Analyse complexe G2

c
      |

FONCTION DǯUNE VARIABLE COMPLEXE |
|||  ||  | | | 
 | |
| ||| 
|  ||

| 
 | |‫ ݓ‬ൌ ‫ ݑ‬൅ ݅‫ ݓ| ||||
||
||ݒ‬ൌ ݂ ሺ‫ݖ‬ሻ| |||

݂ ሺ‫ݖ‬ሻ ൌ ‫ ݑ‬ሺ‫ݔ‬ǡ ‫ ݕ‬ሻ ൅ ݅‫ݒ‬ሺ‫ݔ‬ǡ ‫ ݕ‬ሻ|  
| | | 
| 

|  |


| | |  | | 
| 

|| 
 |

|| | |

!|
|݂ | |  |uniforme 
|||  |||| || |  || |
| 
|  |||  ||| 
 ||||
||


||݂ |
| 
||  || | || | 
 |||
|multiforme :|| 
 || | 
| 
 | | | |

| |

ñ |
    | 
|‫ ݓ‬ൌ ‫ݖ‬ଶ | |‫ ݓ‬ൌ ‫ ݑ‬൅ ݅‫ ݒ‬ൌ ‫ݖ‬ଶ ൌ ሺ‫ ݔ‬൅ ݅‫ݕ‬ሻଶ ൌ ‫ ݔ‬ଶ െ ‫ ݕ‬ଶ ൅ ʹ݅‫|ݕݔ‬

!| 
|

|| |
|
| ||| ||‫ ݑ‬ൌ ‫ ݔ‬ଶ െ‫ ݕ‬ଶ | | 
|


| | |‫ ݒ‬ൌ ʹ‫ݕݔ‬Ǥ|"| 
   
| 
|| 
| |
||
||| ‫ ݔ‬ൌ ‫ݔ‬଴ | |‫ ݕ‬ൌ ‫ݕ‬଴ || ||| 

|‫ ݓ‬ൌ ‫ ݖ‬ଶ |!
|| |
||‫ ݔ‬ൌ ‫ݔ‬଴ | |‫ ݕ‬ൌ ‫ݕ‬଴ | | |||| 

||‫ ݓ‬ൌ ‫ ݖ‬ଶ | !
|| |
|‫ ݔ‬ൌ ‫ݔ‬଴ ǡ െλ ൏ ‫ ݕ‬൏ λ| || | |
||‫ ݑ‬ൌ ‫ݔ‬଴ଶ െ ‫ ݕ‬ଶ| |‫ ݒ‬ൌ ʹ‫ݔ‬଴‫|ݕ‬  | |  
|  
|| # ||||| | | | |

?
  |




c
      | |

| ௩మ

$| 

 | | # | | |  | | ‫ݔ‬଴ ് Ͳǡ ‫ ݑ‬ൌ ‫ݔ‬଴ଶ െ ସ௫ మ|  
| 
| | బ

 |  |||| | ||‫|||
|ݓ‬%
| | ||‫| ݑ‬

&| '| 
| | | 
| ‫ ݕ‬ൌ ‫ݕ‬଴ ǡ െλ ൏ ‫ ݔ‬൏ λ|  | | | | 
‫ ݑ‬ൌ ‫ݔ‬ଶ െ ‫ݕ‬଴ଶ  | ௩మ

ൌ ʹ‫ݕ‬଴‫ | |ݔ‬$| 

 | | # | | | |  | |‫ݕ‬଴ ് Ͳǡ ‫ ݑ‬ൌ ସ௬ మ െ ‫ݕ‬଴ଶ |  
| 
| బ

|  |  ||
|| | ||‫||||ݓ‬%
| | ||‫| ݑ‬

ß |
  | ଵ

ଵ

ଵ

௫ି௜௬


| |
|

| |‫ ݓ‬ൌ ||||‫ ݓ‬ൌ ‫ ݑ‬൅ ݅‫ ݒ‬ൌ ௓ ൌ ௫ା௜௬ ൌ ௫ మ ା௬ మ ሺͳሻ| &|‫ ݑ‬ൌ

௫

௫ మ ା௬ మ

ሺʹሻ||| |‫ ݒ‬ൌ െ

௭

௬

௫ మ ା௬ మ

ሺ͵ሻ|

௫ మ ା௬మ ା௬ మ ሻమ

!| 
| ሺʹሻ|  | ሺ͵ሻ|  ( | |  
| ‫ ݑ‬ଶ ൅ ‫ ݒ‬ଶ ൌ ሺ௫ మ  |ሺʹሻ| | ሺ͵ሻ‫ ݔ‬ൌ

௨

௨మ ା௩ మ

ǡ ‫ ݕ‬ൌ

ି௩

௨మ ା௩ మ

ൌ

ଵ

௫ మ ା௬ మ

| | )| | 

 | | ଵ

|| | 
   
| 
|| 

‫ ݓ‬ൌ | ௭

| || 
||
|  |  || |
|
|||

||  
|
 |)|ߣ|
| | #  |‫ ݔ‬ൌ ‫ݔ‬଴ ൅ ߣܿ‫ ݕ| |ߙݏ݋‬ൌ ‫ݕ‬଴ ൅ ߣ‫| |ߙ݊݅ݏ‬ ௨ െλ ൏ ߣ ൏ λǤ| 
 | | 
 | ‫ | ݕ | ݔ‬ሺͶሻ|  | ሺͷሻ| | 
 | ௨మା௩ మ ൌ ‫ݔ‬଴ ൅ ߣܿ‫|  ߙݏ݋‬ ି௩

௨మ ା௩ మ

ቀ

௨

ൌ ‫ݔ‬଴ ൅ ߣ‫ | ߙ݊݅ݏ‬$| 

 | | # | ߣ | | | ‫ ߙ݃ݐ‬ൌ

௨మ ା௩ మ

ି௩

షೡ ି௫బ ೠమ శೡమ ೠ ି௫బ ೠమ శೡమ

| |  
|||||||||||||||||||

െ ‫ݔ‬଴ ቁ ‫ ߙ݊݅ݏ‬ൌ ቀ௨మା௩ మ െ ‫ݔ‬଴ ቁ ܿ‫ ݑ| | 

||||ߙݏ݋‬ଶ ൅ ‫ݒ‬ଶ |

ሺ‫ ݑ‬ଶ ൅ ‫ ݒ‬ଶ ሻሺ‫ݕ‬଴ ܿ‫ ߙݏ݋‬െ ‫ݔ‬଴ ‫ߙ݊݅ݏ‬ሻ ൅ ‫ ߙ݊݅ݏݑ‬൅ ‫ ߙݏ݋ܿݒ‬ൌ Ͳሺ͸ሻ|

 | 
| ሺ͸ሻ| 
||
| | ‫ݕ‬଴ ܿ‫ݔ ് ߙݏ݋‬଴ ‫| | | ||
|
 | ߙ݊݅ݏ‬



||‫ݕ|
| ݓ݈݊ܽ݌||

 | |  | ||| |ݖ݈݊ܽ݌‬଴ ܿ‫ ߙݏ݋‬ൌ ‫ݔ‬଴ ‫*| |ߙ݊݅ݏ‬ 
| | 
| | | 

| | ‫| |

 | |  |
 | | |
| |ݖ݈݊ܽ݌‬ ‫ ݔ | | | | | | | |  | |
 | | |
 & | ݓ݈݊ܽ݌‬ൌ ‫ݔ‬଴ ൅

?
  |




c
      | ‫ ݕ–‡߮ݏ݋ܿݐ‬ൌ ‫ݕ‬଴ ൅ ‫ Ͳ |  |߮݊݅ݏݐ‬൏ ߮ ൏ ʹߨ| |  #| 


| | # | ሺ‫ ݔ‬െ ‫ݔ‬଴ ሻଶ ൅ ሺ‫ ݕ‬െ ‫ݕ‬଴ ሻଶ ൌ ‫ ݐ‬ଶ | ௨

ଶ

ି௩

ଶ


| | 
 |ሺ͸ሻ‡–ሺ͹ሻ||  |‫ ݐ‬ଶ ൌ ቀ௨మ ା௩ మ െ ‫ݔ‬଴ ቁ ൅ ቀ௨మ ା௩ మ െ ‫ݕ‬଴ ቁ |)| ‫ ݒ‬ଶ ሺ‫ ݒ‬ଶ ൅ ‫ ݑ‬ଶሻଶ ൌ ሾ‫ ݑ‬െ ሺ‫ ݑ‬ଶ ൅ ‫ ݒ‬ଶ ሻ‫ݔ‬଴ሿଶ ൅ ሾെ‫ ݒ‬െ ሺ‫ ݑ‬ଶ ൅ ‫ ݒ‬ଶ ሻ‫ݕ‬଴ ሿଶ | ൌ ‫ ݑ‬ଶ ൅ ‫ ݒ‬ଶ െ ʹሺ‫ݔݑ‬଴ െ ‫ݕݒ‬଴ሻሺ‫ ݑ‬ଶ ൅ ‫ ݒ‬ଶ ሻ ൅ ሺ‫ݔ‬଴ଶ ൅ ‫ݕ‬଴ଶሻሺ‫ ݑ‬ଶ ൅ ‫ ݕ‬ଶ ሻ|

$|

 |

|
#|
| |ሺ‫ݑ‬ଶ ൅ ‫ݒ‬ଶ ሻ|| ||

|ሺ‫ݑ‬ଶ ൅ ‫ ݒ‬ଶ ሻ ് Ͳǡ| ሺ‫ ݐ‬ଶ െ ‫ݔ‬଴ଶ െ ‫ݕ‬଴ଶ ሻሺ‫ ݑ‬ଶ ൅ ‫ݒ‬ଶ ሻ ൅ ʹሺ‫ݔݑ‬଴ െ ‫ݕݒ‬଴ ሻ ൌ ͳ|

& #| 
| ሺͻሻ| 
| |  |  | ሺ‫ ݐ‬ଶ െ ‫ݔ‬଴ଶ െ ‫ݕ‬଴ଶሻ ് Ͳ|  
| 
| | | | | 

||‫| |ݓ݈݊ܽ݌| || ||| |ݖ݈݊ܽ݌‬

à    | + | J || |
|݂ ሺ‫ݖ‬ሻ ൌ Ž‘‰ ‫|| || 
 |||ݖ‬ |

| ,||Ž‘‰ ‫ ݖ‬ൌ Ž‘‰ ‫ ݎ‬൅ ݅ߠ |  || |  || 
 |‫ݖ‬ଵ ് Ͳ|| | | |‫ ݎ‬ൌ ‫ݎ‬ଵ ǡ ߠ ൌ ߠଵ Ǥ,| ||Ž‘‰ ‫ݖ‬ଵ ൌ Ž‘‰ ‫ݎ‬ଵ ൅ ݅ߠଵǤ|- | #| 
|
|| |  | || 

|| | | 
 | |  | |  | | ‫ݖ‬ଵ ǡ ‫ ݎ‬ൌ ‫ݎ‬ଵ ǡ ߠ ൌ ߠଵ •‹„‹‡“—‡ Ž‘‰ ‫ݖ‬ଵ ൌ Ž‘‰ ‫ݎ‬ଵ ൅ ݅ ሺߠଵ ൅ ʹߨሻ| .|||| ||| |
| ||‫ ݖ‬ൌ Ͳ|| || 
 ||  | ,| | 
| | Ž‘‰ ‫| |


| |  |
 |
 | |  |ݖ‬  

 |!|| 

#||Ž‘‰ ‫|

| |ݖ| 
 | ||

| |
|ݖ‬  | 

| |  | | 

| 

  |"| 
|

|| 

 ||
 |ൌ Ͳ|| |‫ ݖ‬൐ Ͳ |
|  |' |
||  |Ž‘‰ ‫ ݖ‬ൌ Ž‘‰ ‫ ݎ‬൅ ݅ߠ ||)|ߠ | |

| | 

|
#||Ͳ ൑ ߠ ൏ ʹߨ‘— െ ߨ ൑ ߠ ൏ ʹߨ|

Í

Ž‘‰ሺ‫ ݖ‬െ ܽሻ ‫ ݖ‬ൌ ܽ|| 
 || |

 |

?
  |




c
      |


 Ã    | ||!|
| 


| |

| |‫ ݓ‬ൌ ݁ ௭ | ,||‫ ݓ‬ൌ ݁ ௭ ൌ ݁ ௫ା௜௬ ൌ ݁ ௫ ሺ…‘• ‫ ݕ‬൅ ݅‫ݕ݊݅ݏ‬ሻ|

"| 
| 
   
| 
|||
 ||
||
||| ‫ ݔ‬ൌ ‫ݔ‬଴  |‫ ݕ‬ൌ ‫ݕ‬଴ |
|‫ ݔ‬ൌ ‫ݔ‬଴ ||  |‫ ݓ‬ൌ ‫ ݑ‬൅ ݅‫ ݒ‬ൌ ሺ…‘• ‫ ݕ‬൅ ݅‫ݕ݊݅ݏ‬ሻ|)|‫ ݑ‬ൌ ݁ ௫బ ܿ‫ ݒ| ||ݕݏ݋‬ൌ ݁ ௫బ ‫|ݕ݊݅ݏ‬

$| 

 | | # | | |  | ‫ ݑ‬ଶ ൅ ‫ݒ‬ଶ ൌ ݁ ଶ௫బ | | 
| |  | | | ݁ ଶ௫బ |  || 

 | "| ‫ ݕ‬ൌ ‫ݕ‬଴ ǡ| | 
 | | ‫ ݓ‬ൌ ‫ ݑ‬൅ ݅‫ ݒ‬ൌ ݁ ௫ ሺ…‘• ‫ݕ‬଴ ൅ ݅‫ݕ݊݅ݏ‬଴ ሻ| )‫ ݑ‬ൌ ݁ ௫ ܿ‫ݕݏ݋‬଴| |  | ‫ ݒ‬ൌ ݁ ௫బ ‫ݕ݊݅ݏ‬଴| ௩

$| 

 | | # | | |  | ௨ ൌ ‫ݕ݃ݐ‬଴| | 
|   | | 
|  | |



||‫ݕ|  | | |ݓ݈݊ܽ݌‬଴ |


| |   |  

| 

 | | ܽ‫ Ͳ݄ݐ݃ݎܽ |

 | |

 |ݕ݄ݐ݃ݎ‬ൌ Ͳ| ଵ

ଵା௭

 ||ܽ‫ ݖ݄ݐ݃ݎ‬ൌ ଶ Ž‘‰ ቀଵି௭ ቁ| ௘ ೢ ି௘ షೢ

ଵା௭

‫ ݓ‬ൌ ܽ‫ ݖ•”‘Žƒݖ݄ݐ݃ݎ‬ൌ ‫ ݓ݄ݐ‬ൌ ௘ೢ ା௘ షೢ )ሺͳ െ ‫ݖ‬ሻ݁ ௪ ൌ ሺͳ ൅ ‫ݖ‬ሻ݁ ି௪  ֜ ݁ଶ௪ ൌ ଵି௭| ‫ ݁ܿ݊݋ܦ‬ଶ௪ ൌ ݁ ଶ௪ାଶ௞గ ൌ

ଵା௭ ଵି௭

ଵ

)‫ ݓ‬ൌ ݇ߨ݅ ൅ Ž‘‰

!|| 

 | | | |݇ ൌ Ͳ|

ଶ

ଵା௭ ଵି௭

|

‫݇׊‬ǡ ݁ ௭ା௜ଶ௞గ ൌ ݁ ௭ ݁ ௜ଶ௞గ ൌ ݁ ௭ | |
| 


| | 

|| 
|‫ ݌‬ൌ ʹ݅ߨ|

|


   | !|
| 

| |

|| 
 || |
| 


 | ‫ ݓ‬ൌ Ž ‫ ݖ‬฻ ‫ ݖ‬ൌ ݁௪ |


 ||| |߮| ||  
|| |

‫ ݖ‬ൌ ‫݁ݎ‬௜ఝ ൌ ݁ ௪ ൌ ݁ ௨ା௜௩ ൌ ݁ ௨ ݁ ௜௩ )| |
|‫ ݎ‬ൌ ݁௨ ||
| 
|‫ ݑ‬ൌ Ž ‫݁ݐ݁ݎ‬௜௩ ൌ ݁ ௜ఝ || 
| 
|‫ ݒ‬ൌ ߮ ൅ ʹ݇ߨ| |/||
 |

|!|
| 

| ||
|


||
 ||  | 

 | |݇ ൌ ͲǤ|

?
  |




c
      | | |

|

?
  |




c
      | |

DERIVEE DǯUNE FONCTION A VARIABLE COMPLEXE |

 | 
|݂ሺ‫ݖ‬ሻ||
|| 
 | |||
|‫ ݈݊ܽ݌||ܦ‬െ ‫ݖ‬Ǥ|

!|
| ݂ሺ‫ݖ‬ሻ| |
|
|| 
 |‫ݖ‬଴ ‫ܦ א‬ǡ|
| ݂ሺ‫ݖ‬଴ ൅ ο‫ݖ‬ሻ| | | ݂ሺ‫ݖ‬଴ ሻ| | ο‫| |ݖ‬ | | | 0|   | ,| 
| 
 | | | ݂ሺ‫ݖ‬ሻ|  | 
| | ‫ݖ‬଴ ൌ ‫ݔ‬଴ ൅ ݅‫ݕ‬଴ | 
|  |   | 
| | 
| 

|  |


| |ሺ‫ݖ‬ሻ| |  
| ‫ ݑ‬ൌ ‫ݑ‬ሺ‫ݔ‬ǡ ‫ ݕ‬ሻ݁‫ ݒݐ‬ൌ ‫ݒ‬ሺ‫ݔ‬ǡ ‫ݕ‬ሻ|  |


|| 
 |ሺ‫ݔ‬଴ ǡ ‫ݕ‬଴ ሻ |,| 

| ݂ሺ‫ݖ‬ሻ|
 || 
 |‫ݖ‬଴ ‫| 
| |
 |ܦ א‬

௙ሺ௭బ ାο௭ሻି௙ሺ௭బ ሻ ο௭

|

#|| || |

| |ο‫||| | |ݖ‬0|  |

|

| |
| la dérivée de ݂ ሺ‫ݖ‬ሻau point ‫ݖ‬଴  | |
|  ݂ ሺ‫ݖ‬଴ ሻ ‫ ݂ݑ݋ݖ‬ᇱ ሺ‫ݖ‬଴ ሻA Si ݂ሺ‫ݖ‬ሻ |
 ||  | 
 ||
|‫݂|ܦ‬ሺ‫ݖ‬ሻ| |  |analytique ||‫| ܦ‬ |

