Laplace_transform

Laplace_transform - 7KH /DSODFH 7UDQVIRUP 'HILQLWLRQ DQG...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview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•  7KH /DSODFH WUDQVIRUP RI I W  GHQRWHG E\ ) V RU / ^I W ` LV JLYHQ E\ / ^I W ` ) V ³ f  H  VW I W GW  3URYLGHG WKDW WKLV LPSURSHU LQWHJUDO H[LVWV LH WKDW WKH LQWHJUDO LV FRQYHUJHQW )RU IXQFWLRQV FRQWLQXRXV RQ > ’  WKH DERYH WUDQVIRUPDWLRQ LV RQHWRRQH 7KDW LV GLIIHUHQW FRQWLQXRXV IXQFWLRQV ZLOO KDYH GLIIHUHQW WUDQVIRUPV ³ / ^I W `   V  WKHQ ) V ([DPSOH /HW I W f  f ³ H  VW I W GW  V !    VW H V H  VW GW f  7KH LQWHJUDO LV GLYHUJHQW ZKHQHYHU V ”  +RZHYHU ZKHQ V !  LW FRQYHUJHV WR    H V   V  V ) V   V  ([DPSOH /HW I W W WKHQ ) V V !  >7KLV LV OHIW WR \RX DV DQ H[HUFLVH@ / ^I W `   VD H DW WKHQ ) V ([DPSOH /HW I W ³ f  H  VW H DW GW ³ f  V ! D H D  V W GW  DV W H DV f  7KH LQWHJUDO LV GLYHUJHQW ZKHQHYHU V ” D +RZHYHU ZKHQ V ! D LW FRQYHUJHV WR    H DV   DV  VD ) V  'HILQLWLRQ $ IXQFWLRQ I W LV FDOOHG SLHFHZLVH FRQWLQXRXV LI LW RQO\ KDV ILQLWHO\ PDQ\ RU QRQH ZKDWVRHYHU ± D FRQWLQXRXV IXQFWLRQ LV FRQVLGHUHG WR EH ³SLHFHZLVH FRQWLQXRXV´ GLVFRQWLQXLWLHV RQ DQ\ LQWHUYDO >D E@ DQG WKDW ERWK RQHVLGHG OLPLWV H[LVW DV W DSSURDFKHV HDFK RI WKRVH GLVFRQWLQXLW\ IURP ZLWKLQ WKH LQWHUYDO 7KH ODVW SDUW RI WKH GHILQLWLRQ PHDQV WKDW I FRXOG KDYH UHPRYDEOH DQGRU MXPS GLVFRQWLQXLWLHV RQO\ LW FDQQRW KDYH DQ\ LQILQLW\ GLVFRQWLQXLW\ 7KHRUHP 6XSSRVH WKDW  I LV SLHFHZLVH FRQWLQXRXV RQ WKH LQWHUYDO  ” W ” $ IRU DQ\ $ !   I W ” . H DW ZKHQ W • 0 IRU DQ\ UHDO FRQVWDQW D DQG DQ\ SRVLWLYH FRQVWDQWV . DQG 0 7KLV PHDQV WKDW I LV ³RI H[SRQHQWLDO RUGHU´ LH LWV UDWH RI JURZWK LV QR IDVWHU WKDQ WKDW RI H[SRQHQWLDO IXQFWLRQV 7KHQ WKH /DSODFH WUDQVIRUP ) V / ^I W ` H[LVWV IRU V ! D 1RWH 7KH DERYH WKHRUHP JLYHV D VXIILFLHQW FRQGLWLRQ IRU WKH H[LVWHQFH RI /DSODFH WUDQVIRUPV ,W LV QRW D QHFHVVDU\ FRQGLWLRQ $ IXQFWLRQ GRHV QRW QHHG WR VDWLVI\ WKH WZR FRQGLWLRQV LQ RUGHU WR KDYH D /DSODFH WUDQVIRUP ([DPSOHV RI VXFK IXQFWLRQV WKDW QHYHUWKHOHVV KDYH /DSODFH WUDQVIRUPV DUH ORJDULWKPLF IXQFWLRQV DQG WKH XQLW LPSXOVH IXQFWLRQ 6RPH SURSHUWLHV RI WKH /DSODFH 7UDQVIRUP  / ^`   / ^I W r J W `  / ^F I W ` / ^I W ` r / ^J W ` F / ^I W ` IRU DQ\ FRQVWDQW F 3URSHUWLHV  DQG  WRJHWKHU PHDQV WKDW WKH /DSODFH WUDQVIRUP LV OLQHDU  >7KH GHULYDWLYH RI /DSODFH WUDQVIRUPV@ / ^ íW I W `  ([DPSOH / ^W ` ) V RU HTXLYDOHQWO\ / ^W I W ` í / ^W`  G GV V    V í) V  V ,Q JHQHUDO WKH