìåáçå=çÑ=ÉîÉåíë=^=~åÇ=_I== ^ ∩ _ =áë=íÜÉ=áåíÉêëÉÅíáçå=çÑ=ÉîÉåíë=^=~åÇ=_K= = 1268. `çåÇáíáçå~ä=mêçÄ~Äáäáíó= m(^ ∩ _) m(^ L _) = m(_) = = 1269. m(^ ∩ _) = m(_)⋅ m(^ L _) = m(^)⋅ m(_ L ^) = 320

CHAPTER 12. PROBABILITY 1270. i~ï=çÑ=qçí~ä=mêçÄ~Äáäáíó= ã m(^) = ∑m(_á )m(^ L_á ) I== á=N ïÜÉêÉ= _á =áë=~=ëÉèìÉåÅÉ=çÑ=ãìíì~ääó=ÉñÅäìëáîÉ=ÉîÉåíëK== = 1271. _~óÉë∞=qÜÉçêÉã= m(^ L _)⋅ m(_) m(_ L ^) = m(^) = = 1272. _~óÉë∞=cçêãìä~= m(_á )⋅ m(^ L _á ) m(_á L ^) = I== ã ∑ m(_á )⋅ m(^ L _á ) â =N ïÜÉêÉ== _á =áë=~=ëÉí=çÑ=ãìíì~ääó=ÉñÅäìëáîÉ=ÉîÉåíë=EÜóéçíÜÉëÉëFI= ^ =áë=íÜÉ=Ñáå~ä=ÉîÉåíI== m(_á )=~êÉ=íÜÉ=éêáçê=éêçÄ~ÄáäáíáÉëI= m(_á L ^) =~êÉ=íÜÉ=éçëíÉêáçê=éêçÄ~ÄáäáíáÉëK= = 1273. i~ï=çÑ=i~êÖÉ=kìãÄÉêë= m på − µ ≥ ε → M =~ë= å → ∞ I== å  m på − µ < ε → N =~ë= å → ∞ I== å  ïÜÉêÉ== på =áë=íÜÉ=ëìã=çÑ=ê~åÇçã=î~êá~ÄäÉëI= å =áë=íÜÉ=åìãÄÉê=çÑ=éçëëáÄäÉ=çìíÅçãÉëK= = 1274. `ÜÉÄóëÜÉî=fåÉèì~äáíó= s(u) m( u − µ ≥ ε)≤ I== εO ïÜÉêÉ= s(u) =áë=íÜÉ=î~êá~åÅÉ=çÑ=uK= 321

CHAPTER 12. PROBABILITY 1275. kçêã~ä=aÉåëáíó=cìåÅíáçå= N ( ñ −µ ) O ( )ϕ ñ = É− OσO I== σ Oπ ïÜÉêÉ=ñ=áë=~=é~êíáÅìä~ê=çìíÅçãÉK= = 1276. pí~åÇ~êÇ=kçêã~ä=aÉåëáíó=cìåÅíáçå= ϕ(ò) = N É−ò O = O Oπ ^îÉê~ÖÉ=î~äìÉ= µ = M I=ÇÉîá~íáçå= σ = NK= = ===== = = Figure 210. = 1277. pí~åÇ~êÇ=w=s~äìÉ= w= u−µ = σ = 1278. `ìãìä~íáîÉ=kçêã~ä=aáëíêáÄìíáçå=cìåÅíáçå= ( ) ∫c ñ = N ñ (í−µ) O É − OσO Çí I== σ Oπ −∞ 322