 !"# $ |  | | | 
| 

|  |


| | | 
|݂ ሺ‫ݖ‬ሻ ൌ ‫ ݑ‬ሺ‫ݔ‬ǡ ‫ݕ‬ሻ ൅ ݅‫ݒ‬ሺ‫ݔ‬ǡ ‫ݕ‬ሻ| 
 | 
 |
 | |  ||| ||| |
||


|| |

Th   !| 
| ݂ሺ‫ݖ‬ሻ ൌ ‫ݑ‬ሺ‫ݔ‬ǡ ‫ݕ‬ሻ ൅ ݅‫ݒ‬ሺ‫ݔ‬ǡ ‫ݕ‬ሻ|  | 
 | | | 
 | ‫| |
 |   |  |
 |ܦ א ݖ‬



|
 | |

|||  ||

డ௨ డ௫

ൌ

డ௩

డ௬

ǡ

డ௨ డ௬

ൌെ

డ௩

డ௫

ሺͳሻ|

|

| |  |

|||1
 | "| || #|| 

||| |

|ሺͳሻ| |
 |,| | ௗ௙ሺ௭ሻ

| 
 | | |

| ݂ ᇱ ሺ‫ݖ‬ሻ ൌ

ௗ௭

ൌ Ž‹ο௭՜଴

௙ሺ௭బ ାο௭ሻି௙ሺ௭బ ሻ ο௭

| )| ο‫ | | | | |ݖ‬0|



| | $|   | | 
 |‫

 |
 | | | |ݖ‬# | | |  |  
| ο‫ ݖ‬ൌ ௗ௙ ௗ௙ డ௙ డ௨ డ௩ ο‫ |  ||ݔ‬ൌ
ቚ ൌ ൌ ൅ ݅ ሺʹሻ || ௗ௭

ௗ௫ ௗ௫ୀ଴

డ௫

డ௫

డ௫

$|  || 
 || | ||
| 

# || |


|| 
|ο‫ ݖ‬ൌ ο‫||ݕ‬ ௗ௙ డ௙ ଵ డ௨ డ௩ ௗ௙ ൌ ൌ ሺ ൅ ݅ ሻ||234|

 | ൌ
ቚ ௗ௭

ௗ௬ ௗ௬ୀ଴

డ௬

௜ డ௬ డ௨

$|  |ሺʹሻ݁‫ݐ‬ሺ͵ሻ|||

డ௫

?
  |

൅݅

డ௬

డ௩

డ௫

ൌ

డ௩

డ௬

െ݅

డ௨ డ௬

|)| |
| |
|ሺͳሻ|




c
      | ,| | |
||
|2+4|| |
#|
 || డ௨

!|



| ݂ | |
|݂ ሺ‫ݖ‬ሻ‡•–†‡Žƒˆ‘”‡ ݂ ൌ ቀ $|
 |

డ௨ డ௬

݁‫ݐ‬

డ௩

డ௬

డ௫

൅݅

డ௩

డ௫

డ௨

ቁ ‫ ݔ‬൅ ቀ

డ௬

൅݅

డ௩

డ௬

ቁ ‫| ݕ‬

|| |
#| ||||
||

|
#| 
|

 | |
ሺͳሻ||  || ݂ ൌ ൬

߲‫ݒ‬ ߲‫ݒ‬ ߲‫ݑ‬ ߲‫ݑ‬ ߲‫ݒ‬ ߲݂ ߲‫ݑ‬ ൅ ݅ ൰ ‫ ݔ‬൅ ൬െ ൅ ݅ ൰ ‫ ݕ‬ൌ ൬ ൅ ݅ ൰ ሺ ‫ ݔ‬൅ ݅ ‫ݕ‬ሻ ൌ ‫| ݖ‬ ߲‫ݔ‬ ߲‫ݔ‬ ߲‫ݔ‬ ߲‫ݔ‬ ߲‫ݔ‬ ߲‫ݔ‬ ߲‫ݔ‬ డ௙

ௗ௙

-

| |

| ௗ௭ |
 | | | |  |డ௫ |  |
|| | || |0| |ο‫| | |ݖ‬  |

M


doutes les notions vues en variable réelle sont valables en variable complexe 1A| ሾ݂ሺ‫ݖ‬ሻ ൅ ݃ሺ‫ݖ‬ሻሿᇱ ൌ ݂ ᇱ ሺ‫ݖ‬ሻ ൅ ݃ᇱ ሺ‫ݖ‬ሻ 2A| ሾ݂ ሺ‫ݖ‬ሻ݃ሺ‫ݖ‬ሻሿᇱ ൌ ݂ ᇱ ሺ‫ݖ‬ሻ݃ሺ‫ݖ‬ሻ ൅ ݂ሺ‫ݖ‬ሻ݃ ᇱ ሺ‫ݖ‬ሻ ௙ሺ௭ሻ ᇱ

3A| ௚ሺ௭ሻቃ ൌ

௙ ᇲሺ௭ሻ௚ሺ௭ሻି௙ሺ௭ሻ௚ᇲ ሺ௭ሻ ௚ሺ௭ሻమ

| |݃ሺ‫ݖ‬ሻ ് Ͳ

|
| ݂ ሺ‫ݖ‬ሻ ൌ ‫ ݑ‬ሺ‫ݔ‬ǡ ‫ݕ‬ሻ ൅ ݅‫ݒ‬ሺ‫ݔ‬ǡ ‫ݕ‬ሻ|  | | 
|  
| | | | | | 
|‫|ݒݐ݁ݑ‬ 
 | | 

| | 1
| ሺͳሻ| | ,|  | 

| | ‫ | | ݒݐ݁ݑ‬# |  
 ||

|
 |!|

|ሺͳሻ|| 
 |

  || డమ ௨

|‫ !||
 | 

 |ݒݐ݁ݑ‬

డ௫ మ

൅

డమ ௨ డ௬మ

ൌ Ͳ5||

డమ ௩

డ௫ మ

൅

డమ ௩

డ௬ మ

ൌ Ͳ|

!| 
|

| |


||
| 
| |||6|
|
|
| 
 ||

||1
|

%$&' + | J || |
||
 | |
||‫|ܦ‬  | ‫ݔ‬ଶ െ‫ ݕ‬ଶ ൅ ʹ‫|ݕ‬  | ‫|ݕ݄ܿݔ݊݅ݏ‬ ߶ሺ‫ݔ‬ǡ ‫ݕ‬ሻ ൌ ‫ ݔ‬ଶ െ ‫ ݕ‬ଶ ൅ ʹ‫ ݕ‬՜ ߶ሺ‫ݔ‬ǡ ‫ݕ‬ሻ ൌ ‫ ݕ݄ܿݔ݊݅ݏ‬՜

߲ଶ߶ ߲ଶ߶ ߲ ଶ ߶ ߲ ଶ߶ ൌ ʹǡ ଶ ൌ െʹ‫ ݑ݋‬ଶ ൅ ଶ ൌ Ͳ| ଶ ߲‫ݔ‬ ߲‫ݕ‬ ߲‫ݔ‬ ߲‫ݕ‬

߲ଶ߶ ൌ െ‫ݕ݄ܿݔ݊݅ݏ‬ǡ ߲‫ݔ‬ଶ

߲ଶ߶ ߲ଶ߶ ߲ଶ߶ ൌ ‫ݑ݋ݕ݄ܿݔ݊݅ݏ‬ ൅ ൌ Ͳ| ߲‫ ݕ‬ଶ ߲‫ ݔ‬ଶ ߲‫ ݕ‬ଶ

6 | J | | | 
| | |  | 
| | | | |   |
| | | | |


|‫ ݖ‬ൌ ‫ ݓ‬ଷ |
| | | ‫ ݖ‬ൌ ‫ ݓ‬ଷ  ՜ ‫ ݔ‬൅ ݅‫ ݕ‬ൌ ሺ‫ ݑ‬൅ ݅‫ݒ‬ሻଷ ൌ ‫ ݑ‬ଷ െ ͵‫ݒݑ‬ଶ ൅ ݅ሺ͵‫ݑ‬ଶ ‫ ݒ‬െ ‫ ݒ‬ଷ ሻ|  | | ‫ ݔ‬ൌ ‫ ݑ‬ଷ െ ଶ ͵‫ ݕ|  ݒݑ‬ൌ ͵‫ݑ‬ଶ ‫ ݒ‬െ ‫ ݒ‬ଷ |  | ߶ሺ‫ݔ‬ǡ ‫ݕ‬ሻ ൌ ‫ݔ‬ଶ െ ‫ ݕ‬ଶ ൅ ʹ‫ ݕ‬ൌ ሺ‫ ݑ‬ଷ െ ͵‫ݒݑ‬ଶ ሻଶ െ ሺ͵‫ݑ‬ଶ ‫ ݒ‬െ ‫ ݒ‬ଷ ሻଶ ൅ ʹሺ͵‫ݑ‬ଶ‫ ݒ‬െ ‫ ݒ‬ଷ ሻ| డమ థ

డమ థ

ᇱ ‫݋‬î డ௨మ ൌ ͵Ͳ‫ ݑ‬ସ െ ͳͺͲ‫ݑ‬ଶ ‫ ݒ‬ଶ ൅ ͵Ͳ‫ ݒ‬ସ ൅ ͳʹ‫| ݒ‬5| డ௩ మ ൌ െ͵Ͳ‫ ݑ‬ସ ൅ ͳͺͲ‫ݑ‬ଶ ‫ݒ‬ଶ െ ͵Ͳ‫ݒ‬ସ െ ͳʹ‫| ݒ‬ ߲ ଶ ߶ ߲ ଶ߶ ֜ ൅ ൌ Ͳ| ߲‫ ݑ‬ଶ ߲‫ ݒ‬ଶ

?
  |




c
      |

 | ,| 
|  | | ߶ ൌ •‹ሺ‫ ݑ‬ଷ െ ͵‫ ݒݑ‬ଶ ሻ ݄ܿሺ͵‫ݑ‬ଶ ‫ ݒ‬െ ‫ ݒ‬ଷ ሻ|  |   | 
|  
|

డమ థ డ௨మ

൅

డమ థ డ௩ మ

ൌ Ͳ|

1
 |
|| | | ||
|
| ||
|
|
|
 || 
|  
|  | |  | | 
| 

 | !| 
|


| |  | | 
| 
|7|||| | 
|| |
#|
 || 
 | | | |

డ௨

ൌ

డ௫

డ௩

డ௬

 |

డ௨

ൌെ

డ௬

డ௩

డ௫

| |  
|

డ௩

డ௫

‫ ݔ‬൅

డ௩

డ௬

‫ ݕ‬ൌ െ

డ௨ డ௫

‫ ݔ‬൅

డ௨ డ௬

‫| |  | ݕ‬





|   | ‫| |

| ||

|
|

||
||| |
"| ݒ‬


 | |  
 | |

డ

డ௬

ቀെ

డ௨ డ௬

ቁൌ

డ

డ௨

ቀ ቁ| | 
|  | | 
| | |  | 
| |
| |

డ௫ డ௫


||


||| |
  || 
  |

| ሺ௫ǡ௬ሻ డ௨ ቀെ ‫ݔ‬ డ௬ బ ǡ௬బ ሻ

‫ ݒ‬ሺ‫ݔ‬ǡ ‫ ݕ‬ሻ ൌ ‫׬‬ሺ௫

డ௨

൅ డ௫ ‫ݕ‬ቁ ሺͷሻ)ሺ‫ݔ‬଴ ǡ ‫ݕ‬଴ ሻ| 
| | 
 | 

| | 
| |




|| ||
  ||
|
 
||
|
|| #||8|| |  | !| 
| ሺͷሻ| 

| | 

| | 1
| | 2+4| | | డమ ௩

డ௫ మ

൅

డమ ௩

డ௬ మ

ൌ

డ

డ௨

ቀ ቁെ

డ௬ డ௫

డ

డ௨

ቀ ቁൌ

డ௫ డ௬

డమ ௨

డ௫డ௬

൅

డమ ௨

డ௫డ௬

ൌ Ͳ|

డ௨ డ௫

ൌ

డ௩

డ௬

|  |

డ௨ డ௬

ൌെ

డ௩

డ௫

|  | | |

-

| |

| | 
|294| ||
|
||7||||5|

| |
| |  
|||  |

| # |
| |
||| |


 || |7| ||| 7|

| |294| || ||
|
 -|
| ||
|

|
|| 


|
 | Áሺࢠሻ ൌ  ࢋࢠ ൌ ࢋ࢞ା࢏࢟ ൌ ࢋ࢞ ሺࢉ࢕࢙ ࢟ ൅ ࢙࢏࢔ ࢟ሻ|

!| 
|


| |

|| §

ൌ ࢋ࢞ ࢉ࢕࢙ ࢟
| ࢜ ൌ ࢋ࢞ ࢙࢏࢔ ࢟

 |
 |
 | |

 ||

||1
2+4||  | 
 || || ì ì࢜ ൌ ࢋ࢞ ࢉ࢕࢙ ࢟ ൌ | ì࢞ ì࢟ ì࢜ ì ൌ െࢋ ࢞ ࢉ࢕࢙ ࢟ ൌ െ | ì࢞ ì࢟

!||
| 


| || 
|| | |*| ||
| | ||

' | Áᇱ ሺࢠሻ ൌ

ìÁ ì ࢞ ሾࢋ ሺࢉ࢕࢙ ࢟ ൅ ࢏ ࢙࢏࢔ ࢟ሻሿ ൌ ࢋ ࢞| ൌ ì࢞ ì࢞

,| 

|
 |||| | | 

 | 
||! | ο ൌ

?
  |

ì ì ൅ ൌ ࢋ ࢞ ࢉ࢕࢙ ࢟ െ ࢋ ࢞ ࢉ࢕࢙ ࢟ ൌ ૙| ì࢟ ì࢞




c
      | ο࢜ ൌ

ì ࢜ ì ࢜ ൅ ൌ ࢋ ࢞ ࢙࢏࢔ ࢟ െ ࢋ࢞ ࢙࢏࢔ ࢟ ൌ ૙| ì࢟ ì࢞

1
 | 
| |  |  
| | 
|  
|Áሺࢠሻ|  | | 
| 

|  | | 
|
||| ൌ ࢋ ࢞ ࢉ࢕࢙ ࢟|

,| #||
|  
| െ ì࢟ ࢊ࢞ ൅ ì࢞ ࢊ࢟ ൌ  ࢋ࢞ ࢙࢏࢔ ࢟ࢊ࢞ ൅ ࢋ࢞ ࢉ࢕࢙ ࢟ࢊ࢟ | | 



|   | ࢊ࢜| | 
| ì

ì


|7|| ࢜ࢊࢋ |||294|

ሺ࢞ǡ࢟ሻ

࢜ሺ࢞ǡ ࢟ሻ ൌ න ሺࢋ ࢞ ࢙࢏࢔ ࢟ࢊ࢞ ൅ ࢋ ࢞ ࢉ࢕࢙ ࢟ࢊ࢟ሻ ൅ ࢉ| ሺ૙ǡ૙ሻ

$ ||  ||
  |||ሺ૙ǡ ૙ሻ|| ሺ૙ǡ ࢟ሻ| | || |
| ࢞ ൌ ૙| | 
||ሺ૙ǡ ࢟ሻ|| ሺ࢞ǡ ࢟ሻ| | || |
|࢟ ൌ ࢉ࢕࢔࢚Ǥ| ࢞

࢞

ሺ࢞ǡ ࢟ሻ ൌ න ࢉ࢕࢙ ࢟ ࢊ࢟ ൅ ࢙࢏࢔ ࢟ න ࢋ ࢞ ࢊ࢞ ൅ ࢉ| ૙

૙

ൌ ࢙࢏࢔ ࢟ ൅ ࢙࢏࢔ ࢟ࢋ࢞ െ ࢙࢏࢔ ࢟ ൅ ࢉ| ൌ ࢋ࢞ ࢙࢏࢔ ࢟ ൅ ࢉ|

| | 
| | 


| | |  | | | ሺ࢞ǡ ࢟ሻ| 
| | | 
| 
  | | |
|࢜| |
| 
|||  |

| || Áሺࢠሻ ൌ ࢋ࢞ ࢉ࢕࢙ ࢟ ൅ ࢏ࢋ࢞ ࢙࢏࢔ ࢟ ൅ ࢉ| ൌ ࢋ࢞ା࢏࢟ ൅ ࢉ| ൌ ࢋ ࢠ ൅ ࢉ|

Ex   | ૜൅ࢇ ൰| Áሺࢠሻ ൌ ࡸ࢕ࢍ ൬ ૜െࢇ



||1
|||  
|  || | 
|

| |


|| |
|| Áሺࢠሻ ൌ ሺ࢘ǡ ࣐ሻ ൅ ࢏࢜ሺ࢘ǡ ࣐ሻ|

?
  |




c
      |

ൌ ࢘ࢋ࢏࣐|

 |
 
 | |  ||࢘ࢋ࢚࣐||| |
|&||


 | ÁሺࢠାοࢠሻିÁሺࢠሻ


|Áሺࢠሻ|
 ||| 
 |||&|| |


οࢠ

|

"#| |  | |  |

| | οࢠ| | | | | |  | | ࢊࢠ ൌ ࢋ ࢏࣐ ሺࢊ࢞ ൅ ࢏࢜ࢊ࣐ሻ| ,|  
οࢠ| | || | ||  ||||: 
|ࢊ࣐ ൌ ૙| ࢊÁ ì ì ࢊÁ ൌ ࢋି࢏࣐ ൌ ࢋ ି࢏࣐ ൬ ൅ ࢏ ൰| ࢊ࢘ ì࢘ ì࢘ ࢊ࢘