GHULYDWLYHV RI /DSODFH WUDQVIRUPV VDWLVI\ / ^ íW QI W ` ) Q V RU HTXLYDOHQWO\ / ^W Q I W ` í Q ) Q V :DUQLQJ 7KH /DSODFH WUDQVIRUP ZKLOH D OLQHDU RSHUDWLRQ LV QRW PXOWLSOLFDWLYH 7KDW LV LQ JHQHUDO / ^I W J W `  / ^I W ` / ^J W ` DW ([HUFLVH D 8VH SURSHUW\  DERYH DQG WKH IDFW WKDW / ^ H ` DW WR GHGXFH WKDW / ^W H `   VD   DW   E :KDW ZLOO / ^W H ` EH" V  D 6ROXWLRQ RI ,QLWLDO 9DOXH 3UREOHPV :H QRZ VKDOO PHHW ³WKH QHZ 6\VWHP´ KRZ WKH /DSODFH WUDQVIRUPV FDQ EH XVHG WR VROYH OLQHDU GLIIHUHQWLDO HTXDWLRQV DOJHEUDLFDOO\ 7KHRUHP >/DSODFH WUDQVIRUP RI GHULYDWLYHV@ 6XSSRVH I LV RI H[SRQHQWLDO RUGHU DQG WKDW I LV FRQWLQXRXV DQG I LV SLHFHZLVH FRQWLQXRXV RQ DQ\ LQWHUYDO  ” W ” $ 7KHQ / ^I W ` V / ^I W ` í I  $SSO\LQJ WKH WKHRUHP PXOWLSOH WLPHV \LHOGV / ^I W ` V / ^I W ` í V I  í I   / ^I W ` V / ^I W ` í V I  í V I  í I     / ^I Q W ` V Q / ^I W ` í V Q í  I  í V Q í  I  í « í V  I Q í  í V I Q í  í I Q í   7KLV LV DQ H[WUHPHO\ XVHIXO DVSHFW RI WKH /DSODFH WUDQVIRUP WKDW LW FKDQJHV GLIIHUHQWLDWLRQ ZLWK UHVSHFW WR W LQWR PXOWLSOLFDWLRQ E\ V DQG DV VHHQ D OLWWOH HDUOLHU GLIIHUHQWLDWLRQ ZLWK UHVSHFW WR V LQWR PXOWLSOLFDWLRQ E\ í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` RI V < V LQ WHUPV  )LQG WKH LQYHUVH WUDQVIRUP RI < V  2U UDWKHU ILQG D IXQFWLRQ \ W ZKRVH /DSODFH WUDQVIRUP PDWFKHV WKH H[SUHVVLRQ RI < V  7KLV LQYHUVH WUDQVIRUP \ W  LV WKH VROXWLRQ RI WKH JLYHQ GLIIHUHQWLDO HTXDWLRQ 7KH QLFH WKLQJ LV WKDW WKH VDPH VWHS SURFHGXUH ZRUNV ZKHWKHU RU QRW WKH GLIIHUHQWLDO HTXDWLRQ LV KRPRJHQHRXV RU QRQKRPRJHQHRXV 7KH ILUVW WZR VWHSV LQ WKH SURFHGXUH DUH UDWKHU PHFKDQLFDO 7KH ODVW VWHS LV WKH KHDUW RI WKH SURFHVV DQG LW ZLOO WDNH VRPH SUDFWLFH /HW¶V JHW VWDUWHG \ í \  \ ([DPSOH  \   \  í >6WHS @ 7UDQVIRUP ERWK VLGHV / ^\ í \  \` / ^`  V / ^\` í V \  í \  í  V/ ^\` í \    / ^\` >6WHS @ 6LPSOLI\ WR ILQG < V / ^ \`  V / ^\` í V í í í  V/ ^\` í   / ^\` V í  V   / ^\` í V   V í  V   / ^\` / ^ \`   Ví V V   V   >6WHS @ )LQG WKH LQYHUVH WUDQVIRUP \ W 8VH SDUWLDO IUDFWLRQV WR VLPSOLI\ / ^ \` V V   V   V V   V   V  D E  V  V   D V   E V    V   V   V   V   D V    E V   D  E V   D  E  (TXDWLQJ WKH FRUUHVSRQGLQJ FRHIILFLHQWV  DE í íD í E +HQFH / ^ \` D E  í   V    V   V   V   V   7KH ODVW H[SUHVVLRQ FRUUHVSRQGV WR WKH /DSODFH WUDQVIRUP RI H W í H W 7KHUHIRUH LW PXVW EH WKDW \ W H W í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íW \  í >6WHS @ 7UDQVIRUP ERWK VLGHV / ^\  \` / ^W HíW` V/ ^\` í \   / ^\` >6WHS @ 