CHAPTER 12. PROBABILITY ïÜÉêÉ== ñ=áë=~=é~êíáÅìä~ê=çìíÅçãÉI== í =áë=~=î~êá~ÄäÉ=çÑ=áåíÉÖê~íáçåK= = 1279. m(α < u < β) = c α−µ  − c β −µ  I=  σ   σ  ïÜÉêÉ= u=áë=åçêã~ääó=ÇáëíêáÄìíÉÇ=ê~åÇçã=î~êá~ÄäÉI= c=áë=Åìãìä~íáîÉ=åçêã~ä=ÇáëíêáÄìíáçå=ÑìåÅíáçåI== m(α < u < β) =áë=áåíÉêî~ä=éêçÄ~ÄáäáíóK= = 1280. m( u − µ < ε)= Oc ε  I==  σ  ïÜÉêÉ== u=áë=åçêã~ääó=ÇáëíêáÄìíÉÇ=ê~åÇçã=î~êá~ÄäÉI= c=áë=Åìãìä~íáîÉ=åçêã~ä=ÇáëíêáÄìíáçå=ÑìåÅíáçåK= = 1281. `ìãìä~íáîÉ=aáëíêáÄìíáçå=cìåÅíáçå= ñ c(ñ) = m(u < ñ) = ∫ Ñ (í)Çí I== −∞ ïÜÉêÉ=í=áë=~=î~êá~ÄäÉ=çÑ=áåíÉÖê~íáçåK= = 1282. _Éêåçìääá=qêá~äë=mêçÅÉëë= µ = åé = I= σO = åéè I== ïÜÉêÉ== å =áë=~=ëÉèìÉåÅÉ=çÑ=ÉñéÉêáãÉåíëI== é =áë=íÜÉ=éêçÄ~Äáäáíó=çÑ=ëìÅÅÉëë=çÑ=É~ÅÜ=ÉñéÉêáãÉåíëI= è =áë=íÜÉ=éêçÄ~Äáäáíó=çÑ=Ñ~áäìêÉI= è = N− é K= = 1283. _áåçãá~ä=aáëíêáÄìíáçå=cìåÅíáçå======= Ä(åI éI è ) =  å  éâ è å−â I== â 323

CHAPTER 12. PROBABILITY µ = åé I= σO = åéè I= Ñ (ñ) = (è + éÉñ )å I== ïÜÉêÉ== å=áë=íÜÉ=åìãÄÉê=çÑ=íêá~äë=çÑ=ëÉäÉÅíáçåëI= é=áë=íÜÉ=éêçÄ~Äáäáíó=çÑ=ëìÅÅÉëëI= è=áë=íÜÉ=éêçÄ~Äáäáíó=çÑ=Ñ~áäìêÉI= è = N− é K= = 1284. dÉçãÉíêáÅ=aáëíêáÄìíáçå= m(q = à) = èà−Né I== µ = N I= σO = è I== é éO ïÜÉêÉ== q=áë=íÜÉ=Ñáêëí=ëìÅÅÉëëÑìä=ÉîÉåí=áë=íÜÉ=ëÉêáÉëI= à=áë=íÜÉ=ÉîÉåí=åìãÄÉêI= é=áë=íÜÉ=éêçÄ~Äáäáíó=íÜ~í=~åó=çåÉ=ÉîÉåí=áë=ëìÅÅÉëëÑìäI== è=áë=íÜÉ=éêçÄ~Äáäáíó=çÑ=Ñ~áäìêÉI= è = N− é K= = 1285. mçáëëçå=aáëíêáÄìíáçå= m(u = â) ≈ λâ É−λ I= λ = åé I== â> µ = λ I= σO = λ I== ïÜÉêÉ== λ =áë=íÜÉ=ê~íÉ=çÑ=çÅÅìêêÉåÅÉI= â=áë=íÜÉ=åìãÄÉê=çÑ=éçëáíáîÉ=çìíÅçãÉëK= = 1286. aÉåëáíó=cìåÅíáçå== Ä m(~ ≤ u ≤ Ä) = ∫ Ñ (ñ)Çñ == ~ = 1287. `çåíáåìçìë=råáÑçêã=aÉåëáíó= Ñ = N I= µ = ~ + Ä I== − O Ä ~ 324