& |  |
|οࢠ| | || | || |||| | || 

|: 
ࢊ࢘ ൌ ૙| |
 ||

ൌ

ࢋష࢏࣐ ࢏࢘

ࢊÁ ࢋି࢏࣐
ࢊÁ ൌ ඐ | ࢊࢠ ࢏࢘ ࢊ࣐ ࢊ࢘ୀ૙ ቀ

ì ì࣐

൅ ࢏ ì࣐ቁ|264| ì࢜

ࢊÁ

$|  | | 
|2+4| |264| | |
ࢊ࢘||
||

||1
| | |||  
|
 ||

૚ ì࢜ ì ൌ | ì࢘ ࢘ ì࣐

ì࢜ ૚ì ൌെ | ì࢘ ࢘ ì࣐

 | | 
|2+4| |
| | ì ì࢜ ࢘ ìÁ ࢊÁ ൌ ࢋି࢏࣐ ൬ ൅ ࢏ ൰ ൌ | ì࢘ ì࢘ ࢠ ì࢘ ࢊࢠ

-|
| ||
| |
| 

| Áሺࢠሻ ൌ ࢒࢔ ࢠ ൌ ࢒࢔ ࢘ࢋ ࢏࣐ ൌ ࢒࢔ ࢘ ൅ ࢏࣐|

!| 
|

| |


|| §

ൌ ࢒࢔ ࢘
| ࢜ൌ࣐

 |
 |
 | |
 ||

||1
||  | 
 || | ૚ ૚ ì࢜ ì ൌ ൌ | ì࢘ ࢘ ࢘ ì࣐

?
  |




c
      |

૚ ì࢜ ì ൌ૙ൌെ | ࢘ ì࣐ ì࢘

!|
| 

| || 
|| | ||| 
|| 
 |;<| ||
| | Áᇱ ሺࢠሻ ൌ

࢘૚ ૚ ൌ | ࢠ࢘ ࢠ

|

   | +| =

|| |
||1
| |

| | | 
|

| |


|| 
|
 || |4Áሺࢠሻ ൌ ࢞ ൅ ૞࢏ࢠ ൅ ૜ െ ࢏| |4Áሺࢠሻ ൌ ࢠࢋ ିࢠ|

|4Áሺࢠሻ ൌ ࢙࢏࢔ ࢞ ൅ ࢏ ࢉ࢕࢙ ࢠ| 6| |J || |
|
 | |
| |4Áሺࢠሻ ൌ ࢞ሺ૚ െ ࢟ሻ|

|4Áሺࢠሻ ൌ ࢟ ൅ ࢞ െ ࢟ |

|4Áሺࢠሻ ൌ ࢏࢞ ൅ ࢞|

3| |& 
| |
|
|7||
|
 || |4૜࢞ ࢟ ൅ ࢞ െ ࢟ |

|4࢞ࢋ࢞ ࢉ࢕࢙ ࢟ െ ࢟ࢋ࢞ ࢙࢏࢔ ࢟| |4ࢋି

࢞࢟

࢙࢏࢔൫࢞ െ ࢟
|

?
  |




c
      | |

INTEÍRATION DANS LE DOMAINE COMPLEXE

()' )*'( |

| 
|24||
|
|||  | 
 |||| || | | |

 | " | | | |
  

| | | | 
 |‫ݖ‬+| ‫ݖ‬6| >| ‫ݖ‬+| 

 | 

|  | | ܽ ൌ ‫ ݈|<ݖ‬ൌ ‫|ݖ‬ 

||||{+ || ||

x

{



{




{ 















 




ž ൌ ෍ ݂ሺߝ௞ ሻሺ‫ݖ‬௞ െ ‫ݖ‬௞ିଵ ሻ ൌ ෍ ݂ሺߝ௞ ሻȟ‫ݖ‬௞ ሺʹሻ| ௞ୀଵ

௞ୀଵ


| |
|
| ||||


||0|| | |ȁȟ‫ݖ‬௞ ȁ|| | ||| | | || | || | | ||

|


| |  |||| 


| ||
 | ௕

න ݂ሺ‫ݖ‬ሻ ‫ ݑ݋‬න ݂ ሺ‫ݖ‬ሻ ‫| ݖ‬ ௔

௖

I
  
  
|mሺ‫ݔ‬ǡ ‫ݕ‬ሻ| |ܳሺ‫ݔ‬ǡ ‫ݕ‬ሻ| ||
|

||‫|| 
 | | ||
|ݕ| |ݔ‬ || 
  |

|m ‫ ݔ‬൅ ܳ ‫
||

| '|  | ܥ|| | | ݕ‬#| 
 ||

?
  |




c
      | න ሾmሺ‫ݔ‬ǡ ‫ ݕ‬ሻ ‫ ݔ‬൅ ܳ ሺ‫ݔ‬ǡ ‫ ݕ‬ሻ ‫ݕ‬ሿ‫ ݑ݋‬න m ‫ ݔ‬൅ ܳ ‫| ݕ‬ ௖

௖


| |
 |

 | || |   |  
| ߯ ൌ ߶ሺ‫ݐ‬ሻ|

‫ ݕ‬ൌ ߰ሺ‫ݐ‬ሻ)| + |?| |?| 6|

௧మ

න mሾ߶ሺ‫ݐ‬ሻǡ ߰ሺ‫ݐ‬ሻሿ߶ ᇱ ሺ‫ݐ‬ሻ ‫ ݐ‬൅ ܳ ሾ߶ሺ‫ݐ‬ሻǡ ߰ሺ‫ݐ‬ሻሿ߰ ᇱ ሺ‫ݐ‬ሻ ‫| ݐ‬ ௧భ

Ã)&)+
|݂ሺ‫ݖ‬ሻ  ൌ ‫ݑ‬ሺ‫ݔ‬ǡ ‫ݕ‬ሻ  ൅ ݅‫ݒ‬ሺ‫ݔ‬ǡ ‫ݕ‬ሻ  ൌ ‫ ݑ‬൅ ݅‫| |

|  
 |ݒ‬2+4|  |' | 
|| |||
  |

|

|| |0|
 || න ݂ ሺ‫ݖ‬ሻ ‫ ݖ‬ൌ  න ሺ‫ ݑ‬൅ ݅‫ݒ‬ሻሺ ‫ ݔ‬൅ ݅ ‫ ݕ‬ሻ ൌ න ‫ ݔ ݑ‬െ ‫ ݕ ݒ‬൅ ݅ න ‫ ݔ ݒ‬൅ ‫|ݕ ݑ‬ ௖

௖

௖


݂ሺ‫ݖ‬ሻ| |݃ሺ‫ݖ‬ሻ| ||
|
  | | ||| |

௖

<| ‫׬‬௖ ൫݂ሺ‫ݖ‬ሻ ൅ ݃ ሺ‫ ݖ‬ሻ
‫ ݖ‬ൌ ‫׬‬௖ ݂ ሺ‫ݖ‬ሻ ‫ ݖ‬൅ ‫׬‬௖ ݃ ሺ‫ݖ‬ሻ ‫| ݖ‬

<| ‫׬‬௖ ‫ ݂ܣ‬ሺ‫ݖ‬ሻ ‫ ݖ‬ൌ ‫׬ ܣ‬௖ ݂ ሺ‫ ݖ‬ሻ ‫| ݖ‬ ௕

௔

<| ‫׬‬௔ ݂ ሺ‫ݖ‬ሻ ‫ ݖ‬ൌ െ ‫׬‬௕ ݂ ሺ‫ݖ‬ሻ ‫| ݖ‬ ௕

௠

௕

<| ‫׬‬௔ ݂ ሺ‫ݖ‬ሻ ‫ ݖ‬ൌ ܾ ‫׬‬௔ ݂ ሺ‫ݖ‬ሻ ‫ ݖ‬൅ ‫׬‬௠ ݂ ሺ‫ ݖ‬ሻ ‫| ߳ ||||)|||||ݖ‬

<| ቚ ‫׬‬௖ ݂ ሺ‫ ݖ‬ሻ ‫ݖ‬ቚ ൑ ܾܾ‫ |)||| ܮܯ‬ȁ݂ሺ‫ݖ‬ሻȁ ൑ ‫ ||
 ||| ܯ‬ȁ݂ሺ‫ݖ‬ሻȁ||| |!|||||| |

|||

%$&' ሺଶǡସሻ

<|   |‫׬‬ሺ଴ǡଷሻ ሺʹ‫ ݕ‬൅ ‫ ݔ‬ଶ ሻ ‫ ݔ‬൅ ሺ͵‫ ݔ‬െ ‫ ݕ‬ሻ ‫||| | || ݕ‬

!|  |§

‫ ݔ‬ൌ ʹ‫ݐ‬ ൠ| ‫ ݕ‬ൌ ‫ ;ݐ‬൅ ͵

| !| 
 |ሺͲǡ͵ሻ| |ሺʹǡͶሻ|| |  |  | 
 |‫ ݐ‬ൌ Ͳ |‫ ݐ‬ൌ ͳ | |!
  ||| ଵ

න ሾʹሺ‫ ݐ‬ଶ ൅ ͵ሻ ൅ ሺʹ‫ݐ‬ሻ;ሿʹ ‫ ݐ‬൅  ሾ͵ሺʹ‫ݐ‬ሻ െ ሺ‫ ݐ‬ଶ ൅ ͵ሻʹ‫ݐ ݐ‬ሿ| ௧ୀ଴

ଵ

ൌ න ሺʹͶ‫ ;ݐ‬൅ ͳʹ െ ʹ‫ ݐ‬ଷ െ ͸‫ݐ‬ሻ ‫ ݐ‬ൌ ͵͵Ȁʹ | ௧ୀ଴

?
  |




c
      | !| || |
|
|| | | ||
ሺͲǡ͵ሻሺʹǡ͵ሻ ሺʹǡ͵ሻሺʹǡͶሻ|

֜ | || ||
| 
|ሺͲǡ͵ሻ ሺʹǡ͵ሻǡ ‫ ݕ‬ൌ ͵ǡ ‫ ݕ‬ൌ Ͳ| | 
  |

|  || න

ଶ

௫ୀ଴

ሺ͸ ൅ ‫ݔ‬ଶ ሻ ‫ ݔ‬൅ ሺ͵‫ ݔ‬െ ͵ሻͲ ൌ  න

ଶ

௫ୀ଴

ሺ͸ ൅ ‫ݔ‬ଶ ሻ ‫ ݔ‬ൌ ͶͶȀ͵|

!| || ||
| 
|ሺʹǡ͵ሻ ሺʹǡͶሻǡ ‫ ݔ‬ൌ ʹǡ ‫ ݔ‬ൌ Ͳ| | 
  |

|  || න

ସ

௬ୀଷ

ሺʹ‫ ݕ‬൅ ͶሻͲ ൅ ሺ͸ െ ‫ ݕ‬ሻ ‫ ݕ‬ൌ න

ସ

௬ୀଷ

ሺ͸ െ ‫ݕ‬ሻ ‫ ݕ‬ൌ ͷȀʹ|

‫ݎ݁ܮ‬±‫ܿ݊݋ ݐݏ݁ݐܽݐ݈ݑݏ‬ͶͶȀ͵ ൅ ͷȀʹ ൌ ͳͲ͵Ȁ͸|

| !| || ||
| 
|ሺͲǡ͵ሻ ሺʹǡ͵ሻ|

!
|| |
|ʹ‫ݕ‬Ȃ ‫ ݔ‬ൌ ͸)||
‫ ݔ‬ൌ ʹ‫ݕ‬Ȃ ͸ |&)| |  || 
  |

|| ସ

න

௬ୀଷ

ሺʹ‫ ݕ‬൅ ሺʹ‫ ݕ‬െ ͸ሻଶ ʹ ‫ ݕ‬൅ ሺ͵ሺʹ‫ݕ‬ሻ െ ͸ሻ െ ‫ ݕ‬ሻ ‫|ݕ‬ ସ

ൌ න ሺͺ‫ ݕ‬ଶ െ ͵ͻ‫ ݕ‬൅ ͷͶሻ ‫ ݕ‬ൌ ͻ͹Ȁ͸| ଷ

<| $ ||

$  |‫׬‬௖ ‫ݖ‬ҧ ‫ ݖ݁  ݖ‬ൌ Ͳ‫ ݖ‬ൌ Ͷ ൅ ʹ݅ | || ||||  | 

| |‫ ݖ‬ൌ ‫ ;ݐ‬൅ ݅‫|ݐ‬  | || |7
 |Ͳʹ݅ ʹ݅Ͷ ൅ ʹ݅ |

|  | !| 
 |‫ ݖ‬ൌ Ͳ݁‫ ݖݐ‬ൌ Ͷ ൅ ʹ݅ |||  ||‫ ݐ‬ൌ Ͳ݁‫ ݐݐ‬ൌ ʹ || !
  |

||  ||| ଶ

ଶ

ଶ

ଶ ൅ ଓ‫ݐ‬ሻ ሺ‫ ݐ‬ଶ ൅ ݅‫ ݐ‬ሻ ൌ න ሺ‫ ݐ‬ଶ െ ݅‫ ݐ‬ሻሺʹ‫ ݐ‬൅ ݅ ሻ ‫ ݐ‬ൌ න ሺʹ‫ ݐ‬ଷ െ ݅ ሻଶ ൅ ‫ݐ‬ሻ ‫| ݐ‬ തതതതതതതതതതത න ሺ‫ݐ‬ ଴

଴

଴

ൌ ͳͲ െ ͺ݅Ȁ͵|

)$! | !
  ||
| න ሺ‫ ݔ‬െ ݅‫ ݕ‬ሻሺ ‫ ݖ‬൅ ݅ ‫ ݕ‬ሻ ൌ න ‫ ݔ ݔ‬൅ ‫ ݕ ݕ‬൅ ݅ න ‫ ݕ ݔ‬െ ‫| ݔ ݕ‬ ௖

?
  |

௖

௖




c
      | !|
|  
||| ‫ ݔ‬ൌ ‫ ݕݐ݁;ݐ‬ൌ ‫ ݐ݁ ݐ‬ൌ Ͳ‫ ݐ‬ൌ ʹ| !
  |

| || ଶ

ଶ

௧ୀ଴

௧ୀ଴

න ሺ‫ ݐ‬ଶ ሻሺʹ‫ݐ ݐ‬ሻ ൅ ሺ‫ݐ‬ሻ ‫ ݐ‬൅ ݅ න ሺ‫ ݐ‬ଶ ሻሺ ‫ݐ‬ሻ െ ሺ‫ݐ‬ሻሺʹ‫ݐ ݐ‬ሻ| ଶ

ଶ

௧ୀ଴

௧ୀ଴

ൌ  න ሺʹ‫ ݐ‬ଷ ൅ ‫ݐ‬ሻ ‫ ݐ‬൅ ݅ න ሺെ‫ ݐ‬ଶ ሻ ‫ ݐ‬ൌ ͳͲ െ

ͺ݅ | ͵

|  | |!
  ||  | න ሺ‫ ݔ‬െ ݅‫ ݕ‬ሻሺ ‫ ݖ‬൅ ݅ ‫ ݕ‬ሻ ൌ න ‫ ݔ ݔ‬൅ ‫ ݕ ݕ‬൅ ݅ න ‫ ݕ ݔ‬െ ‫| ݔ ݕ‬ ௖

௖

௖

!|
|
|7
 |<||6
|7
 | | 
 |ሺͲǡͲሻ݁‫ݐ‬ሺͲǡʹሻ|||||

|
|‫ ݔ‬ൌ Ͳǡ ‫ ݔ‬ൌ Ͳ| | |  || 
  | || න

ଶ

௬ୀ଴

ଶ

ሺͲሻሺͲሻ ൅ ‫ ݕ ݕ‬൅ ݅ න

ሺܱሻ ‫ ݕ‬െ ‫ݕ‬ሺͲሻ ൌ  න

௬ୀ଴

| | ||
|ʹ݅ǡ Ͷ ൅ ʹ݅|||‫ ݕ‬ൌ ʹǡ ‫ ݕ‬ൌ Ͳ|)| න

ସ

௫ୀ଴

‫ ݔ ݔ‬൅ ʹǤͲ ൅ ݅ න

ଶ

௫ୀ଴

ସ

ଶ

௬ୀ଴

‫ ݕ ݕ‬ൌ ʹ|

ସ

‫ݔ‬Ǥ Ͳ െ ʹ ‫ ݔ‬ൌ  න ‫ ݔ ݔ‬൅ ݅ න െʹ ‫ ݔ‬ൌ ͺ െ ͺ݅ | ଴

଴

| |

M h   
|

!($*),' | න ܿ‫݃ݐ݋‬ሺʹ‫ ݖ‬൅ ͷሻ ‫ ݔ‬ൌ න

…‘•ሺʹ‫ ݖ‬൅ ͷሻ ‫ݖ‬ | •‹ሺʹ‫ ݖ‬൅ ͷሻ

|,| |‫ ݑ‬ൌ •‹ሺʹ‫ ݖ‬൅ ͷሻ ฺ ‫ ݑ‬ൌ ʹ …‘•ሺʹ‫ ݖ‬൅ ͷሻ  ‫•‘…| |||ݖ‬ሺʹ‫ ݖ‬൅ ͷሻ  ‫ ݖ‬ൌ  ֜න

ௗ௨ ଶ

|

ͳ ͳ …‘•ሺʹ‫ ݖ‬൅ ͷሻ ‫ݑ ͳ ݖ‬ ൌ න ൌ  ‫ ݑ݃݋ܮ‬൅ ‫ ܥ‬ൌ ‫‹• ݃݋ܮ‬ሺʹ‫ ݖ‬൅ ͷሻ ൅ ‫| ܥ‬ ʹ ‫ݑ‬ ʹ ʹ •‹ሺʹ‫ ݖ‬൅ ͷሻ

()&)&) |

?
  |




c
      |

න ‫ܨ‬ሺ‫ݖ‬ሻ‫ ܩ‬ᇱ ሺ‫ݖ‬ሻ ‫ ݖ‬ൌ ‫ ܨ‬ሺ‫ݖ‬ሻ‫ ܩ‬ሺ‫ݖ‬ሻ െ න ‫ ܨ‬ᇱ ሺ‫ݖ‬ሻ‫ܩ‬ሺ‫ݖ‬ሻ ‫| ݖ‬

Ex  <|   ‫ ݁ݖ ׬‬ଶ௭ ‫| ݖ‬

,||‫ܨ‬ሺ‫ݖ‬ሻ  ൌ ‫ݖ‬ǡ ‫ܩ‬ǯሺ‫ݖ‬ሻ  ൌ  ݁ ଶ௓ |

‫ܨ‬ǯሺ‫ݖ‬ሻ  ൌ ͳ‫ܩ‬ሺ‫ݖ‬ሻ  ൌ Φ݁ ଶ௓ |

| - || න ‫ ݁ݖ‬ଶ௭ ‫ ݖ‬ൌ න ‫ܨ‬ሺ‫ݖ‬ሻ‫ ܩ‬ᇱ ሺ‫ݖ‬ሻ ‫ ݖ‬ൌ ‫ ܨ‬ሺ‫ݖ‬ሻ‫ ܩ‬ሺ‫ݖ‬ሻ െ න ‫ ܨ‬ᇱ ሺ‫ݖ‬ሻ‫ܩ‬ሺ‫ݖ‬ሻ ‫|ݖ‬ ͳ ͳ ൌ  ሺ‫ݖ‬ሻ ൬ ݁ ଶ௭ ൰ െ න ͳǤ ݁ ଶ௭ ‫ݖ‬ ʹ ʹ ͳ ͳ ൌ  ‫ ݁ݖ‬ଶ௭ െ ݁ଶ௭ ൅ ‫ܥ‬ Ͷ ʹ

| <|   |‫׬‬

ௗ௭

௭ మ ା௔మ

|

,|||| ͳ ͳ ͳ ͳ ͳ ൌ ൌ ൬ െ ൰| ‫ ;ݖ‬൅ ܽ; ሺ‫ ݖ‬െ ܽ݅ ሻሺ‫ ݖ‬൅ ܽ݅ሻ ʹܽ݅ ሺ‫ ݖ‬െ ܽ݅ሻ ‫ ݖ‬൅ ܽ݅

න

‫ݖ‬ଶ

‫ݖ‬ ‫ݖ‬ ͳ ͳ ͳ ͳ ‫ݖ‬ න න ൌ െ ൌ ‫݃݋ܮ‬ሺ‫ ݖ‬െ ܽ݅ ሻ െ ‫݃݋ܮ‬ሺ‫ ݖ‬൅ ܽ݅ሻ ൅ ‫ܥ‬ଶ | ଶ ʹܽ݅ ‫ ݖ‬െ ܽ݅ ʹܽ݅ ‫ ݖ‬൅ ܽ݅ ʹܽ݅ ʹܽ݅ ൅ܽ ൌ