6LPSOLI\ WR ILQG < V / ^W H íW` / ^ \`  V    V/ ^\` í í  / ^\` V   / ^ \`   V   / ^\` / ^ \`  V     V     í V       V    V      V    V     V    V   V    >6WHS @ )LQG WKH LQYHUVH WUDQVIRUP \ W %\ SDUWLDO IUDFWLRQV / ^ \`  V    V   V    D E F   V    V    V     V    V   V    D E V   F V      V    V    V    D  EV  E  FV   FV  F V    í FV   E  F V  D  E  F V    F D E F í E  F í D   E   F / ^ \`  V    V   V      í    V    V    7KLV H[SUHVVLRQ FRUUHVSRQGV WR WKH /DSODFH WUDQVIRUP RI W Hí W í H íW 7KHUHIRUH \ W Q DW 1RWH / ^W H ` W Hí W í H íW Q V  D Q  ([DPSOH \ í \  \ H W  \   \   >6WHS @ 7UDQVIRUP ERWK VLGHV  V / ^\` í V \  í \  í  V/ ^\` í \   / ^\` >6WHS @ 6LPSOLI\ WR ILQG < V / ^ \`  V / ^\` í V í  í  V/ ^\` í   / ^\`  V í  V   / ^\` í V    V í  V   / ^\` / ^ \` / ^HW`   V í    V í   Ví V V    V   V   V   V   V      V V    V   V   V   V   >6WHS @ )LQG WKH LQYHUVH WUDQVIRUP \ W %\ SDUWLDO IUDFWLRQV / ^ \` V    V   V   V   V   7KHUHIRUH \ W        V  V   V  W  H  H  W  H  W    )RU WKH QH[W H[DPSOH ZH ZLOO QHHG WKH IROORZLQJ /DSODFH WUDQVIRUPV V V   E / ^FRV EW` / ^VLQ EW` E V   E  V!  V! DW VD V  D   E   V!D DW E V  D   E   V!D / ^H FRV EW` / ^H VLQ EW` 1RWH 7KH YDOXHV RI D DQG E LQ WKH ODVW WZR H[SUHVVLRQV¶ GHQRPLQDWRUV FDQ EH GHWHUPLQHG ZLWKRXW XVLQJ WKH PHWKRG RI FRPSOHWLQJ WKH VTXDUHV $Q\ LUUHGXFLEOH TXDGUDWLF SRO\QRPLDO V  %V  & FDQ DOZD\V EH ZULWWHQ LQ WKH UHTXLUHG IURP RI V í D   E E\ XVLQJ WKH TXDGUDWLF IRUPXOD WR ILQG QHFHVVDULO\ FRPSOH[YDOXHG V 7KH YDOXH D LV WKH UHDO SDUW RI V DQG WKH YDOXH E LV MXVW WKH DEVROXWH YDOXH RI WKH LPDJLQDU\ SDUW RI V 7KDW LV LI V r L WKHQ D DQG E  \ í \  \ ([DPSOH FRV W  \   \   >6WHS @ 7UDQVIRUP ERWK VLGHV V / ^\` í V \  í \  í  V/ ^\` í \    / ^\` >6WHS @ 6LPSOLI\ WR ILQG < V / ^ \` V / ^\` í V í  í  V/ ^\` í   / ^\` V í  V   / ^\` í V   V í  V   / ^\` / ^ \` / ^FRV W ` V  V   V  V   V Ví  V  V   V   V    V V    V    V   V   V     V V  V   V   V   V    V    V   >6WHS @ )LQG WKH LQYHUVH WUDQVIRUP \ W %\ SDUWLDO IUDFWLRQV / ^ \` ª V V   º V   V   V    V    V    V    « V    V   V   » ¬ ¼ ª V   V    º      « V   V   V  V   V  V   » ¬ ¼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­ ® ¯ WF WtF  F •  2IWHQ WKH XQLW VWHS IXQFWLRQ XF W LV DOVR GHQRWHG DV X W í F  +F W  RU + W í F  7KH VWHS FRXOG DOVR EH QHJDWLYH JRLQJ GRZQ  7KH FRPSOHPHQW IXQFWLRQ LV   XF W ­ ® ¯ WF WtF  F •  7KH /DSODFH WUDQVIRUP RI WKH XQLW VWHS IXQFWLRQ LV H  FV / ^XF W ` V V !   