CHAPTER 12. PROBABILITY ïÜÉêÉ=Ñ=áë=íÜÉ=ÇÉåëáíó=ÑìåÅíáçåK= = 1288. bñéçåÉåíá~ä=aÉåëáíó=cìåÅíáçå= Ñ (í) = λÉ−λí I= µ = λ I= σO = λO = ïÜÉêÉ=í=áë=íáãÉI= λ =áë=íÜÉ=Ñ~áäìêÉ=ê~íÉK= = 1289. bñéçåÉåíá~ä=aáëíêáÄìíáçå=cìåÅíáçå= c(í) = N− É−λí I== ïÜÉêÉ=í=áë=íáãÉI= λ =áë=íÜÉ=Ñ~áäìêÉ=ê~íÉK= = 1290. bñéÉÅíÉÇ=s~äìÉ=çÑ=aáëÅêÉíÉ=o~åÇçã=s~êá~ÄäÉë= å µ = b(u) = ∑ ñáéá I== á=N ïÜÉêÉ= ñ á =áë=~=é~êíáÅìä~ê=çìíÅçãÉI= éá =áë=áíë=éêçÄ~ÄáäáíóK= = 1291. bñéÉÅíÉÇ=s~äìÉ=çÑ=`çåíáåìçìë=o~åÇçã=s~êá~ÄäÉë= ∞ µ = b(u) = ∫ ñÑ(ñ)Çñ == −∞ = 1292. mêçéÉêíáÉë=çÑ=bñéÉÅí~íáçåë= b(u + v) = b(u)+ b(v) I== b(u − v) = b(u)− b(v)I= b(Åu) = Åb(u) I== b(uv) = b(u)⋅ b(v) I== ïÜÉêÉ=Å=áë=~=Åçåëí~åíK= = ( )1293. b uO = s(u)+ µO I== ïÜÉêÉ== µ = b(u)=áë=íÜÉ=ÉñéÉÅíÉÇ=î~äìÉI=== s(u) =áë=íÜÉ=î~êá~åÅÉK= = = = 325

CHAPTER 12. PROBABILITY 1294. j~êâçî=fåÉèì~äáíó= m(u > â)≤ b(u) I== â ïÜÉêÉ=â=áë=ëçãÉ=Åçåëí~åíK= = 1295. s~êá~åÅÉ=çÑ=aáëÅêÉíÉ=o~åÇçã=s~êá~ÄäÉë= [ ]å σO = s(u) = b (u − µ)O ∑= (ñá − µ)O éá I== á=N ïÜÉêÉ== ñ á =áë=~=é~êíáÅìä~ê=çìíÅçãÉI= éá =áë=áíë=éêçÄ~ÄáäáíóK= = 1296. s~êá~åÅÉ=çÑ=`çåíáåìçìë=o~åÇçã=s~êá~ÄäÉë= [ ]∞ σO = s(u) = b (u − µ)O = ∫ (ñ − µ)OÑ (ñ)Çñ == −∞ = 1297. mêçéÉêíáÉë=çÑ=s~êá~åÅÉ= s(u + v) = s(u)+ s(v) I== s(u − v) = s(u)+ s(v) I= s(u + Å) = s(u)I= s(Åu) = ÅOs(u) I= ïÜÉêÉ=Å=áë=~=Åçåëí~åíK= = 1298. pí~åÇ~êÇ=aÉîá~íáçå= [ ]a(u) = s(u) = b (u − µ)O = = 1299. `çî~êá~åÅÉ= Åçî(uIv) = b[(u − µ(u))(v − µ(v))]= b(uv)− µ(u)µ(v)I== ïÜÉêÉ== u =áë=ê~åÇçã=î~êá~ÄäÉI== s(u) =áë=íÜÉ=î~êá~åÅÉ=çÑ=uI== µ =áë=íÜÉ=ÉñéÉÅíÉÇ=î~äìÉ=çÑ=u=çê=vK= 326

CHAPTER 12. PROBABILITY = 1300. `çêêÉä~íáçå= Åçî(uI v ) ρ(uIv) = s(u)s(v ) I== ïÜÉêÉ== s(u) =áë=íÜÉ=î~êá~åÅÉ=çÑ=uI== s(v) =áë=íÜÉ=î~êá~åÅÉ=çÑ=vK= = = = = = = = = = = = = = = = = = = = = = = = = = = = = 327

= iççâ=Ñçê=çíÜÉê=Ü~åÇÄççâë=~åÇ=ëçäîÉÇ=éêçÄäÉã=ÖìáÇÉë=~í= ïïïKã~íÜ-ÉÄççâëKÅçãK= = = = =