ሺ‫ ݖ‬െ ܽ݅ሻ ͳ ‫݃݋ܮ‬ሺ ൅ ‫ܥ‬ଶ  ሺ‫ ݖ‬൅ ܽ݅ሻ ʹܽ݅

| <|   |‫݁ ׬‬௔௭ •‹ ܾ‫ ݁ ׬ ݐ݁ݔ‬௔௭ …‘• ܾ‫|ݔ ݔ‬

| ,|  |
||| න ݁ ሺ௔ା௜௕ሻ௫ ‫ ݔ‬ൌ

݁ ሺ௔ା௜௕ሻ௫ | ܽ ൅ ܾ݅

,|  |
|
||

?
  |




c
      | න ݁ ௔௫ ሺܿ‫ ݔܾݏ݋‬൅ ݅‫ݔܾ݊݅ݏ‬ሻ ‫| ݔ‬ ݁ ௔௫ ሺܿ‫ ݔܾ ݏ݋‬൅ ݅ •‹ ܾ‫ݔ‬ሻሺܽ െ ܾ݅ሻ ݁௔௫ ሺܿ‫ ݔܾ ݏ݋‬൅ ݅ •‹ ܾ‫ݔ‬ሻ ൌ  ܽ ൅ ܾ݅ ܽ; ൅ ܾ; ฺ න ݁ ௔௫ …‘• ܾ‫ ݔ ݔ‬ൌ ฺ න ݁ ௔௫ ‫ ݔ ݔܾ݊݅ݏ‬ൌ

݁௔௫ ሺƒ…‘• ܾ‫ ݔ‬൅ ܾ‫ݔܾ݊݅ݏ‬ሻ | ܽ; ൅ ܾ;

݁௔௫ ሺܽ‫ ݔܾ݊݅ݏ‬െ ܾܿ‫ݔܾݏ݋‬ሻ | ܽ; ൅ ܾ;

 | ,|| 

|  ||| |
|
  ||
|  ||||||||||
  |  |' | 
| |
|
||| 
 |||| | |*|*|
|  ||| 
 |&| ||  
|| |


 | | |

| | @ ||

|
|| 
 | || |

||


| |  |||A
| B|||A
||||| |
||
 |!  ||| ȁ‫ݖ‬ȁ ൏ ‫| ܯ‬

|

*-)- ))$ | ,|
|| |
||  || |
|| ||
 |
||  ||  0 || |||

?
  |




c
      | |

|||- |6௖ ݂ ሺ‫ݖ‬ሻ ‫|ݖ‬

|


)$') 
|mሺ‫ݔ‬ǡ ‫ݕ‬ሻ| |ܳሺ‫ݔ‬ǡ ‫ݕ‬ሻ||
|
| ||
|| 


|
|||
|࣬|  ||||
#|| ර m ‫ ݔ‬൅ ܳ ‫ ݕ‬ൌ ඵ ൬ ௖

߲ܳ ߲m െ ൰ ‫| ݕ ݔ‬ ߲‫ݕ߲ ݔ‬

  $&'%')$')  |
‫ܨ‬ሺ‫ݖ‬ǡ ‫ݖ‬ҧሻ||
|
||
| 


|
|| |‫ ݖ‬ൌ ‫ ݔ‬൅ ݅‫ݖ|ݕ‬ ഥ ൌ ‫ ݔ‬െ ݅‫|ݕ‬ డி

!| ||8|
||‫׬‬௖ ‫ܨ‬ሺ‫ݖ‬ǡ ‫ݖ‬ҧሻ ‫ ݖ‬ൌ ʹ݅ ‫ ׭‬డ௭ ‫||||ܣ‬,)| ‫ ܣ‬ൌ  ‫| ݕ ݔ‬

!).$ !" 
||
| 
|||  ||×| |||
| || ර ݂ ሺ‫ݖ‬ሻ ‫ ݖ‬ൌ Ͳ| ௖

!).$)) 
|24||
|
|||  ||×|| ||6௖ ݂ሺ‫ݖ‬ሻ ‫ ݖ‬ൌ Ͳ| "|  ||

|
 ||×| |24| | 
||×Ǥ|

()'++
|݂ሺ‫ݖ‬ሻ| |‫ܨ‬ሺ‫ݖ‬ሻ| | 
|||  ||×| | 

|‫ܨ‬ǯሺ‫ݖ‬ሻ ൌ ݂ሺ‫ݖ‬ሻ| |‫ܨ‬ሺ‫ݖ‬ሻ| |   |
  |


| ‫ܨ‬ሺ‫ݖ‬ሻ ൌ  න ݂ሺ‫ݖ‬ሻ ‫|ݖ‬

| |

?
  |




c
      |

,'/'/+&)$*+ න …‘–Š ‫ ݖ ݖ‬ൌ ‫|ݖ݄ݏ݃݋ܮ‬

|

න ୬ †œ ൌ න

 | ൅ͳ ୬ାଵ

න

†œ ൌ ‘‰ሺœሻ| œ

න ܽ ௭ ‫ ݖ‬ൌ

‫ݖ‬ ൌ െܽ‫|ݖ݄ ܿ݃ݐ݋ܿܿݎ‬ ‫ݖ݄ݏ‬

න

න ݁ ௭ ‫ ݖ‬ൌ ݁ ௭ |

න

ܽ௭ | ‫ܽ݃݋ܮ‬

න

න •‹ ‫ ݖ ݖ‬ൌ െ …‘• ‫|ݖ‬

න

න …‘• ‫ ݖ ݖ‬ൌ •‹ ‫|ݖ‬ න ‫ ݖ ݖ ݃ݐ‬ൌ  െ‫|ݖ •‘… ݃݋ܮ‬ න ܿ‫ ݖ ݖ݃ݐ݋‬ൌ ‫|ݖ ‹• ݃݋ܮ‬ න

න

න

‫ݖ‬ ൌ ‫|ݖ݃ݐ‬ ܿ‫ݏ݋‬ଶ ‫ݖ‬

‫ݖ‬ ൌ െܿ‫|ݖ݃ݐ݋‬ ‫݊݅ݏ‬ଶ ‫ݖ‬

‫ݖ݃ݐ‬ ͳ න ‫ ݖ‬ൌ | …‘• ‫ݖ‬ …‘• ‫ݖ‬

ͳ ܿ‫ݖ݃ݐ݋‬ න ‫ ݖ‬ൌ െ | •‹ ‫ݖ‬ •‹ ‫ݖ‬ න ‫ ݖ ݖ ݄ݏ‬ൌ ݄ܿ‫|ݖ‬ න ݄ܿ‫ ݖ ݖ‬ൌ ‫|ݖ݄ݏ‬ න ‫ ݖ ݖ݄ݐ‬ൌ ‫|ݖ݄ܿ݃݋ܮ‬

|

?
 

‫ݖ‬

ඥ‫ ݖ‬ଶ

න

‫ݖ‬ଶ න

න

‫ݖ‬

‫ݖ‬ඥܽଶ

න

േ

‫ݖ‬ ൌ ‫|ݖ݄ݐ‬ ݄ܿଶ ‫ݖ‬

‫ݖ‬ ൌ െ …‘–Š ‫|ݖ‬ ‫ ݄ݏ‬ଶ

ͳ ‫ݖ݄ݐ‬ ‫ ݖ‬ൌ  െ | ݄ܿ‫ݖ‬ ݄ܿ‫ݖ‬

ͳ …‘–Š ‫ݖ‬ ‫ ݖ‬ൌ െ | ‫ݖ݄ݏ‬ ‫ݖ݄ݏ‬ ൌ ‫ ݃݋ܮ‬ቀ‫ ݖ‬൅ ඥ‫ ݖ‬ଶ േ ܽଶ ቁ|

േ ܽଶ

න

ͳ ‫ݖ‬ ‫ݖ‬ ൌ ‫ ݃݋ܮ‬൬ െ ܿ‫ݖ݃ݐ݋‬൰ ൌ ‫ ݃ݐ݃݋ܮ‬ቀ ቁ| •‹ ‫ݖ‬ ʹ •‹ ‫ݖ‬ න

|

න

ͳ ‫ߨ ݖ‬ ‫ݖ‬ ൌ ‫ ݃݋ܮ‬൬ ൅ ‫ݖ݃ݐ‬൰ ൌ ‫ ݃ݐ݃݋ܮ‬ቀ ൅ ቁ| …‘• ‫ݖ‬ ʹ Ͷ …‘• ‫ݖ‬

න

‫ݖ‬ ൌ ܽ‫|ݖ݄ܿ݃ݐܿݎ‬ ݄ܿ‫ݖ‬

ͳ ‫ݖ‬ ‫ݖ‬ ൌ ܽ‫| ݃ݐܿݎ‬ ܽ ‫ ݖ‬ଶ ൅ ܽଶ ܽ

ͳ ‫ݖ‬െܽ ‫ݖ‬ ൌ ‫ ݃݋ܮ‬ቀ ቁ| ଶ ʹܽ ‫ݖ‬൅ܽ െܽ ‫ݖ‬

ξ‫ ݖ‬ଶ ‫ݖ‬ଶ

െ

ൌ

‫ݖ‬

‫ݖ‬ඥ‫ ݖ‬ଶ

ܽଶ

‫ݖ‬ ൌ ܽ‫| ‹• ܿݎ‬ ܽ

‫ݖ‬ ͳ ቇ| ‫ ݃݋ܮ‬ቆ ܽ ܽ ൅ ඥܽଶ േ ‫ ݖ‬ଶ

േ

ܽଶ

ൌ

ͳ ܽ ܽ‫| •‘… ܿݎ‬ ܽ ‫ݖ‬

‫ݖ‬ න ඥ‫ ݖ‬ଶ േ ܽଶ ‫ ݖ‬ൌ ඥ‫ ݖ‬ଶ േ ܽ ଶ ܽ ܽ ൌ ‫ ݃݋ܮ‬ቀ‫ ݖ‬൅ ඥ‫ ݖ‬ଶ േ ܽଶ ቁ| ʹ

‫ݖ‬ ܽଶ ‫ݖ‬ න ඥܽ ଶ േ ‫ݖ‬ଶ  ‫ ݖ‬ൌ ඥܽ ଶ േ ‫ ݖ‬ଶ ൅ ܽ‫| ‹• ܿݎ‬ ʹ ܽ ‫ݖ‬ න ݁ ௔௭ •‹ ܾ‫ ݖ ݖ‬ൌ න ݁ ௔௭ …‘• ܾ‫ ݖ ݖ‬ൌ

݁ ௔௭ ሺƒ•‹ ܾ‫ݖ‬Ȃ ܾܿ‫ݖܾݏ݋‬ሻ | ܽ ଶ ൅ ܾଶ

݁ ௔௭ ሺƒ…‘• ܾ‫ ݖ‬൅ ܾ‫ݖܾ݊݅ݏ‬ሻ ܽଶ ൅ ܾଶ




c
      | | |

+ +/ +!).$)  Th   A |

| ,| 
#| | | | 
 | | ܿ| 

| | 

|   | | | 
| 
|5|  || ||| 
 | || 

# ||||| |!| ||
| ž
|  | 
 
|| |||
|࣬ଵ  | ࣬ଶ |
| ||  |
|| | || 8| | | 

| | ||8|
 | ߲ܳ ߲m ሺͳሻ න m ‫ ݔ‬൅ ܳ ‫ ݕ‬ൌ ඵ ൬ െ ൰  ‫| ݕ ݔ‬ ߲‫ݕ߲ ݔ‬ ࣬భ

ௌ்௎ௌ

߲ܳ ߲m ሺʹሻ න m ‫ ݔ‬൅ ܳ ‫ ݕ‬ൌ ඵ ൬ െ ൰  ‫| ݕ ݔ‬ ߲‫ݕ߲ ݔ‬ ௌ௏்ௌ

࣬మ

"|

|| 
|||ሺͳሻ| |ሺʹሻ| ||

 | |
|m ‫ ݔ‬൅ ܳ ‫|||| ݕ‬ න ൅ න ൌ න൅ න൅ න൅ න ൌ න൅ න ൌ

ௌ்௎௏

ௌ௏்ௌ

ௌ்

்௎ௌ

ௌ௏்

்ௌ

்௎ௌ

Ž̵ƒ‹†‡†‡ න ൌ െ න

ௌ௏்

න

|

்௎ௌ௏்

|

்ௌ

ௌ்

&| |'|0| |

|||||ሺͳሻ| |ሺʹሻ| ඵ൅ඵ ൌ ඵ

࣬భ

?
  |

࣬మ

|

࣬




‫     
‪c‬‬ ‫|‬ ‫|)&‬ ‫‪߲ܳ ߲m‬‬ ‫| ݕ ݔ  ‪െ ൰‬‬ ‫ݕ߲ ݔ߲‬

‫‪ ൌ ඵ ൬‬ݕ ܳ ‪ ൅‬ݔ ‪න m‬‬ ‫࣬‬

‫்‪்௎ௌ௏‬‬

‫| ‪|
| | | #‬‬ ‫|
|
| |||  
|   | |‪.| | | | ||8‬‬ ‫| ܿ| 
||||| |

‬ ‫|
|||| ||

|
| |
|
| |
||| |&‬ ‫|  |

| |  |
ž||  ‬ ‫| | 
|
|||| ||  |   | |‪!| ||8‬‬

‫‪Th   B‬‬ ‫|| | |||||||‪$ | | 
|| | ||8‬‬ ‫| | 
 |‪ 

# |||||| ||6‬‬ ‫|‬

‫|‬ ‫|| 
|  || |
|   | |‪.|

| || | ||8‬‬ ‫| 
|
| ||  |
 ||  | ࣬‬

‫|
| |ܦܩܨܧܦ |
 
|‪ || || |
#‬ܣܮܭܬܪܣ |‪!|
#|| ࣬||| |
#‬‬ ‫| |
 
 |
 | | | | ‪| | | 
 | | 

| 0| |   | |  0‬‬ ‫| |||
‬ ‫|  |
  |‪| | 
 | | 
#‬ܦܣ| |

 |
| |
 |‪"|  
| | #‬‬ ‫|
 ||  |||  
| | ܣܪܬܭܮܣܦܩܨܧܦܣ|‪
 
 |!  || |
#‬‬ ‫|‪
| | #||8‬‬ ‫‪߲ܳ ߲m‬‬ ‫| ݕ ݔ ‪ ൌ ඵ ൬ െ ൰‬ݕ ܳ ‪ ൅‬ݔ ‪m‬‬ ‫ݕ߲ ݔ߲‬ ‫࣬‬

‫‪ර‬‬

‫‪஺஽ாிீ஽஺௅௄௃ு஺‬‬

‫|||
|  |||||  
!‬

‫
‬

‫ 
?‬ ‫|‬

c
      | න൅

஺஽

න ൅ න൅

஽ாிீ஽

஽஺

න

஺௅௄௃ு஺

ൌ

න ൅

஽ாிீ஽

න

|

஺௅௄௃ு஺

ܿଵ| 
| | | ‫ܿ | | ܣܪܬܭܮ‬ଶ | 
| ‫
 | | ܿ |  |ܦܩܨܧܦ‬#| | ࣬5|

m‫׬ ݁ݑݍݏ݅ݑ‬஺஽ ൌ െ ‫׬‬஽஺

 ||

߲ܳ ߲m ර m ‫ ݔ‬൅ ܳ ‫ ݕ‬ൌ ඵ ൬ െ ൰ ‫| ݕ ݔ‬ ߲‫ݕ߲ ݔ‬ ࣬

௖

Th   C 
|mሺ‫ݔ‬ǡ ‫ݕ‬ሻ| |ܳሺ‫ݔ‬ǡ ‫ݕ‬ሻ||
#|
| ||
| 


| 
#|
||  | 
 ||  |࣬|
  | |

J ||

|
| |
 | ||6௖ m ‫ ݔ‬൅ ܳ ‫ ݕ‬ൌ Ͳ| |  | | |||࣬| ||| ߲m ߲ܳ ൌ | ߲‫ݔ߲ ݕ‬ <| condition suffisante డ௉

,| ||

డ௬

ൌ

డொ డ௫

||| | | ||8|| ර m ‫ ݔ‬൅ ܳ ‫ ݕ‬ൌ ඵ ൬ ௖

࣬

‫݋‬î࣬  |   |

| | | <| condition nécessaire

߲ܳ ߲m െ ൰ ‫ ݕ ݔ‬ൌ Ͳ| ߲‫ݕ߲ ݔ‬ డ௉

,| ||6௖ m ‫ ݔ‬൅ ܳ ‫ ݕ‬ൌ Ͳ||  | ||||࣬| || ൌ డ௬ డ௉

|࣬ |" | 
 | || డ௉

డ௬

డொ

െ

డொ డ௫

൐ Ͳ|| 
 |ሺ‫ݔ‬଴ǡ ‫ݕ‬଴ ሻ|

డொ డ௫

||| 
 |

"|  #|డ௬ | | డ௫ | |
||࣬|
|
|
|
 ||  ||Ȟ| | డ௉

డொ

ሺ‫ݔ‬଴ ǡ ‫ݕ‬଴ ሻ| | 
 |
 
| |  ||డ௬ െ డ௫ ൐ Ͳ |
|Ȟ|| | |
#||߬ | | | |

 ||8|

ර m ‫ ݔ‬൅ ܳ ‫ ݕ‬ൌ ඵ ൬ ୻

߲ܳ ߲m െ ൰ ‫ ݕ ݔ‬൐ Ͳ| ߲‫ݕ߲ ݔ‬

…‡
| | 
||   #6௖ m ‫ ݔ‬൅ ܳ ‫ ݕ‬ൌ Ͳ| |  ||||࣬ | డொ

డ௫ డொ డ௫

െ

డ௉ డ௬ డ௉

|  ||' | 

|5|| |'|0||  | ||

െ డ௬ ||  |' |
||

,||
||

| 
|
|' |

 |

| 
|| ߲m ߲ܳ ൌ | ߲‫ݔ߲ ݕ‬ !|

 |   |' | ||
|


 | |

?
  |




c
      |

Th   D |

EXERCICES <|   ||"| || 
 | ‫݃ݐ݋ܿ ׬|||ݖ ݖ •‘… ݖ ‹• ׬||||| ݖ ݖ͵ •‘… ݖ͵ ‹• ׬‬ሺʹ‫ ݖ‬൅ ͷሻ ‫|ݖ‬ <| "|
 
| | 
| ଵ

ௗ௭

‫׬‬଴ ‫ ݖݖ‬ଶ௭ ‫ ׬|||ݖ‬௭;ା௔;| <|   |6௖ ȁ‫ݖ‬ȁ; ‫||| | |ݖ‬ሺͲǡͲሻǢሺͳǡͲሻǢሺͳǡͳሻǢሺͲǡͳሻ| ௗ௭