F• 1RWLFH WKDW ZKHQ F  X W KDV WKH VDPH /DSODFH WUDQVIRUP DV WKH FRQVWDQW IXQFWLRQ I W  :K\" 7KHUHIRUH IRU RXU SXUSRVH X W  .HHS LQ PLQG WKDW WKH /DSODFH WUDQVIRUPV PHWKRG RQO\ ZRUNV IRU W •  1RWH 7KH FDOFXODWLRQ RI / ^XF W ` JRHV DV IROORZ JLYHQ WKDW F •   / ^XF W ` ³ f  XF W H  VW GW    H  FV V H  FV V  ³ f F  ˜ H  VW GW V !     VW H V f F 7KH XQLW VWHS IXQFWLRQ LV PXFK PRUH XVHIXO WKDQ LW ILUVW DSSHDUV WR EH :KHQ LQ D SURGXFW ZLWK D VHFRQG IXQFWLRQ WKH XQLW VWHS IXQFWLRQ DFWV OLNH D VZLWFK WR WXUQ WKH RWKHU IXQFWLRQ RQ RU RII XF W I W   XF W I W ­ ® ¯ I W  ­ I W  ® ¯ WF WtF WF W t F DQ ³RQ´ VZLWFK DQ ³RII´ VZLWFK  %\ FRPELQLQJ WZR XQLW VWHS IXQFWLRQV ZH FDQ DOVR VHOHFWLYHO\ PDNH D IXQFWLRQ DSSHDUV RQO\ IRU D ILQLWH GXUDWLRQ WKHQ LW GLVDSSHDUV 7KDW LV WKH IXQFWLRQ LV VZLWFKHG ³RQ´ DW D WKHQ LV VZLWFKHG ³RII´ DW D ODWHU WLPH E X D W  XE W I W ­  ° ® I W  °  ¯ WD DdW E  W tE ZKHUH  ” D  E :H FRXOG WKLQN WKLV FRPELQDWLRQ DV DQ ³RQRII´ WRJJOH VZLWFK WKDW FRQWUROV WKH DSSHDUDQFH RI WKH VHFRQG IXQFWLRQ I W  %\ FDVFDGLQJ WKH DERYH W\SHV RI SURGXFWV ZH FDQ QRZ ZULWH DQ\ SLHFHZLVH GHILQHG IXQFWLRQ LQ D VXFFLQFW IRUP LQ WHUPV RI XQLW VWHS IXQFWLRQV 6XSSRVH ) W ­ I W  ° I W  ° ° ® I  W  °  ° ° I Q W  ¯ WD DdWE EdW F  WtG  7KHQ ZH FDQ UHZULWH ) W  VXFFLQFWO\ DV ) W  í XD W I W  XD W í XE W I W  XE W í XF W I W  «  XG W IQ W  ([DPSOH ) W ­ W    ° W ®H  W  ° FRV W  ¯ W d W  W t  7KHQ ) W  í X W W í   X W í X W HW  W  X W FRV W  7KH GLIIHUHQFH EHWZHHQ XF W I W DQG XF W I W í F ([DPSOH X  W VLQ W DQG X  W VLQ W í  )LJ *UDSK RI  X  W VLQ W )LJ *UDSK RI  X  W VLQ W í  7KH HIIHFWV RI WKH /DSODFH WUDQVIRUP RQ WUDQVODWLRQV $UJXDEO\ WKH PRVW LPSRUWDQW IRUPXOD IRU WKLV FODVV LW LV XVXDOO\ FDOOHG WKH 6HFRQG 7UDQVODWLRQ 7KHRUHP RI WKH /DSODFH WUDQVIRUP  7KHRUHP ,I ) V / ^I W ` DQG LI F LV DQ\ SRVLWLYH FRQVWDQW WKHQ HíFV / ^I W ` / ^XF W I W í F ` HíFV ) V  1RWH (TXLYDOHQWO\ HíFV / ^J W  F ` / ^XF W J W ` &RQYHUVHO\ LI I W / í ^) V ` WKHQ XF W I W í F / í ^HíFV ) V ` ([DPSOH )LQG WKH LQYHUVH WUDQVIRUP RI ) V 6LQFH ) V F /  DQG I W í ^) V ` H  V  V   H  V V    HíV / ^HíW` WKHUHIRUH HíW $SSO\ WKH DERYH WKHRUHP DQG ZH KDYH X W Hí W í   X W Hí W í   7KLV HTXLYDOHQW IRUPXOD LV PRUH H[SOLFLW DERXW ZKDW QHHGV WR EH GRQH ZKHQ WUDQVIRUPLQJ D SURGXFW W  F EHIRUH WUDQVIRUP WKH FRQWDLQLQJ D XQLW VWHS IXQFWLRQ ,W WHOOV \RX WR WUDQVODWH WKH IXQFWLRQ W IXQFWLRQ 5HPHPEHU ZKHQ WUDQVIRUPLQJ D SURGXFW FRQWDLQLQJ D VWHS IXQFWLRQ ³WUDQVODWH EHIRUH WUDQVIRUP´ ([DPSOH )LQG WKH /DSODFH WUDQVIRUPV RI X  W VLQ W DQG / ^X  W VLQ W ` V H SV     V  Hí V X  W VLQ W í   Hí / ^VLQ W   ` V / ^FRV W ` 5HFDOO WKH DGGLWLRQ IRUPXOD RI VLQH VLQ D r E VLQ D FRV E r FRV D VLQ E / ^X  W VLQ W í  `  H SV     V  Hí V ) V Hí V / ^VLQ W ` W ([DPSOH )LQG WKH /DSODFH WUDQVIRUP RI X W H  / ^X W HW` HíV / ^H W   ` H  V     V  HH  V V HíV / ^HW  ` HíV H/ ^HW`  ([DPSOH )LQG WKH /DSODFH WUDQVIRUP RI X W W  W    / ^X W W  W   ` HíV / ^ W      W    ` HíV / ^ W  W   W   `  · § H V ¨       ¸ V V¹ ©V HíV / ^W  W  ` §   · H V ¨     ¸ V V¹ ©V ([DPSOH )LQG WKH /DSODFH WUDQVIRUP RI ) W ­ W    ° W ®H  W  ° FRV W  ¯ W d W  W t  :H KDYH VHHQ HDUOLHU WKDW ) W  í X W W í   X W í X W HW  W  X W FRV W W í   X W HW  W í W    X W FRV W í HW í W  7UDQVIRUP WKH ODVW H[SUHVVLRQ DERYH DSSO\LQJ ZKHUH DSSURSULDWH WKH íFV IRUPXOD / ^XF W J W ` H / ^J W  F ` / ^) W ` / ^W í `  HíV / ^H W    W   í  W     `  HíV / ^FRV W   í H W   í W   ` / ^W í `  HíV / ^HW    W   í  W  W    `  HíV / ^FRV W   í HW   í W í ` / ^W í `  HíV / ^HHW í W í W í `  HíV / ^FRV W FRV  í VLQ W VLQ  í HHW í W í `      ·  V § H  H ¨      ¸ ¨ V V V V V V¸ © ¹ H  V §   · V H  ¨ FRV    VLQ      ¸ ¨ V  V  V  V V¸ © ¹ 7KHRUHP ,I ) V / ^I W ` DQG LI F LV DQ\ SRVLWLYH FRQVWDQW WKHQ / ^H FW I W ` &RQYHUVHO\ LI I W / í ) V í F  ^) V ` WKHQ H FW I W / í ^) V í F ` 7KHUHIRUH LQ WKH ZRUOG RI /DSODFH WUDQVIRUPV WUDQVODWLRQV EHFRPH PXOWLSOLFDWLRQV E\ H[SRQHQWLDO IXQFWLRQV 7KLV WKHRUHP LV XVXDOO\ FDOOHG WKH )LUVW 7UDQVODWLRQ 7KHRUHP V DQG / ^VLQ EW` V  E D DQG UHSODFH V E\ V í F V í D ([DPSOH %HFDXVH / ^FRV EW` WKHQ OHWWLQJ F VD V  D   E  DQG E V  D   E   DW / ^H FRV EW` / ^H DW VLQ EW` Q 6LPLODUO\ VLQFH / ^W `  Q V Q  WKHUHIRUH Q DW / ^W H ` Q V  D Q   E  V  E  'LIIHUHQWLDO (TXDWLRQV ZLWK 'LVFRQWLQXRXV )RUFLQJ )XQFWLRQV :H DUH QRZ UHDG\ WR WDFNOH OLQHDU GLIIHUHQWLDO HTXDWLRQV ZKRVH ULJKWKDQG VLGH LV SLHFHZLVH FRQWLQXRXV $V PHQWLRQHG EHIRUH WKH PHWKRG RI /DSODFH WUDQVIRUPV ZRUNV WKH VDPH ZD\ WR VROYH DOO W\SHV RI HTXDWLRQV 7KHUHIRUH WKH VDPH VWHSV VHHQ SUHYLRXVO\ DSSO\ KHUH DV ZHOO \  \ ([DPSOH ZKHUH ) W ) W  ­ ® ¯ 7KH UHTXLUH HTXDWLRQ LV \  \ \   \   W S W tS  X W  7UDQVIRUP WKH HTXDWLRQ DQG VLPSOLI\ ZH KDYH V / ^\` í V \  í \   / ^\` V / ^\` í   / ^\` / ^ \` / ^X W ` V  / ^\` í  H SV V H SV   V V    V   7KH VHFRQG SDUW FDQ EH LQYHUWHG GLUHFWO\ LQWR VLQ W  íV 7KH ILUVW SDUW FDQ EH LQYHUWHG E\ ILUVW VHWWLQJ DVLGH H DQG WKHQ XVH SDUWLDO IUDFWLRQV