<| $  |6௖ | | ௭;ିଶ

| || |ȁ‫ ݖ‬െ ʹȁ ൌ Ͷ|

| |||||||||||ʹ ൅ ʹ݅ǡ െʹ ൅ ʹ݅|

<|   |‫׬‬௖ ሺ‫ ݖ‬ଶ ൅ ͵‫ݖ‬ሻ ‫|ݖ‬

| || |ȁ‫ݖ‬ȁ ൌ ʹ||ሺʹǡͲሻሺͲǡʹሻ|| ||
 |

!| || ||
|7
 |ሺʹǡͲሻሺͲǡʹሻ|

!| || |   || | | ||
|7
 | ሺʹǡͲሻሺʹǡʹሻ݁‫ݐ‬ሺʹǡ ʹሻሺͲǡʹሻ| ଶି௜

<|   |‫׬‬௜

ሺ͵‫ ݕݔ‬൅ ݅‫ ݕ‬ଶ ሻ ‫| ݖ‬

!| ||
|7
 |‫ ݖ‬ൌ ݅ ‫ ݖ‬ൌ ʹȂ ݅ |

!| || ||‫ ݔ‬ൌ ʹ‫ݐ‬Ȃ ʹ| ௗ௭

ଵ

‫ ݕ‬ൌ ͳ ൅ ‫ݐ‬Ȃ ‫|;ݐ‬

ሺ௭ି௔௜ሻ

ൌ ‫݃݋ܮ‬ሺሺ௭ା௔௜ሻ ൅ ‫ܥ‬ଶ | <| J ||‫׬‬ ଶ௔௜ ௭;ା௔;

|

Th   1 | ௭


||| || || 
 | ||1| |‫׬‬௔ ݂ ሺ‫ݖ‬ሻ ‫| |

|
||  
| ݖ‬ 

|||| |

?
  |




c
      | |

Th   2 | ௭


|| || || 
 | ||1| |
|824|;‫׬‬௔ ݂ሺ‫ݖ‬ሻ ‫| |ݖ‬24| | 
||1| | 824|;|24 |

Th   3 | ௕


||| || || 
 | ||1| |
|24|;|24| |‫׬‬௔ ݂ ሺ‫ݖ‬ሻ ‫ ݖ‬ൌ ‫ ܨ‬ሺܾሻ െ ‫ܨ‬ሺܽሻ|

Th   4 | 
|24||
| 
|||  ||

| ||||
 ||  |6|)|6| || 
 
||||| | 
|| |
 | |

- |‫׬‬஼ ݂ሺ‫ݖ‬ሻ ‫ ݖ‬ൌ ‫׬‬஼ ݂ ሺ‫ݖ‬ሻ ‫|ݖ‬ మ

| | | | | | |

Th   5 |

?
  |




c
      | 
|24||
| 
|||  ||

| | |||
 || |  | ||+|6|>|)|+|6|>| |
 
||| || |

6஼ ݂ ሺ‫ݖ‬ሻ ‫ ݖ‬ൌ  6஼ ݂ሺ‫ݖ‬ሻ ‫ ݖ‬൅ 6஼ ݂ ሺ‫ݖ‬ሻ ‫ ݖ‬൅ ‫ ڮ‬൅ 6஼ ݂ ሺ‫ݖ‬ሻ ‫|ݖ‬ భ

మ

೙

| ||$  |

ර

஼

‫ݖ‬ ǡ ݊ ൌ ʹǡ͵ǡͶǡ ǥ | ሺ‫ ݖ‬െ ܽሻ

,)||;|| |
 
|| ||
 || | ଶగ

ൌර

଴

ଶగ ݅ߝ݁ ௜ఏ ߠ ݅ න ݁ ሺଵି ൌ  ߝ ݁௜ ఏ ߝ ିଵ ଴

ൌ

ͳ ሺͳ െ ݊ሻߝ

ିଵ

ሻ௜ఏ

ൣ݁ ଶሺଵି

ଶగ

݁ ሺଵି ሻఏ ቉ | ߠ ൌ ିଵ ቈ ሺͳ െ ݊ ሻ݅ ߝ ଴

ሻగ௜ିଵ

݅

൧ ൌ Ͳ|

Ex  |
|| | |||
||‫ ݕ‬ൌ ‫ ݔ‬ଷ െ ͵‫ ;ݔ‬൅ Ͷ‫ ݔ‬െ ͳ|7
 | | 
 |2++4| |2634 |@ || $  |6௖

ௗ௭

௭ି௔

||)||
||||
 | ||;|| ||

-|  
||| -| 
 
||| |
|| ||  
||| |24|;|

#|||6஼ ݂ሺ‫ݖ‬ሻ ൌ Ͳ|

ଵ

ሺ௭ି௔ሻ

| | 
|| 
 
||| ||| | #| |

 ||
 
||||Ȟ|| |||{| |||;||  ||Ȟ|
|| 
 
|| |

?
  |




c
      |

‫ݖ‬ ‫ݖ‬ ൌර | ‫ݖ‬െܽ ୻ ‫ݖ‬െܽ

ර

஼

|Ȟ||ȁ‫ ݖ‬െ ܽ ȁ ൌ ߝ |)||‫ ݖ‬െ ܽ ൌ ߝ݁ ௜ఏ ֜ ‫ ݖ‬ൌ ܽ ൅ ߝ݁ ௜ఏ ǡ Ͳ ൑ ߠ ൑ ʹߨ|

&)|  | || ‫ ݖ‬ൌ ݅ߝ݁ ௜ఏ |

ଶగ ݅ߝ݁௜ఏ ߠ න ൌ ݅ ߠ ൌ ʹߨ݅ | ௜ఏ ఏୀ଴ ߝ݁ ଴

න

ଶగ

| න ሺͳʹ‫ݖ‬ଶ െ Ͷ݅‫ݖ‬ሻ ‫|ݖ‬ ஼

ñ



න

ଶାଷ௜

ଵା௜

ß



ଶାଷ௜ ሺͳʹ‫ ݖ‬ଶ െ Ͷ݅‫ݖ‬ሻ ‫ ݖ‬ൌ ሺͶ‫ ݖ‬ଷ െ ʹ݅‫ݖ‬ଶ ሻሾሺͶ‫ ݖ‬ଷ െ ʹ݅‫ݖ‬ଶ ሻሿଵା௜ ൌ െͳͷ͸ ൅ ͵ͺ݅ |





!|
|7
 |2++4||2634|  |' | ||2++4||26+4| 
|26+4||2634| +||| | || | 
|2++4| |26+4||;|+||;|<|
|
||||||||||||||;||C
|;||C|
||| |;|| ଶ

න ሾͳʹሺ‫ ݔ‬൅ ʹሻଶ െ Ͷ݅ሺ‫ ݖ‬൅ ݅ ሻሿ ‫ ݖ‬ൌ ሾͶሺ‫ ݔ‬൅ ͵ሻଷ െ ʹ݅ ሺ‫ ݔ‬൅ ݅ ሻଶሿଶଵ ൌ ʹͲ ൅ ͵Ͳ݅ | ௭ୀଵ

6#||| | ||
|26+4||2634||;|6||;|<|
|
|||;||C|
|;|6|C|
||;|
| න

ଷ

௬ୀଵ

ሾͳʹሺʹ ൅ ݅‫ݕ‬ሻଶ െ Ͷ݅ሺʹ ൅ ݅‫ݕ‬ሻሿ ‫|ݕ‬

| | | | | | | | |

?
  |




c
      |

    
   | J || න

ஶ

଴

‫ݔ‬

ξ‫ݔ‬ሺͳ ൅ ‫ݔ‬ሻ

,|
#| 
  |

||6஼

ௗ௭

ξ௭ሺଵା௭ሻ

ൌ ߨ|

|)|| | | |


 || |
|

|

{



  


  
{



  
 


{










|

!|
||
 | ||
|


| ||;|<| || 
 || | ||| 

|| || || 
  |
|| |
|

| | || 
 
 || !| 
 ||;|+| | | |
 
|| 
 
|||
|
|| 
 |
 ||  | |
|  || ܴ±‫ݏ‬௭ୀିଵ‫ ‹Ž ׷‬ቈሺ‫ ݖ‬൅ ͳሻǤ ௭՜ିଵ

ͳ

ഏ

቉ ൌ ݁ ି௜మ ൌ െ݅ | ξ‫ݖ‬ሺ‫ ݖ‬൅ ͳሻ

& #|| | #||
|||| ර

஼

‫ݖ‬

ξ‫ݖ‬ሺ‫ ݖ‬൅ ͳሻ

ൌ ʹߨ|

$|  |
  | |

‫ ݖ‬ൌ ‡š’ሺ݅߮ ሻ ǡ Ͳ ൏ ߮ ൏ ʹߨ|| 
|| 
  |D{| ଵ

ξ௭ሺ௭ାଵሻ

| | || |{|||{| | || |{l| ƒ ൌ െන

ଶగ

ߝ݅݁ ௜ఝ  ߮ ೔ക

ξߝ݁ మ ሺͳ ൅ ߝ݁௜ఝ ሻ

଴

ൌ ξߝ݅ න

&|0| || 

|| 
  |D1|| 
| || |1l>| ƒ
ൌ ξܴ݅ න

ଶగ

଴

?
  |

ଶగ

଴

ଵ

ξ௭ሺ௭ାଵሻ

݁௜ఝȀଶ ߮ ՜ Ͳ‫ ߝ݅ݏ‬՜ Ͳ| ͳ ൅ ߝ݁ ௜ఝ

| | || |1| | || |

݁ ௜ఝȀଶ ͳ ଶగ ݁ ି௜ఝȀଶ න ߮ ൌ ߮ ՜ Ͳ‫ ܴ݅ݏ‬՜ λ| ͳ ൅ ܴ݁ ௜ఝ ξܴ ଴ ͳ ൅ ݁ ି௜ఝȀଶ




c
      |
!| || | 
|ሾߝǡ ܴ ሿ| 
  |
 ||‫׬‬

ௗ௫

ξ௫ሺଵା௫ሻ

|

!| || |

||ሾߝǡ ܴሿ| #|| |  | || 

| 
|ξ‫ ݔ‬ൌ െξ‫| ݔ‬ െන

‫ݔ‬




ሺെξ‫ݔ‬ሻሺͳ ൅ ‫ݔ‬ሻ

|

-

||  | |{ll>| ஶ

6‫׬‬଴

ௗ௫

ξ௫ሺଵା௫ሻ

ൌ 6஼

ௗ௭

ξ௭ሺଵା௭ሻ

ൌ ʹߨ|

$ ||  ||
  |


| ஶ ୱ୧୬ ௫

J ||‫׬‬଴

௫

గ

ൌ ଶ|

,|  |
| 
  |

||6஼

௘ ೔೥ ௭

‫|ݖ‬

,)|| | | |


 || |
 |+ |D || ||
 |1|||1||
 | {|||{| || |ൣȂ ܴǡ െߝ൧݁‫ݐ‬ሾߝǡ ܴ ሿǤ| " | 
 | |

!| 
 ||;|<| || E |
 || |
||
 ||


||

 || | | "
| |
| |2
4F| | 
|| | |
||

| | |,|| | |

#|||| ݁ ௜௭ ‫ ݖ‬ൌ Ͳ| ‫ݖ‬

ර

஼

ି ௘ ೔ೣ

,||‫
ି׬‬

௫

‫ ݔ‬൅ ‫׬‬஼

ഄ

௘ ೔೥ ௭


௘ ೔ೣ

‫ ݖ‬൅ ‫׬‬

௫

‫ ݔ‬൅ ‫׬‬஼

ೃ

௘ ೔೥ ௭

‫ ݖ‬ൌ Ͳ| |

$| |||*|| | 
#|
  | ||
 | | | 

#||  | න

?
  |




݁ ௜௫ െ ݁ ି௜௫ ݁௜௭ ݁ ௜௭ ‫ ݔ‬൅ න ‫ ݖ‬൅ න ‫ ݖ‬ൌ Ͳ| ‫ݔ‬ ஼ഄ ‫ݖ‬ ஼ೃ ‫ݖ‬




c
      | $| |||*|| | 
#|
  | ||
 | | | 

#||  |
| | |‫ ݖ‬ൌ ߝ‡š’ሺ݅߮ሻ| | ||{||
 | | |
#|
  || ݁

గ ௜ ሺ௖௢௦ఝା௜௦௜ ఝሻ

െන

଴

ߝ݁௜ఝ

݅ߝ݁ ௜ఝ ߮ ՜ െʹߨ݈‫ ߝ݁ݑݍݏݎ݋‬՜ Ͳ|

" ||‫ ݖ‬ൌ ܴ ‡š’ሺ݅߮ሻ| | ||1||  | | |  | || | 

#|
  || గ

ቤන

଴

݁ ௜
ሺ௖௢௦ఝା௜௦௜ ܴ݁ ௜ఝ

ఝሻ

గ

ܴ݅ ߮ቤ ൑ න ݁ ି
௦௜ ଴

ఝ

߮ |

|
| | ||
|1l> 
||
 | |{G<| |1GH| ݁

ஶ ௜௫

න ஶ ୱ୧୬ ௫

&)|‫׬‬଴

௫

଴

గ

െ ݁ ି௜௫ ‫ ݔ‬െ ݅ߨ ൌ Ͳ| ‫ݔ‬

‫ ݔ‬ൌ | ଶ

| ஶ ௖௛௔௫

J |||‫׬‬଴

௖௛௫

‫ ݔ‬ൌ

గ

ଶ ୡ୭ୱሺ

ഏೣ ሻ మ

‫݋‬îȁܽȁ ൏ ͳ|

| ,|
#|||||||||||||||||||||||||||||||||||||||| 6஼ ߨ݅ǡ െܴ ൅ ߨ݅ |

௘ ೌ೥

௖௛௭

‫݋ݖ‬î‫ ݏݐ݁݉݉݋ݏݏ݁ ݈݁݃݊ܽݐܿ݁ݎ݊ݑݐݏ݁ܥ‬െ ܴǡ ܴǡ ܴ ൅

| 
|3 | | | | !| E |

௘ ೌ೥

௖௛௭

ଵ

|| |
 | | | | ||||;|<‫ ݖ‬ൌ ቀ݊ ൅ ቁ ߨ݅ǡ| ݊ ൌ Ͳǡ േͳǡ േʹǡ ǥ|

ଶ

| ௜గ

!| || E |
|| 
 
||| | ଶ Ǥ| !|
|||

௘ ೌ೥

௖௛௭

||‫ ݖ‬ൌ

௜గ ଶ

| ||

|

?
  |




c
      | ሺ‫ ݖ‬െ Ž‹ ೔ഏ

௭՜

మ

݅ߨ ݁௔௭ ݁௔గ௜Ȁଶ ݁ ௔గ௜Ȁଶ ௔గ௜Ȁଶ ሻ ൌ ൌ  | గ ൌ െ݅݁ ʹ ݄ܿ‫݄ݏ ݖ‬ሺ௜గሻ ݅‫݊݅ݏ‬ሺ ሻ ଶ

ଶ

|

THEOREME DES RESIDUS | ೌഏ೔ ೌഏ೔ ݁ ௔௭ ‫ ݖ‬ൌ ʹߨ݅ ൬െ݅݁ మ ൰ ൌ ʹߨ݁ మ | ݄ܿ‫ݖ‬

ර

஼

| |
|  |
|| | గ
଴ ݁ ௔ሺ௫ାగ௜ሻ ݁ ௔ሺି
ା௜௬ሻ ݁ ௔ሺ
ା௜௬ሻ ݁௔௫ ݅ ‫ ݕ‬൅ න ‫ ݔ‬൅ න ݅ ‫ ݕ‬ൌ ʹߨ݁ ௔௜గȀଶ | ‫ ݔ‬൅ න ି
݄ܿ‫ݔ‬ ଴ ݄ܿሺܴ ൅ ݅‫ݕ‬ሻ ି
݄ܿሺ‫ ݔ‬൅ ߨ݅ሻ గ ݄ܿሺെܴ ൅ ݅‫ݕ‬ሻ

න




| I|1l>
  

!
   | 
||  | | |"|  | |
| |
#|
  || | ݁
ା௜௬ ൅ ݁ ି
ି௜௬ ͳ ͳ ͳ ȁ݄ܿሺܴ ൅ ݅‫ݕ‬ሻȁ ൌ ቤ ቤ ൒ ൣห݁
ା௜௬ ห െ ห݁ ି
ି௜௬ ห൧ ൌ ሺ݁
െ ݁ ି
ሻ ൒ ݁
| ʹ Ͷ ʹ ʹ

| గ ௘ ೌሺೃశ೔೤ ሻ

,|
|ቚ‫׬‬଴

௖௛ሺ
ା௜௬ሻ

గ ௘ ೌೃ

݅ ‫ ݕ‬ቚ ൑ ‫׬‬଴

భ ೃ ௘ ర

‫ ݕ‬ൌ Ͷߨ݁ ሺ௔ିଵሻ
|

| $ | |

 || |
| ||| ||| | |||1l>
 ȁܽȁ ൏ ͳ| |'|
#||  | || | 
#|
  || 
|||2+4|

| |||1l>"# 
$%& 
  | |
௔௫ ೌഏ೔ ݁ ݁௔௫ ‫ ݔ‬൅ ݁ ௔గ௜ න ‫ ݔ‬቉ ൌ ʹߨ݁ మ | ି
݄ܿ‫ݔ‬ ି
݄ܿ‫ݔ‬