WR VLPSOLI\  V V      V   V  V      FRV W    íV +HQFH WKH ILUVW SDUW LV UHDOO\ H / ^  FRV W ` ,W LV LQYHUWHG  ,W KDV DV DQ LQYHUVH WUDQVIRUP YLD WKH IRUPXOD HíFV / ^I W ` XF W I W í F  §  · XS W ¨  FRV  W  S ¸ ©  ¹ ZLWK F  WR  XS W   FRV W  S   7KHUHIRUH WKH VROXWLRQ LV WKH VXP RI WKH  SDUWV \  VLQ W  XS W   FRV W  1RWH FRV W í  FRV W W S ­VLQ W  ° ®VLQ W     FRV W  W t S °  ¯ \  \ ([DPSOH \  FRV W í X W FRV W   \   7UDQVIRUP DQG VLPSOLI\ V / ^\` í V \  í \   / ^\` X W FRV W `  / ^FRV W í  V V  H SV  V   V  V / ^\` í   / ^\` V  / ^\` 1RWH / ^X W FRV W ` Hí V / ^FRV  W   ` H í V / ^FRV W   ` H í V / ^FRV W ` H SV V V   7KHUHIRUH / ^ \` V V  H  SV  V    V    V   V    8VH SDUWLDO IUDFWLRQV WR VLPSOLI\ WKH ILUVW SDUW V V    V    ,W KDV DQ LQYHUVH WUDQVIRUP V V   V    V     FRV W  FRV W   7KH VHFRQG KDOI FRQVLVWV RI WKH QHJDWLYH RI WKH VDPH H[SUHVVLRQ ZLWK V DQ DGGLWLRQDO WHUP RI íHí  7KH H[WUD WHUP ZLOO LQGXFH WKH VWHS IXQFWLRQ X W  DQG WKH WUDQVODWLRQ WKDW FKDQJHV W LQWR W í   +HQFH WKH VHFRQG SDUW LV LQYHUWHG WR   X S W FRV  W  S  FRV  W  S   6XPPLQJ XS WKH  SDUWV WKH VROXWLRQ LV WKHUHIRUH \  FRV W  FRV W   X S W FRV  W  S  FRV  W  S   ­ ° FRV W  FRV W  W  S ® °  W t S ¯ \  \  \ ([DPSOH X W  \   \   7UDQVIRUP DQG VLPSOLI\ V/ ^\` í V \  í \    V/ ^\` í \   / ^\` V / ^\` í V í    V/ ^\` í   / ^\` V  V   / ^\` í V í  V   V   / ^\` / ^X W ` H  V V H  V V H  V V V +HQFH H  V V  / ^\` V V   V   V   V   7KH VHFRQG KDOI LV VLPSOHU ,W FDQ EH EURNHQ GRZQ E\ SDUWLDO IUDFWLRQV LQWR V V   V       V   V    ,W KDV DQ LQYHUVH WUDQVIRUP RI   W   W HH   7KH ILUVW KDOI ZLWKRXW WKH HíV ±WHUP KDV SDUWLDO IUDFWLRQV GHFRPSRVLWLRQ RI  V V   V   ,W KDV DQ LQYHUVH WUDQVIRUP RI        V  V    V      W   W H H   :H WKHQ PXVW DSSO\ WKH HIIHFWV RI WKH HíV ±WHUP QDPHO\ WKH SUHVHQFH RI WKH VWHS IXQFWLRQ X W  DQG WKH WUDQVODWLRQ WKDW FKDQJHV W LQWR W í  +HQFH WKLV SDUW UHDOO\ UHSUHVHQWV  W   · §  X W ¨  H W   H ¸  ©  ¹ &RPELQLQJ WKH WZR SDUWV ZH QRZ KDYH WKH VROXWLRQ \  W   W  §  · H  H  X W ¨  H W   H W   ¸    ©  ¹ ([DPSOH $ PDVV ZHLJKLQJ  OEV LV DWWDFKHG WR D VSULQJ ZLWK VSULQJ FRQVWDQW N  7KH PDVV LV LQLWLDOO\ DW UHVW LQ LWV HTXLOLEULXP SRVLWLRQ $W W  DQ H[WHUQDO IRUFH RI ) W FRV W LV DSSOLHG WR WKH PDVV 7KH IRUFH LV WKHQ DEUXSWO\ GLVFRQWLQXHG DW W   7KHUH LV QR GDPSLQJ LQ WKH V\VWHP )LQG WKH GLVSODFHPHQW IXQFWLRQ RI WKLV PDVVVSULQJ V\VWHP )URP WKH SUREOHP¶V GHVFULSWLRQ ZH GHGXFH WKDW P   N  DQG ) W  í X W FRV W  7KHUHIRUH WKH LQLWLDO YDOXH SUREOHP ZH QHHG WR VROYH LV X  X FRV W í X W FRV W  X   X   7UDQVIRUP WKH HTXDWLRQ DQG VLPSOLI\ V/ ^X` í V X  í X   / ^X`  V / ^X` í   / ^X` +HQFH / ^X `  V  / ^X` V V  H SV  V   V  V V  H SV  V     V     7KH ILUVW SDUW KDV LQYHUVH WUDQVIRUP 7KH VHFRQG SDUW YLD WKH IRUPXOD Hí V / ^I W ` EHFRPHV / ^FRV W í X W FRV W `  W VLQ W   X W I W í   X S W W  S VLQ  W  S   X S W W  S VLQ W   7KHUHIRUH WKH VROXWLRQ LV X W   W VLQ W  X S W W  S VLQ W   ­ °  W VLQ W  ®S ° VLQ W  ¯  d W  S W t S  1RWLFH WKDW WKH V\VWHP ZDV XQGHUJRLQJ UHVRQDQFH XQWLO WKH IRUFLQJ IXQFWLRQ ZDV VKXW RII 7KHQ LW RVFLOODWHV DW FRQVWDQW DPSOLWXGH 1RWH / ^W VLQ DW ` DV V   D    7KH JUDSK RI WKH DERYH VROXWLRQ ,PSXOVH )XQFWLRQV ,Q WKLV VHFWLRQ )RUFLQJ IXQFWLRQV WKDW PRGHO LPSXOVLYH DFWLRQV í H[WHUQDO IRUFHV RI YHU\ VKRUW GXUDWLRQ DQG XVXDOO\ RI YHU\ ODUJH DPSOLWXGH  7KH LGHDOL]HG LPSXOVLYH IRUFLQJ IXQFWLRQ LV WKH 'LUDF GHOWD IXQFWLRQ RU WKH XQLW LPSXOVH IXQFWLRQ  GHQRWHV W  ,W LV GHILQHG E\ WKH WZR SURSHUWLHV W f ³ f  LI W   DQG G W GW   7KDW LV LW LV D IRUFH RI ]HUR GXUDWLRQ WKDW LV RQO\ QRQHUR DW WKH H[DFW PRPHQW W  DQG KDV VWUHQJWK WRWDO LPSXOVH RI  XQLW 7UDQVODWLRQ RI 7KH LPSXOVH FDQ EH ORFDWHG DW DUELWUDU\ WLPH UDWKHU WKDQ MXVW DW W LPSXOVH DW W F ZH MXVW KDYH W í F ³ f f  L I W  F  )RU DQ DQG G W  F GW   ,W ZDV LQWURGXFHG E\ DQG LV QDPHG DIWHU WKH %ULWLVK SK\VLFLVW 3DXO $ 0 'LUDF  ±   FRZLQQHU RI WKH 1REHO 3UL]H LQ 3K\VLFV LQ  $ SLRQHHU RI TXDQWXP PHFKDQLFV 'LUDF LV SHUKDSV EHVW NQRZQ EHVLGHV IRU WKH GHOWD IXQFWLRQ IRU IRUPXODWLQJ WKH 'LUDF HTXDWLRQ ZKLFK SUHGLFWHG WKH H[LVWHQFH RI DQWLPDWWHU /DSODFH WUDQVIRUPV RI 'LUDF GHOWD IXQFWLRQV / ^ W `  / ^ W í F ` HíFV F •  +HUH LV DQ LPSRUWDQW DQG LQWHUHVWLQJ SURSHUW\ RI WKH 'LUDF GHOWD IXQFWLRQ ,I I W LV DQ\ FRQWLQXRXV IXQFWLRQ WKHQ ³ f f G W  F I W GW I F 7KHUHIRUH IRU F •  / ^ W í F ` ³ f  G W  F H VW GW ³ f f G W  F H  VW GW H FV  1RWH 6LQFH WKH LQWHJUDQG LQ WKH DERYH LQWHJUDOV LV ]HUR HYHU\ZKHUH H[FHSW DW WKH VLQJOH SRLQW W F •  WKH FRQWULEXWLRQ IURP WKH LQWHUYDO í’  WR WKH GHILQLWH LQWHJUDO LV WKHUHIRUH ]HUR +HQFH WKH WZR GHILQLWH LQWHJUDOV DERYH ZLOO KDYH WKH VDPH YDOXH GHVSLWH WKHLU GLIIHUHQW ORZHU OLPLWV RI LQWHJUDWLRQ $QG / ^ W í F I W ` ³ f  G W  F I W H  VW GW I F H FV  ([DPSOH \  \  \ W  W í   \   \  7UDQVIRUP ERWK VLGHV DQG VLPSOLI\ V / ^\` í V \  í \    V/ ^\` í \    / ^\` / ^ W  W í  ` V / ^\` í V   V/ ^\` í   / ^\` V   V   / ^\` í V í  V   V   / ^\`   HíV   