Ž‹ ቈන


՜ஶ




| |݄ܿሺ‫ ݔ‬൅ ߨ݅ሻ ൌ െ݄ܿ‫||||ݔ‬

?
  |




c
      | | ೌഏ೔

ஶ ௔௫ ݁ ʹߨ݁ మ ݁ ௔௫ ‫ ݔ‬ൌ න ‫ ݔ‬ൌ ൌ Ž‹ න
՜ஶ ି
݄ܿ‫ݔ‬ ͳ ൅ ݁ ௔గ௜ ିஶ ݄ܿ‫ݔ‬


݁

ೌഏ೔ మ

ʹߨ

൅݁

ೌഏ೔ ି మ

ൌ

ߨ

గ௔

…‘•ሺ ሻ ଶ

|

| ଴

ஶ ௘ ೌೣ

௘ ೌೣ

&|‫ି׬‬ஶ ௖௛௫ ‫ ݔ‬൅ ‫׬‬଴

‫ ݔ‬ൌ

௖௛௫

గ

ୡ୭ୱሺ

ഏೌ ሻ మ

|

| ,|
|| |||*|| | 
#|
   | | න

ஶ

଴

ஶ ௔௫ ஶ ݁ ݄ܿܽ‫ݔ‬ ߨ ݁ ି௔௫ ‫ ݔ‬൅ න ‫ ݔ‬ൌ ʹ න ‫ ݔ‬ൌ | గ௜ ݄ܿ‫ݔ‬ ݄ܿ‫ݔ‬ ݄ܿ‫ݔ‬ ଴ ଴ ܿ‫ݏ݋‬ሺ ሻ ଶ

| ଵ ௫ ೛షభ

J ||6଴

ଵା௫

గ

‫ ݔ‬ൌ ୱ୧୬ ௣ Ͳ ൏ ‫ ݌‬൏ ͳ|

| 
|6஼

௭ ೛షభ ଵା௭

‫ݖ‬ǡ| | 
 ||;|<|  || ||

| | ||| |
 |

| | 
| | | | | | ,)| | | 

| | | | |)|-J| |8K|%
 | | |||
| | |  | ||


| 
 | | !|
|| |
 #||| E |
 ||;|+|
 
|| | |

?
  |




c
      | !|
| |‫ ݖ‬ൌ െͳ ൌ ݁గ௜ | || | Ž‹ ሺ‫ ݖ‬൅ ͳሻ

௭՜ିଵ

‫ݖ‬௣ିଵ ‫ ݖ‬ൌ ሺ݁గ௜ ሻ௣ିଵ ൌ ݁ ሺ௣ିଵሻగ௜ | ͳ൅‫ݖ‬

| ,|||6௖

௭೛షభ ଵା௭

‫ ݖ‬ൌ ʹߨ݅݁ ሺ௣ିଵሻగ௜ |

| ,||||‫׬‬஺ÿ ൅ ‫׬‬ÿ஽ாிீ ൅ ‫ீ׬‬ு ൅ ‫׬‬ு௃஺ ൌ ʹߨ݅݁ ሺ௣ିଵሻగ௜ | | ,|  ||
|| | න




ଶగ ଴ ‫ݔ‬௣ିଵ ሺܴ݁௜ఏ ሻ௣ିଵ ܴ݅݁ ௜ఏ  ߠ ሺ‫ ݁ݔ‬ଶగ௜ ሻ௣ିଵ ሺߝ݁ ௜ఏ ሻ௣ିଵ ݅ߝ݁ ௜ఏ  ߠ න න ‫ ݔ‬൅ න ൅ ‫ݔ‬ ൅ | ଶగ௜ ͳ൅‫ݔ‬ ͳ ൅ ܴ݁ ௜ఏ ͳ ൅ ߝ݁ ௜ఏ ଴
ͳ ൅ ‫݁ݔ‬ ଶగ

| ൌ ʹߨ݅݁ ሺ௣ିଵሻగ |

| |||,)| || |‫ ݖ‬ൌ ‫ ݁ݔ‬ଶగ௜ | | 
  | | ||8K|  ||| | ||6ߨ ||  | | |J&$8 || |
| | | |

||{ll>| || |
#| | | 

#|
  |

 | |||  || | ஶ

න

଴

଴ ଶగ௜ሺ௣ିଵሻ ௣ିଵ ‫ݔ‬௣ିଵ ݁ ‫ݔ‬ ‫ ݔ‬൅ න ‫ ݔ‬ൌ ʹߨ݁ ሺ௣ିଵሻగ௜ | ͳ൅‫ݔ‬ ͳ൅‫ݔ‬ ஶ

|
|
||| | න

ஶ

଴

?
  |

ʹߨ݅݁ ሺ௣ିଵሻగ௜ ʹߨ݅ ߨ ‫ ݔ‬௣ିଵ ‫ ݔ‬ൌ ൌ ൌ | ͳ െ ݁ ଶగ௜ሺ௣ିଵሻ ݁ ௣గ௜ െ ݁ ି௣గ௜ •‹ ‫ߨ݌‬ ͳ൅‫ݔ‬




c
      | | ஶ ௅௢௚ሺ௫ାଵሻ

J ||‫׬‬଴

,|
#|6஼

௫;ାଵ

௅௢௚ሺ௭ା௜ሻ ௭;ାଵ

‫ ݔ‬ൌ ߨ‫| ʹ݃݋ܮ‬ ‫*|| | ||
 ||| || | | ݖ‬1||C1| ||


 |Ȟ|||1 | 
| | | | | | ௅௢௚ሺ௭ା௜ሻ

!| | E |

௭;ାଵ

||
 
||| | | E |
 ||;|
| | |
| | Ž‹ሺ‫ ݖ‬൅ ݅ሻ ௭՜௜

‫݃݋ܮ‬ሺ‫ ݖ‬൅ ݅ ሻ ‫݃݋ܮ‬ሺʹ݅ሻ ൌ | ሺ‫ ݖ‬െ ݅ ሻሺ‫ ݖ‬൅ ݅ ሻ ʹ݅

,||| #| | #||
| ර

஼

‫݃݋ܮ‬ሺʹ݅ሻ ‫݃݋ܮ‬ሺ‫ ݖ‬൅ ݅ሻ ͳ ൠ ൌ ߨ‫݃݋ܮ‬ሺʹ݅ ሻ ൌ ߨ‫ ʹ݃݋ܮ‬൅ ߨ;݅ | ‫ ݖ‬ൌ ʹߨ݅ § ʹ݅ ʹ ‫ ;ݖ‬൅ ͳ గ௜

$|
 ||‫݃݋ܮ‬ሺʹ݅ ሻ ൌ ‫ ʹ݃݋ܮ‬൅ ‫ ݅݃݋ܮ‬ൌ ‫ ʹ݃݋ܮ‬൅ ‫݁݃݋ܮ‬గ௜Ȁଶ ൌ ‫ ʹ݃݋ܮ‬൅ ଶ |

B

 | | 

| 

 || 
||

 |  |
|| || ‫݃݋ܮ‬ሺ‫ ݖ‬൅ ݅ሻ ͳ ‫݃݋ܮ‬ሺ‫ ݔ‬൅ ͳሻ ‫ ݔ‬൅ න ‫ ݖ‬ൌ ߨ݈‫ ʹ݃݋‬൅ ߨ;݅ | ʹ ‫ ;ݖ‬൅ ͳ ‫ ;ݔ‬൅ ͳ ି
୻

න





‫݃݋ܮ‬ሺ‫ ݔ‬൅ ݅ሻ ‫݃݋ܮ‬ሺ‫ ݔ‬൅ ݅ሻ ‫݃݋ܮ‬ሺ‫ ݖ‬൅ ݅ሻ ͳ ‫ ݔ‬൅ න ‫ ݔ‬൅ න ‫ ݖ‬ൌ ߨ‫ ʹ݃݋ܮ‬൅ ߨ;݅ | ʹ ‫ ;ݔ‬൅ ͳ ‫ ;ݔ‬൅ ͳ ‫ ;ݖ‬൅ ͳ ି
଴ ୻ ଴

න

$| |||*|| | 
#|
  |||| න


‫݃݋ܮ‬ሺ݅ ൅ ‫ݔ‬ሻ ‫݃݋ܮ‬ሺ‫ ݖ‬൅ ݅ሻ ͳ ‫݃݋ܮ‬ሺ݅ െ ‫ݔ‬ሻ ‫ ݔ‬൅ න ‫ ݔ‬൅ න ‫ ݖ‬ൌ ߨ‫ ʹ݃݋ܮ‬൅ ߨ;݅ | ʹ ‫ ;ݔ‬൅ ͳ ‫ ;ݖ‬൅ ͳ ‫ ;ݔ‬൅ ͳ ଴ ୻






,| |‫݃݋ܮ‬ሺ݅ െ ‫ݔ‬ሻ ൅ ‫݃݋ܮ‬ሺ݅ ൅ ‫ݔ‬ሻ ൌ ‫݃݋ܮ‬ሺ݅ ଶ െ ‫ ݔ‬ଶ ሻ ൌ ‫݃݋ܮ‬ሺ‫ݔ‬ଶ ൅ ͳሻ ൅ ߨ݅ |


න




‫݃݋ܮ‬ሺ‫ݔ‬ଶ ൅ ͳሻ ߨ݅ ‫݃݋ܮ‬ሺ‫ ݖ‬൅ ݅ሻ ͳ ‫ ݔ‬൅ න ‫ ݔ‬൅ න ‫ ݖ‬ൌ ߨ‫ ʹ݃݋ܮ‬൅ ߨ;݅ | ʹ ‫ ;ݔ‬൅ ͳ ‫ ;ݖ‬൅ ͳ ଴ ‫ ;ݔ‬൅ ͳ ୻

?
  |




c
      | I|1l> |  | || 
  | | ||L| | |||| ||  | | 
|

|| 
  | ஶ ‫݃݋ܮ‬ሺ‫ ݔ‬ଶ ൅ ͳሻ ‫݃݋ܮ‬ሺ‫ݔ‬ଶ ൅ ͳሻ ‫ ݔ‬ൌ න ൌ ߨ‫|ʹ݃݋ܮ‬
՜ஶ ‫ ;ݔ‬൅ ͳ ‫ ;ݔ‬൅ ͳ ଴

Ž‹

| ஶ

ଵ

గ

J ||‫׬‬଴ •‹ ‫ ݔ ;ݔ‬ൌ ଶ

ଶ

|)| | || | |
|> | |||| | ||<| |

||1 || | 
| | | | | | | & #| | #||| | ර ݁ ௜௭; ‫ ݖ‬ൌ Ͳ| ஼

| න ݁ ௜௭; ‫ ݖ‬൅ න ݁௜௭; ‫ ݖ‬൅ න ݁ ௜௭; ‫ ݖ‬ൌ Ͳ| ஺

஺ÿ

ÿ

| & |  ||,-||;||2;<|||;|14|5|| గ

|-J|‫ ݖ‬ൌ ܴ݁ ௜ఏ ቀ ߠ ൌ Ͳߠ ൌ ସቁ Ǣ|

|&,|‫ ݖ‬ൌ ”݁ ௜గȀସ ሺ ‫ ݎ‬ൌ ܴ‫ ݎ‬ൌ Ͳሻ|

| !
  |2+4|| |

?
  |




c
      |


න ݁௜௭; ‫ ݖ‬൅ න ଴

గȀସ

଴

݁ሺ௜
;௘

మ೔ÿሻ

଴

ߠ ൅ න ݁ ௜

;௘ ഏ೔Ȁమ ௘ ഏ೔Ȁర




‫ ݎ‬ൌ Ͳ ሺʹሻ|

: 
|| ഏ

‫׬‬଴ ሺ…‘• ‫ ;ݔ‬൅ ݅‫;ݔ݊݅ݏ‬ሻ ‫ ݔ‬ൌ ݁ గ௜Ȁସ ‫׬‬଴ ݁ ି ; ‫ ݎ‬െ ‫׬‬଴ర ݁ ௜
;௖௢௦ଶఏି
;௦௜





ଶఏ

ܴ݅݁ ௜ఏ ߠ |234|

!|| || |

||1l> 
    
  ||
 || ஶ

݁గ௜Ȁସ න ݁ ି ; ‫ ݎ‬ൌ ଴

ξߨ ഏ೔ ͳ ߨ ݅ ߨ ݁ర ൌ ൅ ሺͶሻ| ʹ ʹ ʹ ʹ ʹ

!|  | || |
#|
  |||| | ቤන

గȀସ

଴

݁௜
;௖௢௦ଶఏି
;௦௜

ଶఏ

గȀସ

ܴ݅݁௜ఏ ߠ ቤ ൑ න

଴

ഏ

ܴ మ ൑ න ݁ ିଶ
;௦௜ ʹ ଴



݁ ି


మ

ୱ୧୬ ଶఏ

ܴ ߠ |

|

ഏ

ܴ మ ൑ න ݁ ିଶ
;Ȁଶ | ʹ ଴ ൑

ߨ మ ሺͳ െ ݁ ି
ሻ| Ͷܴ

,)| | | ൌ ʹߠ| |

| 
  |‫ ݊݅ݏ‬൒ ʹȀߨͲ ൑  ൑ ߨȀʹ| 
| ||
|1l> 
  
   ' $%&  | ஶ

න ሺ…‘• ‫ ;ݔ‬൅ ݅‫;ݔ݊݅ݏ‬ሻ ‫ ݔ‬ൌ ଴

ͳ ߨ ݅ ߨ ൅ | ʹ ʹ ʹ ʹ

$|  | 
|

| | 
|


|||| ஶ

ஶ

න …‘• ‫ ݔ ;ݔ‬ൌ න •‹ ‫ ݔ ;ݔ‬ൌ ଴

଴

ͳ ߨ | ʹ ʹ

| |

?
  |

|




c
      |

   | @ | |||
|| |
|| ʹ‫ݖ‬ଷ െ Ͷ‫ ;ݖ‬൅ ͷ | ͵‫ݖ‬ÿ െ ͺ‫ ݖ‬൅ ͳͲ

  |‫׬‬௖ ‫ ݖ‬ଷ ݁ଵȀ௭ ‫|
| || | |ݖ‬ȁ‫ ݖ‬െ ͳȁ ൌ Ͷ| ஶ

J |||‫׬‬଴

௫

௦௛గ௫

ଵ

‫ ݔ‬ൌ | ସ

>> | ,)|+| |
|| 
 
| |||||||||||||||||| 6| |
||  
|| |> | ݃ҧ ൌ ଵ

ͳ ͳ ‫ݖ‬ ݂ሺ‫ݖ‬ሻ ර ൌ ර ‫|ݖ‬ ݅ߨ ௖ ሺ‫ ݖ‬െ ‫ݖ‬ଵሻሺ௭ି௭మ ሻ ݅ߨ ஼ ‫ ݖ‬െ ‫ݖ‬ଵ

,)| |
|݂ሺ‫ݖ‬ሻ ൌ ௭ି௭ | | 
|| 
 
||| || || మ

$|

 | ||> |||||||||݃ҧ ൌ

ଶగ௜ ௜గ

݂ሺ‫ݖ‬ଵ ሻ ൌ ʹǤ ௭

ଵ

భ ି௭మ

ൌ ʹ|

$ ||$  
||
  |
  ||| $|
 | 
  |

|| ݁ ଶ௭ ‫|ݖ‬ ͳ ൅ ‫;ݖ‬

ර ௖

!| || |
|| |
|264 |,|
|| 
| 
  |

| …‘• ‫ݔ‬ ‫| ݔ‬ ିஶ ͳ ൅ ‫;ݔ‬ ஶ

ƒൌන


| | | | | ଵ

!|
|ଵା௭; | | 
|| 
 
|| |||| ||||| 
 |+;|
| "| | #| ||
  |||

?
  |




c
      | ර ௖

݁ ଶ௭ ݁ ௜௭ ݂ሺ‫ݖ‬ሻ ‫ ݖ‬ൌ ර ‫ ݖ‬ൌ ර ‫ ݖ‬ൌ ʹߨ݂݅ሺ‫ݖ‬ሻ| ሺ ሻ ‫ ݖ‬൅ ݅ ሺ‫ ݖ‬െ ݅ሻ ͳ ൅ ‫;ݖ‬ ௖ ௖ ‫ݖ‬െ݅

௘ ೔೥

- |݂ሺ‫ݖ‬ሻ ൌ ଵା௭;| | -

||  | ර ௖

ʹߨ݅݁ ିଵ ߨ ݁௜௭ ‫ ݖ‬ൌ ൌ ሺʹሻ| ʹ݅ ʹ ͳ ൅ ‫;ݖ‬

$ | 
| 
  || |
 |1| | || |1l #  | ஶ ݁௜௫ ߨ …‘• ‫ݔ‬ ߨ ‫ ݔ‬ൌ ܴ݁ ቈන ‫ ݔ‬቉ ൌ ܴ݁ ቀ ቁ ൌ | ݁ ݁ ିஶ ͳ ൅ ‫;ݔ‬ ିஶ ͳ ൅ ‫;ݔ‬

ƒൌන

ஶ

!
  || |
 |1|| 
  || 
|264| | || |1l> | $|   | |
 | |‫ ݖ‬ൌ ݁ ௜ఝ ൌ ܴܿ‫ ߮ݏ݋‬൅ ܴ݅‫|| 
|

| | 
||߮݊݅ݏ‬ ቤන

஼ೃ

గ ௜
௖௢௦ఝ ି
௦௜ ݁ ௜௭ ݁ ݁ ‫ݖ‬ቤ ൌ ቤන ͳ ൅ ‫;ݖ‬ ͳ ൅ ܴ;݁ ଶ௜ఝ ଴

ఝ

ܴ݅݁ ௜ఝ ߮ ቤ ൑ න

గ

଴

గ ି
௦௜ ݁ ݁ ି
௦௜ ఝ ܴ න ߮ ൑ ʹ ଶ௜ఝ ȁͳ ൅ ܴ;݁ ȁ ܴ ଴

ఝ

߮|

|
| | || |1l| "| |
#|

||
 |

| |
|| หͳ ൅ ܴ;݁ ଶ௜ఝ ห ൒

ܴଶ | ʹ

-

||@
||K
 | 
 |24||
| 
|| |
 |
|1||M|<| ||;| || 
 |> ||
|
|| '|
  | & #| | |||||| ݂ ሺ‫ ݌‬ሻ ൌ

ͳ ݂ሺ‫ݖ‬ሻ ර ‫| ݖ‬ ʹߨ݅ ௖ ‫ ݖ‬െ ‫݌‬

,)| ||
 
|| |
|| |
|| 8 
| | | |

?
  |




c
      | | ,|
#|
  | | 
 | |;| | 
|||  ||  
||
 | |

    0  | 
|24||
| 
|| 
 
||| || | |||||N|<| |||;|| 8 
|

|

|

$|| 
 |

||;||| 
 
||||
 || |  |
  ||| ሺ ሻ

݂ሺ௔ሻ ൌ

݊ ݂ሺ‫ݖ‬ሻ ර ʹߨ݅ ሺ‫ ݖ‬െ ܽሻ

ାଵ

‫ ݖ‬ǡ݊ ൌ Ͳǡͳǡʹ ǥ|

,|
#|
  | |    |
 || ͳ ͳ ͳ ͳ ൌ ൌ | ‫ ݖ‬െ ܽ ‫ ݖ‬െ ܾ െ ሺܽ െ ܾሻ ‫ ݖ‬െ ܾ ͳ െ ௔ି௕ ௭ି௕

ൌ

ͳ ܽ െ ܾ ሺܽ െ ܾሻ; ቈͳ ൅ ൅ ൅ ‫ ڮ‬቉ሺͳሻ| ‫ݖ‬െܾ ‫ ݖ‬െ ܾ ሺ‫ ݖ‬െ ܾሻ;

$ ||| || 
 ||   || | ||||ȁ‫ ݖ‬െ ܾȁ ൐ ȁܽ െ ܾȁ|: 
| |
|  
| | ||||
|| | 
|2O4| |   | 
 || ||||||| ݂ ሺܽ ሻ ൌ