HíV   HíV  V   HíV  V   H  V V  / ^\` V   V   V   V    )XUWKHU VLPSOLI\LQJ WKH WZR SDUWV XVLQJ SDUWLDO IUDFWLRQV 3DUW  ZLWKRXW WKH H íV WHUP EHFRPHV  V   V       V    V     W  W HH ,W KDV DQ LQYHUVH WUDQVIRUP   1RZ DSSO\ WKH HIIHFWV RI WKH H íV WHUP UHVXOWLQJ LQ  X  W H W   H W     3DUW  EHFRPHV V V   V   ,W¶V LQYHUVH WUDQVIRUP LV     V    V     W  W HH   7KHUHIRUH WKH VROXWLRQ LV \  W   W  H  H  X  W H W    H W      1RWH WKDW WKH ILUVW WHUP LQ WKH IRUFLQJ IXQFWLRQ W  KDV WKH VDPH HIIHFW WR WKH VROXWLRQ RI WKLV LQLWLDO YDOXH SUREOHP DV LW ZRXOG LI WKH VHFRQG LQLWLDO FRQGLWLRQ ZDV \   &KHFN WKLV UHPRYH WKH ILUVW WHUP RI WKH IRUFLQJ IXQFWLRQ DQG FKDQJH WKH VHFRQG LQLWLDO FRQGLWLRQ WR \   DQG VHH WKDW WKLV QHZ LQLWLDO YDOXH SUREOHP ZLOO KDYH WKH VDPH VROXWLRQ DV WKH DERYH SUREOHP ([DPSOH \  \  \ í W í   \   \   7UDQVIRUP ERWK VLGHV DQG VLPSOLI\ V / ^\` í V \  í \    V/ ^\` í \    / ^\` / ^í W í  ` V / ^\` í    V/ ^\` í   / ^\` V   V   / ^\` í  V   V   / ^\` í Hí  í Hí í Hí V V V    H  SV  / ^\` V    V   V   V    í V 1RWLFH WKDW RWKHU WKDQ WKH H[WUD IDFWRU í H  WKH VHFRQG SDUW FRQWDLQV WKH VDPH H[SUHVVLRQ DV WKH ILUVW SDUW 7KLV IDFW VLPSOLILHV RXU WDVN VRPHZKDW &RPSOHWH WKH VTXDUHV LQ WKH GHQRPLQDWRU DQG UHZULWH WKH H[SUHVVLRQ DV  V    V   ,WV LQYHUVH WUDQVIRUP LV    V        W H VLQ W  ZKLFK FRUUHVSRQGV H[DFWO\ WR  WKH ILUVW SDUW RI WKH H[SUHVVLRQ í V 7KH VHFRQG SDUW ZLWK WKH H[WUD í H WHUP JDLQV LWV HIIHFWV SOXV D QHJDWLYH VLJQ RI D XQLW VWHS IXQFWLRQ DQG D WUDQVODWLRQ E\ F   ,W EHFRPHV   X S W H  W S VLQ  W  S   7KHUHIRUH \  W  H VLQ W  X S W H  W S VLQ  W  S    W  H VLQ W  X S W H W  S VLQ W   5HVRQDQFH LQGXFHG E\ D GLVFRQWLQXRXV IRUFLQJ IXQFWLRQ ([DPSOH X X X  W  W í   W í   «  W í    X   7UDQVIRUP DQG VLPSOLI\ WKH HTXDWLRQ V / ^X` í V í   / ^X` V   / ^X`   Hí V  Hí V  «  Hí V   Hí V  Hí V  «  Hí V  §    QSV · ¸ ¨  ¦ H ¸ / ^X ` V    ¨ ¹ © Q 7KH LQYHUVH WUDQVIRUP RI HDFK WHUP FRQWDLQV VLQ W  7KH WHUPV LQVLGH WKH VXPPDWLRQ DIWHU DSSO\LQJ WKH UHTXLUHG VWHS IXQFWLRQV DQG WUDQVODWLRQV EHFRPH XQ W VLQ W í Q XQ W VLQ W  7KHUHIRUH X  VLQ W  ¦ X  QS W VLQ W Q ­ VLQ W   d W  S °  VLQ W  S d W  S ° ° ®  VLQ W  S d W  S °   ° ° VLQ W  S d W ¯ 1RWLFH WKH LQFUHDVH LQ DPSOLWXGH QRW FRQWLQXRXVO\ EXW UDWKHU LQ GLVFUHWH VWHSV WKDW RFFXUV OLQHDUO\ ZLWK HODSVHG WLPH 7KLV LV D GLVFUHWH YHUVLRQ RI WKH UHVRQDQFH EHKDYLRU 7KH JUDSK RI WKH SUHYLRXV VROXWLRQ ...
View Full Document

Ask a homework question - tutors are online