$| | 

ଵ

௭ି௔

| |||    || |
| 
|2+4||
 |

?
  |

݂ሺ‫ݖ‬ሻ ͳ ර ‫| ݖ‬ ʹߨ݅ ௖ ‫ ݖ‬െ ܽ




c
      | ݂ ሺܽ ሻ ൌ

݂ሺ‫ݖ‬ሻ ݂ሺ‫ݖ‬ሻ ݂ሺ‫ݖ‬ሻ ܽെܾ ሺܽ െ ܾሻ; ͳ ර ර ර ‫ ݖ‬൅ ‫ ݖ‬൅ ‫ ݖ‬൅ ‫|ڮ‬ ଷ ʹߨ݅ ௖ ሺ‫ ݖ‬െ ܾሻ; ʹߨ݅ ʹߨ݅ ௖ ‫ ݖ‬െ ܾ ௖ ሺ‫ ݖ‬െ ܾሻ

&)||

 | ||
  ||| |    || |
|24||
||@ | | 
 ||;|| ݂ ሺܽ ሻ ൌ ݂ ሺܾሻ ൅ ݂ ᇱ ሺܾሻሺܽ െ ܾሻ ൅

݂ ᇱᇱ ሺܾሻ ሺܽ െ ܾሻଶ ൅ ‫ڮ‬ሺʹሻ| ʹ

!|    | |   | | | | 
 || 
 
|| | |

Ex  | ||&    ||
||@ || |
|
 || &   |݂ሺ‫ݖ‬ሻ ൌ •‹ ‫ ݖ| 
 || @||
|| ݖ‬ൌ ߨȀͶ| $|
 || | | 
 || ||264| | ||
 | ݂ ሺ‫ݖ‬ሻ ൌ ݂ ሺߨȀͶሻ ൅ ݂ ᇱ ሺߨȀͶሻሺ‫ ݖ‬െ ߨȀͶሻ ൅

݂ ᇱᇱ ሺߨȀͶሻ ሺ‫ ݖ‬െ ߨȀͶሻଶ ൅ ‫|ڮ‬ ʹ

I
|| గ

గ ଷ

ሺ‫ ݖ‬െ ସ ሻ; ሺ‫ ݖ‬െ ସ ሻ ߨ ξʹ ݂ ሺ‫ݖ‬ሻ ൌ ÿͳ ൅ ቀ‫ ݖ‬െ ቁ െ െ ൅‫| ڮ‬ ͵ ʹ ʹ Ͷ

P
  ǯ  
 | 
||
|


|: 
| |
 |||  E|"24| |I24| ݂ ሺ‫ ݖ‬ሻ ൌ

mሺ‫ݖ‬ሻ ሺͳሻ| ܳሺ‫ݖ‬ሻ

- | 
  | 

| | | 24F24| | | | | | 
 | | 7| | E |
   || |

|| | 

|| #|| | 
 | 
 | ‫ ݖ‬ൌ ߙ௞ | ||2||




4|/|| |"24| ||‫ ݖ‬ൌ ÿ௜ | ||| |




|
||
 |I24 | ,| |


| |
|2+4|  ||ߙ௞ ് ÿ௜ | | | |
|/| 
  |

!|ߙ௞ | || |||/| | |ÿ௜ || | E ||
|| |
|2+4 | !|
|2+4| | 
|| 
||| E | |

|  | |  
|
 |

?
  |




c
      | ݂ ሺ‫ ݖ‬ሻ ൌ 

ߨሺ‫ ݖ‬െ ܽ௞ ሻ  mሺ‫ݖ‬ሻ ൌܿ ‫݋‬î‫|ܧ
ܣ
ž ܱܥ݁݊ݑݐݏ݁ܥ‬ ߨሺ‫ ݖ‬െ ÿ௜ ሻ௠೔ ܳሺ‫ݖ‬ሻ

,|  | | ߙ௞ |  | | ÿ௜ | | | |  | | | | | 
 |  | - | | 
 | |
| 
 | ௙ ᇲ ሺ௭ሻ ௙ሺ௭ሻ

ൌ

ௗ

ௗ௭

ሾŽ ݂ሺ‫ݖ‬ሻሿ ൌ ÿ௞



௭ି 

ൌ ÿ௜

௠೔

௭ିÿ೔

|

,|
 | ||
  | | ||||
 | | |
  |||| ݂ ᇱ ሺ‫ݖ‬ሻ ‫ݖ‬ ͳ ර ൌ ෍ ݊௞ െ ෍ ݉௜ ሺʹሻ| ݂ሺ‫ݖ‬ሻ ʹߨ݅ ௖ ௞

௜

,)|݊௞ | |݉௜ | | 
 | |




|| | || | | E || |
|24| 
|| 
 
|| |||
 | |
| |
 
| |

|| ෍ ݊௞ ൌ ௞

݁‫ ݐ‬෍ ݉௜ ൌ m | ௜

!| |264| | || ݂ ᇱ ሺ‫ݖ‬ሻ ‫ݖ‬ ͳ ර ൌ ݂ሺ‫ݖ‬ሻ ʹߨ݅ ௖

െ mሺ͵ሻ|

,)|.| | |||| |"| ||| E |  | || |




|| 
 
| | |

Th    M Mx 

|

Th   | 
|24||
| 
|||
|| ||| || | | |ȁ݂ሺ‫ݖ‬ሻȁ| ||
||||
|24| |  |

Ex   | ሺ௭మ ାଵሻ;


|݂ሺ‫ݖ‬ሻ ൌ ሺ௭;ାଶ௭ାଶሻ ݈ܿܽܿ‫ݎ݈݁ݑ‬

,)|| | | |ȁ‫ݖ‬ȁ ൌ Ͷ|

௙ᇲ ሺ௭ሻௗ௭ ଵ 6 ௙ሺ௭ሻ | ଶగ௜ ௖


|݂ሺ‫ݖ‬ሻ ൌ ‫ ݖ‬ସ െ ʹ‫ݖ‬ଷ ൅ ‫ ;ݖ‬െ ͳʹ‫ ݖ‬൅ ʹͲ| |
|| | | | ȁ‫ݖ‬ȁ ൌ ͷ|  |6௖

?
  |

௭௙ ᇲᇲ ሺ௭ሻௗ௭ ௙ሺ௭ሻ

|




c
      |
| N<| |
||
||||
 > |  6௖

௭௘ ೥ÿ ௗ௭ ሺ௭ାଵሻ;

|

Th    Í    

 |
|24| | 
|| 
 
|| ||
|ȁ‫ ݖ‬െ ܽȁ ൌ ‫| ||| |ݎ‬24| | | ||  ||24|||| ||
| ݂ሺܽሻ ൌ

ͳ ଶగ න ݂൫ܽ ൅ ‫݁ݎ‬௜ఏ
ߠሺͳሻ| ʹߨ ଴

Th     x 

|
|24| | 
|| 
 
||||
 || |||
|| |24| | |   | | |
||ȁ݂ሺ‫ݖ‬ሻȁ| |


 || | |

Th     
 

|
|24| ||
| 
|| 
 
||||
 || |||
|| | 24P<|| 
 
||| |ȁ݂ሺ‫ݖ‬ሻȁ|


 ||

|| |

Th    ǯ  
 | 
|24||
| 
|| 
 
||||
 |||||
|  
|| 
 
||||
 || |||| 
|||

|| E |
 
|| ||| || ݂ ᇱ ሺ‫ݖ‬ሻ ‫ݖ‬ ͳ ර ൌ ݂ሺ‫ݖ‬ሻ ʹߨ݅ ௖

െ mሺͷሻ|

,)|.| |"|
 | 
 | |||| | ||| E ||24|
 
|| |

Th    R h |
|24| |24| | 
|| |||||
 || |
|ȁ݃ሺ‫ݖ‬ሻȁ ൏ ȁ݂ሺ‫ݖ‬ሻȁ|||  |24|C|24| |24| | ||||| 
 
|| |

?
  |




c
      |

Th    L
 | 
|+| ||6|| | 
||| || ||| 
|1+| ||16||| || ݂ሺ‫ݖ‬ሻ| || 
| |
|+| ||6| |  || ||R|2
| 
|R4|



| |+| ||6 | 
|ܽ ൅ ݄|| 
 | ||R |||| ݂ ሺܽ ൅ ݄ ሻ ൌ ܽ଴ ൅ ܽଵ ݄ ൅ ܽଶ݄ ଶ ൅ ‫ ڮ‬൅

,)| ܽ ൌ

௙ሺ௭ሻ ଵ ‫݊ ݖ‬ 6 ଶగ௜ ௖భ ሺ௭ି௔ሻ ೙శభ

ൌ ͳǡʹǡ͵ǡͶǡͷ ǥ ܽି ൌ

| || 

|
||+| |6 | ܽ ൌ

ଵ 6 ଶగ௜ ௖

௙ሺ௭ሻ ሺ௭ି௔ሻ ೙శభ

ଵ 6 ଶగ௜ ௖మ

݂ሺ‫ݖ‬ሻሺ‫ ݖ‬െ ܽሻ

ଵ

ାଵ

‫ | ݖ‬+|  | 6|  | 
|

‫ ݊ ݖ‬ൌ േͳǡ േʹǡ േ͵ ǥ || || 
|

݂ ሺ‫ݖ‬ሻ ൌ ܽ଴ ൅ ܽଵ ሺ‫ ݖ‬െ ܽሻ ൅ ܽଶ ሺ‫ ݖ‬െ ܽሻଶ ൅ ‫ ڮ‬൅

)ܽ ൌ ଶగ௜ 6௖

ܽିଵ ܽିଶ ൅ ଶ ൅ ‫|ڮ‬ ݄ ݄

௙ሺ ሻ
ሺͻሻ| ሺ ି௔ሻ ೙శభ

ܽିଵ ܽିଶ ൅ ൅ ‫ ڮ‬ሺͺሻ| ሺ‫ ݖ‬െ ܽሻ ሺ‫ ݖ‬െ ܽሻଶ

ሺͺሻ |  |
||! | |ሺͻሻ| |  |

 || ||
||! |

!| 
| ܽ଴ ൅ ܽଵ ሺ‫ ݖ‬െ ܽሻ ൅ ܽଶሺ‫ ݖ‬െ ܽሻଶ ൅ ‫|||
|

|
| ||
 |
 | | |ڮ‬ 
|
|| ሺ‫ ݖ‬െ ܽሻ| | | 
| 

  |
| | 
| 

 | |

| |
|| ! ||
|||
+||@  | ஶ

ஶ

ୀ଴

ୀଵ

݂ሺ‫ݖ‬ሻ ൌ ෍ ܽ ሺ‫ ݖ‬െ ܽሻ ൅ ෍ ܽି ሺ‫ ݖ‬െ ܽሻି |

)ܽ ൌ

ଵ ௙ሺ௪ሻ ‫ିܽ| | ݓ‬ 6 ଶగ௜ ௖భ ሺ௪ି௔ሻ ೙శభ

ൌ

ଵ

ଵ ௙ሺ௪ሻ ‫| ݓ‬ 6 ଶగ௜ ௖మ ሺ௪ି௔ሻ ೙శభ

&   |݂ሺ‫ݖ‬ሻ ൌ ሺ௭ାଵሻሺ௭ାଷሻ||
||! | |ͳ ൏ ȁ‫ݖ‬ȁ ൏ ͵| ଵ

ଵ

ଵ

ଵ

ଵ

 | "| 

|݂ ሺ‫ݖ‬ሻ ൌ ሺ௭ାଵሻሺ௭ାଷሻ ൌ ቀ௭ାଵቁ െ ଶ ቀ௭ାଷቁ| ଶ


|ͳ ൏ ȁ‫ݖ‬ȁ| ͳ ͳ ͳ ͳ ͳ ͳ ͳ ͳ ͳ ൬ ൰ൌ ൌ ൬ͳ െ ൅ ଶ െ ‫ ڮ‬൰ ൌ െ ଶ ൅ ଷ െ ‫|ڮ‬ ଵ ʹ ‫ݖ‬൅ͳ ʹ‫ݖ‬ ‫ݖ‬ ‫ݖ‬ ʹ‫ݖ‬ ʹ‫ݖ‬ ʹ‫ݖ‬ ʹ‫ ݖ‬ቀͳ ൅ ቁ ௭


ȁ‫ݖ‬ȁ ൏ ͵|

| ͳ ‫ݖ ݖ‬ଶ ‫ ݖ‬ଷ ͳ ͳ ͳ ͳ ‫ݖ‬ ‫ݖ‬ଶ ‫ݖ‬ଷ ൬ ൰ൌ ൌ ቆͳ െ ൅ െ ൅‫ڮ‬ቇ ൌ െ ൅ െ ൅ ‫|ڮ‬ ௭ ʹ ‫ݖ‬൅͵ ͵ ͻ ʹ͹ ͸ ͳͺ ͷͶ ͳ͸ʹ ͸ ቀͳ ൅ ቁ ͸ ଷ

| | ௭ି௦௜ ௭ |‫ ݖ‬ൌ Ͳ| &   |݂ሺ‫ݖ‬ሻ ൌ ௭

?
  |




c
      |

‫ݖ‬ଷ ‫ݖ‬ÿ œ ͳ ‫ݖ‬ଶ œସ ‫ ݖ‬െ ‫ͳ ݖ݊݅ݏ‬ ቈ‫ݖ‬ ቆ‫ݖ‬ ቇ቉ െ െ ൌ ൅ െ ൅ ‫ڮ‬ ൌ െ െ | ͵ ͷ ͹ ͷ ͹ ‫ݖ‬ଷ ͵ ‫ݖ‬ଷ Q;<| ||
 
|   | ௭ &   |݂ሺ‫ݖ‬ሻ ൌ ሺ௭ାଵሻሺ௭ାଶሻ|| 
 |‫ ݖ‬ൌ െʹ|

|,| |‫ ݖ‬൅ ʹ ൌ ‫| ݑ‬ ‫ݑ‬െʹ ʹെ‫ͳ ݑ‬ ʹെ ‫ݑ‬ ‫ݖ‬ ሺͳ ൅ ‫ ݑ‬൅ ‫ ݑ‬ଶ ൅ ‫ ݑ‬ଷ ൅ ‫ ڮ‬ሻ| ൌ ൌ ൌ ሺ‫ ݖ‬൅ ͳሻሺ‫ ݖ‬൅ ʹሻ ‫ݑ‬ሺ‫ ݑ‬െ ͳሻ ‫ͳ ݑ‬െ‫ݑ‬ ‫ݑ‬ ʹ ൅ ͳ ൅ ሺ‫ ݖ‬൅ ʹሻ ൅ ሺ‫ ݖ‬൅ ʹሻଶ ൅ ‫|ڮ‬ ൌ ‫ݖ‬൅ʹ ‫ ݖ‬ൌ െʹ | | E ||| 
  | !| 
|  | |  | |  | | |  | | Ͳ ൏ ȁ‫ ݖ‬൅ ʹȁ ൏ ͳ| & 
| |     | | 
| | ! | | 
| 
 | | 

||
 
|

| ݂ ሺ‫ ݖ‬ሻ ൌ

௘ మ೥

ሺ௭ିଷሻ

|| |‫ ݖ‬ൌ ͳ|


‫ ݖ‬െ ͳ ൌ ‫ ݖ ֜ ݑ‬ൌ ͳ ൅ ‫|ݑ‬ ʹ‫ ݑ‬ሺʹ‫ ݑ‬ሻଶ ሺʹ‫ݑ‬ሻଷ ݁ ଶାଶ௨ ݁ ଶ ݁ ଶ௨ ݁ ଶ ݁ଶ௭ ൌ ൌ ൌ ଷ ቆͳ ൅ ൅ ൅ ൅ ‫ ڮ‬ቇ| ଷ ଷ ଷ ሺ‫ ݖ‬െ ͳሻ ͳ ‫ݑ‬ ‫ݑ‬ ‫ݑ‬ ʹ ͵ ʹ݁ଶ ʹ݁ ଶ Ͷ݁ଶ ʹ݁ ଶ ݁ଶ ሺ‫ ݖ‬െ ͳ ሻ ൅ ‫|ڮ‬ ൅ ൅ ൅ ൅ ൌ ଶ ሺ‫ ݖ‬െ ͳሻଷ ሺ‫ ݖ‬െ ͳሻ ሺ‫ ݖ‬െ ͳ ሻ ͵ ͵ ‫ ݖ‬ൌ ͳ ||  || 
| !|
| | |  |  |‫|ͳ ് ݖ‬ |

EXERCICES | + | &   | |
|
 ||
||@ || 
 |

| |  | |
||  |  | ݁ ି௭ ǡ ‫ ݖ‬ൌ Ͳ| గ  | …‘• ‫ݖ‬ǡ ‫ ݖ‬ൌ | ଶ |

ଵ

ଵା௭ ଷ

ǡ ‫ ݖ‬ൌ ͳ|

 | ‫ ݖ‬െ ͵‫ ݖ‬ଶ ൅ Ͷ‫ ݖ‬െ ʹǡ ‫ ݖ‬ൌ ʹ|  | ‫݁ݖ‬ଶ௭ |‫ ݖ‬ൌ െͳ| 6 | &   ||
||! || 

||‫ ݖ‬ൌ Ͳ| |

|

ଵି௖௢௦௭ ଶ௭

మ ௘೥

|

|

ଶ௭ ଶ ି௭ ర

| ‫| ݁ ݖ‬  | ‫ݖ݄ݏݖ‬ξ‫|ݖ‬ |

 
      | D | | 
 || | |
 
||
|݂ሺ‫ݖ‬ሻ| | || |
||! ||

?
  |




c
      |

1.| P  |
| ݂ሺ‫ݖ‬ሻ|| | | | ሺͺሻ| | 

| | 
| 

 | | #| | | 

| | || ௔ ௔ ௔ | | ݂ ሺ‫ݖ‬ሻ ൌ ܽ଴ ൅ ܽଵሺ‫ ݖ‬െ ܽሻ ൅ ܽଶ ሺ‫ ݖ‬െ ܽሻଶ ൅ ‫ ڮ‬൅ షభ ൅ షమ మ ൅ ‫ ڮ‬ష೙ ೙ | C| | )| ܽି ് Ͳ| ሺ௭ି௔ሻ

 |‫ ݖ‬ൌ ܽ| |  |  |||

ሺ௭ି௔ሻ

ሺ௭ି௔ሻ

|
|݊ ൌ ͳ|||
|||  |
 ||


|‫ ݖ‬ൌ ܽ| ||  ||݂ሺ‫ݖ‬ሻ| Ž‹௭՜௔ ݂ሺ‫ݖ‬ሻ ൌ λ|

2.| S
 
 |
| | 
| 
| ݂ሺ‫ݖ‬ሻ|  | | 

| | ‫ ݖ‬ൌ ܽ| 
| 
| Ž‹௭՜௔ ݂ሺ‫ݖ‬ሻÿ|  | ‫ ݖ‬ൌ ܽ|  |   | | | 
 
|    | &| |  | | | | 

|݂ሺ‫ݖ‬ሻ| | ‫ ݖ‬ൌ ܽ | |  | || Ž‹௭՜௔ ݂ሺ‫ݖ‬ሻ|
|݂ሺ‫ݖ‬ሻ ൌ 

௦௜ ௭ ௭

|‫ ݖ‬ൌ Ͳ|

݈݅݉௭՜௔

‫ݖ݊݅ݏ‬ ൌ ͳǡ ‫ݖ‬

ˆሺͲሻ ൌ ݈݅݉௭՜௔

‫ݖ݊݅ݏ‬ ൌ ͳ| ‫ݖ‬

‫ݖ ݖ ͳ ݖ݊݅ݏ‬ଷ ‫ݖ‬ÿ ‫ݖ‬ଶ ‫ݖ‬ସ ൌ ቆ െ ൅ ൅ ‫ ڮ‬ቇ ൌ ͳ െ ൅ ൅ ‫|ڮ‬ ‫ݖ‬ ͵ ͷ ͵ ͷ ‫ͳ ݖ‬

¥ |


  
| |
| ݂ሺ‫ݖ‬ሻ| |
|| |  |
 
|
| |
|  |
|
 
|  | |  | 
 
|


 |
| ‫ ݖ‬ൌ ܽ|  | | 
 
| 


| | ሺ‫ݖ‬ሻ| | | 
| 

 | |     | | ! | #||


||  |

Ex  భ

݁೥ ൌ ͳ ൅

ଵ

ଵ௭

൅

ଵ

ଶ ௭మ

൅

ଵ

ଷ௭

൅ ‫ ݖ ڮ‬ൌ Ͳ ||
 
|


|

4.| P
  b
h 
 | B| 
 | ‫ ݖ‬ൌ ‫ݖ‬଴ || |  | 
 || || |
|


|݂ሺ‫ݖ‬ሻ||
| ||| |݂ሺ‫ݖ‬ሻ| |||
|| || ||- | | | | | 
| 


|  |  
|  | | | | #| | | 
| 
| |

?
  |




c
      |

Ex  భ

!|||݂ሺ‫ݖ‬ሻ ൌ ‫ ݖ‬మ |
| | |  |+| |ൌ ͳ|| #||    ||||
|| @ || ||ܽ଴ ൅ ܽଵ ሺ‫ ݖ‬െ ܽሻ ൅ ܽଶ ሺ‫ ݖ‬െ ܽሻଶ ൅ ‫|ڮ‬ | |

5.| S
 à ǯ

 | ଵ

ଵ

$|  | ‫ ݖ‬ൌ ௪ †ƒ•݂ሺ‫ݖ‬ሻ| | 
 | | 
| ݂ ቀ௪ቁ ൌ ‫ ܨ‬ሺ‫ݓ‬ሻ|  | |  | | | 
 
| | ‫ ݖ‬ൌ λ|2 || 
 || 


4| |

||  | |'|| 
|||‫ ܨ‬ሺ‫ݓ‬ሻ|‫ ݓ‬ൌ Ͳ|

Ex  ݂ ሺ‫ݖ‬ሻ ൌ ‫ݖ‬ଷ ||  |
 ||‫ ݖ‬ൌ λ| ଵ

ଵ

݂ ቀ௪ቁ ൌ ‫ ܨ‬ሺ‫ݓ‬ሻ ൌ ௪ | | |  | 
 | ൌ Ͳ| | &| | '| 
#| ‫ݖ‬ଷ | #| | 
 
| భ

ଵ




||‫ ݖ‬ൌ λ||݂ ቀ௪ቁ ൌ ‫ ܨ‬ሺ‫ݓ‬ሻ ൌ ݁ ೢ ||||
 
|


||‫ ݓ‬ൌ Ͳ| |

& 
| |     | | 
| | ! | | | 
| 
 | | 

| | | 
 
|

| ௘ మ೥

ሺ௭ିଵሻ

|‫ ݖ‬ൌ ͳ|||||| |‫ ݖ‬െ ͳ ൌ ‫ ݑ‬՜ ‫ ݖ‬ൌ ͳ ൅ ‫| ݑ‬ ݁ ଶାଶ௨ ݁ ଶ ଶ௨ ݁ଶ ݁ଶ௭ ʹ‫ ݑ‬ሺʹ‫ݑ‬ሻଶ ሺʹ‫ݑ‬ሻଷ ቆͳ ൌ ൌ ݁ ൌ ൅ ൅ ൅ ‫ ڮ‬ቇ| ൅ ሺ‫ ݖ‬െ ͳሻଷ ʹ ͵ ‫ݑ‬ଷ ‫ݑ‬ଷ ‫ݑ‬ଷ ͳ ௘మ

ଶ௭ మ

ଶ௭ మ

ൌ ሺ௭ିଵሻ ൅ ሺ௭ିଵሻమ ൅ ሺ௭ିଵሻ ൅

ସ௭ మ ଷ

൅

ଶ௭ మ ሺ௭ିଵሻ ଷ


| | |  |  ||‫| ͳ ് ݖ‬

ሺ‫ ݖ‬െ ͵ሻ‫݊݅ݏ‬

ଵ

௭ାଶ

൅ ‫ ݖ ڮ‬ൌ ͳ | |  | 
| |  | | 3 | !|

‫ ݖ‬ൌ െʹǡ| |‫ ݖ‬൅ ʹ ൌ ‫ ݖ||| ݑ‬ൌ ‫ ݑ‬െ ʹǡ| ሺ‫ ݖ‬െ ͵ሻ‫݊݅ݏ‬

ͳ ͳ ͳ ͳ ͳ ൌ ሺ‫ ݑ‬െ ͷሻ‫ ݊݅ݏ‬ൌ ሺ‫ ݑ‬െ ͷሻ ൬ െ ൅ ൅ ‫ ڮ‬൰| ଷ ‫ݑ‬ ‫ݑ ͵ ݑ‬ ͷ ‫ݑ‬ÿ ‫ݖ‬൅ʹ ൌ ൬ͳ െ

ൌ ൬ͳ െ

ͳ ͷ ͳ ͷ െ ൅ ൅ ൅ ‫ ڮ‬൰| ଶ ଷ ͵ ‫ݑ‬ ͷ ‫ݑ‬ସ ‫ݑ ͵ ݑ‬

ͷ ͳ ͷ ͳ െ ൅ ൅ ൅ ‫ ڮ‬൰| ଶ ଷ ሺ‫ ݖ‬൅ ʹሻ ͵ ሺ‫ ݖ‬൅ ʹሻ ͵ ሺ‫ ݖ‬൅ ʹሻ ͷ ሺ‫ ݖ‬൅ ʹሻସ

‫ ݖ‬ൌ െʹ ||
 
|


|

?
  |




c
      | | ௭ି௦௜ ௭ ௭

|||||‫ ݖ‬ൌ Ͳ| ͳ ‫ݖ‬ଷ ‫ݖ‬ÿ ‫ݖ‬ ‫ ݖ‬െ ‫ݖ݊݅ݏ‬ ቆ‫ݖ‬ ൌ ൭œ െ െ ൅ െ ൅ ‫ ڮ‬ቇÿ| ͵ ͷ ͹ ‫ݖ‬ଶ ‫ݖ‬ଷ ൌ

‫ݖ‬ଷ ‫ݖ‬ÿ ‫ݖ‬ ͳ ൭ቆ െ ൅ െ ‫ ڮ‬ቇÿ| ͷ ͹ ‫ݖ‬ଶ ͵ ൌ

‫ ݖ‬ൌ Ͳ ||
 
|  |

ͳ ‫ݖ‬ଶ ‫ݖ‬ÿ െ ൅ | ͵ ͷ ͹

| ଵ

&   |݂ሺ‫ݖ‬ሻ ൌ ሺ௭ାଵሻሺ௭ାଷሻ||
||! | ||

ͳ ൏ ȁ‫ݖ‬ȁ ൏ ͵| ȁ‫ݖ‬ȁ ൐ ͵| Ͳ ൏ ȁ‫ ݖ‬൅ ʹȁ ൏ ʹ| | | ଵ  | !|    ||݂ሺ‫ݖ‬ሻ||  ||
 ||ሺ௭ାଵሻሺ௭ାଷሻ ൌ

| | |


|ȁ‫ݖ‬ȁ ൐ ͳ|

ž
|ȁ‫ݖ‬ȁ ൏ ͵|

ଵ

ଶሺ௭ାଵሻ

െ

ଵ

ଶሺ௭ାଷሻ

|

ͳ ͳ ͳ ͳ ͳ ൌ ൌ െ ଶ ൅ ଷ െ ‫|ڮ‬ ͳ ʹሺ‫ ݖ‬൅ ͳሻ ʹ‫ݖ‬൫ͳ ൅ ‫ݖʹ ݖʹ
ݖ‬ ʹ‫ݖ‬

ͳ ͳ ‫ݖ‬ ‫ݖ‬ଶ ‫ݖ‬ଷ ͳ ൌ ൌ െ ൅ െ ൅ ‫|ڮ‬ ʹሺ‫ ݖ‬൅ ͵ሻ ͸൫ͳ ൅ ‫
ݖ‬͸ ͳͺ ͷͶ ͳͳʹ ͵ ֜"ͳ ൏ ȁ‫ݖ‬ȁ ൏ ͵||| |    | ͳ ͳ ͳ ͳ ͳ ͳ ‫ݖ‬ ‫ݖ‬ଶ ‫ݖ‬ଷ ൌ െ ଶ൅ ଷെ ସെ ൅ െ ൅ ൅| ሺ‫ ݖ‬൅ ͳሻሺ‫ ݖ‬൅ ͵ሻ ʹ‫ݖʹ ݖ‬ ʹ‫ݖ‬ ʹ‫ݖ‬ ͸ ͳͺ ͷͶ ͳͳʹ

|

?
  |

|




c
      |

       1 ÃÃ      

   |

R  | 
| ݂ሺ‫ݖ‬ሻ||
| 
| |
|| 
 
|| |ܿ| ||||| || 
 | ‫ ݖ‬ൌ ܽ| ||ܿ| |- |||  |݂ሺ‫ݖ‬ሻ| #||    ||
||! | | | 

||‫ ݖ‬ൌ ܽ||| | ஶ

݂ ሺ‫ݖ‬ሻ ൌ ෍ ܽ ሺ‫ ݖ‬െ ܽሻ ሺͳሻ| ୀିஶ

ൌ ൅ܽଵ ሺ‫ ݖ‬െ ܽሻ ൅ ܽଶ ሺ‫ ݖ‬െ ܽሻଶ ൅ ‫ ڮ‬൅ ܽ ൌ

ͳ ݂ሺ‫ݖ‬ሻ ර ʹߨ݅ ௖ ሺ‫ ݖ‬െ ܽሻ

ାଵ

ܽିଶ ܽି ܽିଵ ൅ ൅‫ڮ‬ | ሺ‫ ݖ‬െ ܽሻ ሺ‫ ݖ‬െ ܽሻ ሺ‫ ݖ‬െ ܽሻଶ

‫ ݖ‬ǡ ݊ ൌ േͳǡ േʹǡ േ͵ ǥ ሺʹሻ|

&| || 

|)|݊ ൌ െͳ|||6௖ ݂ ሺ‫ݖ‬ሻ ‫ ݖ‬ൌ ʹߨ݅ܽିଵ ሺ͵ሻ| ,|  | 
|234||| 
||2+4| |
 
| || ||

 | |

 | ර ௖

‫ݖ‬ ʹߨ݅ǡ‫ ݌‬ൌ ͳ
| ൌ§ Ͳǡ‫ ݌‬ൌ ݁݊‫ͳ ് ݎ݁݅ݐ‬ ሺ‫ ݖ‬െ ܽሻ௣

!
  || ሺ͵ሻ| 
|| 
|| |

 |ܽିଵ | ሺͳሻ| |,| 

| ܽିଵ | |

 || ݂ ሺ‫ݖ‬ሻ||‫ ݖ‬ൌ ܽ|

' '+)++ | "|  
| | 
| | 
|݂ሺ‫ݖ‬ሻ| | ‫ ݖ‬ൌ ܽ| | 
| 
|  #|ሺͳሻ| | | 
|| 
| |    ||݂ሺ‫ݖ‬ሻ||
||! || | 

||ൌ ܽ| |$|
||| || )|‫ ݖ‬ൌ ܽ| ||  ||/||
|
 || |
 |
||ܽିଵ | ͳ ௞ିଵ ሺ‫ ݖ‬െ ܽሻ௞ ݂ሺ‫ݖ‬ሻሺͷሻ| ௭՜௔ ሺ݇ െ ͳሻ ‫ ݖ‬௞ିଵ

ܽିଵ ൌ Ž‹


| ݇ ൌ ͳ|2  |
 4| |

 | 

# |
 || ܽିଵ ൌ Ž‹௭՜௔ ሺ‫ ݖ‬െ ܽ ሻ݂ሺ‫ݖ‬ሻ|
| ||| 

||ሺͷሻ| |݇ ൌ ͳ|
| | | ൌ ͳ |

?
  |




c
      |

Ex  ௭


|݂ሺ‫ݖ‬ሻ ൌ ሺ௭ିଵሻሺ௭ାଵሻమ||| ‫ ݖ‬ൌ ͳ| |‫ ݖ‬ൌ െͳ| | 
 || E |
 | | | ||

| #|ሺ͸ሻ| |ሺͷሻ| |݇ ൌ ʹ||

௭

ଵ

1
||;+|Ž‹௭՜ଵ ሺ‫ ݖ‬െ ͳሻ ÿሺ௭ିଵሻሺ௭ାଵሻమ ൌ | 1
||‫ ݖ‬ൌ െͳ|Ž‹௭՜ିଵ

ଵ ௗ

ଵ ௗ௭

ସ

௭

ଵ

ሺ‫ ݖ‬൅ ͳሻଶ ሺ௭ିଵሻሺ௓ାଵሻమ ቃ ൌ െ | ସ


| ‫ ݖ‬ൌ ܽ| |  | | 
 | 

| 
 | | 
|  | 
| ' |  | | 

 | |     ||
| |

Ex  భ


|݂ሺ‫ݖ‬ሻ ൌ  ݁ ି ೥ || |‫ ݖ‬ൌ Ͳ|; || 
 |

|
 | | #| |    ||| భ

ଵ

݁ ௨  |ൌ െ |||  |݁ ି೥ ൌ ͳ െ ௭

ଵ



 ||| |||||  | |െͳ ||

ଵ

ଵ௭

൅

ଶ ௭ భమ

െ

ଵ

ଷ௭

൅ ‫ ݖ||
| |  | |)|||ڮ‬ൌ Ͳ|  | |

௭

|

!).$+)++ | 
| ݂ሺ‫ݖ‬ሻ| | 
| 
|  |  
| | 
 
| | | | 
 | |  | | | | | | | 
 
>|
 
| | | | | 

| | 
| | ݂ሺ‫ݖ‬ሻ|  | ܽିଵ ǡ ܾିଵ ǡ ܿିଵ ǡ ǥ| | | #||
| 
|| ර ݂ ሺ‫ݖ‬ሻ ‫ ݖ‬ൌ ʹߨ݅ሺ ܽିଵ ൅ ܾିଵ ൅ ܿିଵ ൅ ‫ ڮ‬ሻሺ͹ሻ| ௖

| |  |
|| 
  ||݂ሺ‫ݖ‬ሻ|| | |||| | ||ʹߨ݅ |
| |||
||݂ሺ‫ݖ‬ሻ| | |
 
| || | |

|

?
  |




c
      | M
ሺ͹ሻ est une généralisation de ሺ͵ሻ A Le théorème de cauchy et les formes intégrales sont des cas particuliers de ce théorèmeA

   | @ | |
|| ௭మ ିଶ௭

ଵସ

 | ݂ ሺ‫ݖ‬ሻ ൌ  ሺ௭ାଵሻమሺ௭ మ ାସሻ|| െ ଶÿ|

„Ǥ| ݂ሺ‫ݖ‬ሻ ൌ

௘೥

ି௜ ||| | ||||||||||||  |‫ݖ‬ ௦௜ మ ௭ ଶÿ

ൌ Ͳǡ േߨǡ േʹߨ ǥ ‫ ݖݐ݅݋ݏ‬ൌ ݉ߨ

 |

     | ஶ

௫ మ ௗ௫

గ

<| J |||‫ି׬‬ஶ ሺ௫ మାଵሻమሺ௫మ ାଶ௫ାଶሻ ൌ ÿ଴ | ௭మ

!| E ||ሺ௭ మ ାଵሻమሺ௭ మ ାଶ௭ାଶሻ|
|| 
 
|| |||| |‫ ݖ‬ൌ ݅|6|

|$ |‫ ݖ‬ൌ െͳ ൅ ݅ ||+| 1‫ ݖ‬ൌ ݅ | |Ž‹௭՜௜

ௗ

ௗ௭

௭మ

ሺ‫ ݖ‬െ ݅ሻଶ ሺ௭ା௜ሻమሺ௭ି௜ሻమ

ሺ௭ మ ାଶ௭ାଵሻ ௭మ

ቃൌ

ÿ௜ିଵଶ ଵ଴଴

|

1‫ ݖ‬ൌ െͳ ൅ ݅ | |Ž‹௭՜ିଵା௜ ሺ‫ ݖ‬൅ ʹ െ ݅ ሻ ሺ௭ା௜ሻమሺ௭ାଶି௜ሻሺ௭ାଵା௜ሻ ൌ ර ௖

ଷିସ௜ ଶÿ

|

͹ߨ ͻ݅ െ ͳʹ ͵ െ Ͷ݅ ‫ݖ‬ଶ ‫ݖ‬ ൅ ൰ൌ | ൌ ʹߨ݅ ൬ ଶ ଶ ଶ ሺ‫ ݖ‬൅ ݅ ሻ ሺ‫ ݖ‬െ ݅ ሻ ሺ‫ ݖ‬൅ ʹ‫ ݖ‬൅ ͳሻ ʹͷ ͷͲ ͳͲͲ

‫ ݔ‬ଶ ‫ݔ‬ ͹ߨ ‫ ݖ‬ଶ ‫ݖ‬ න ൌ | ൅ ଶ ଶ ଶ ଶ ଶ ଶ ͷͲ ି
ሺ‫ ݔ‬൅ ͳሻ ሺ‫ ݔ‬൅ ʹ‫ ݔ‬൅ ʹሻ ୻ ሺ‫ ݖ‬൅ ͳሻ ሺ‫ ݖ‬൅ ʹ‫ ݖ‬൅ ʹሻ


‫ ݑ݋‬න

?
  |




c
      |

$|  | |

| | |  
| |ܴ ՜ λ|  | |  | | | 
#|
  | | |||
 | |

 | | ଶగ ௖௢௦ଷఏ

<| |J ||‫׬‬଴

ÿିସ௖௢௦ఏ

గ

ߠ ൌ

ଵଶ

| ௭ା௭ షభ

ฺ ܿ‫ ߠ͵ݏ݋‬ൌ ଶ ‫ݖ‬ ‫ ݖ‬ൌ ݅݁ ௜ఏ ߠ ᇱ ‫݋‬î ߠ ൌ | ݅‫ݖ‬

,| |‫ ݖ‬ൌ ݁௜ఏ | || ||ܿ‫ ߠݏ݋‬ൌ

௘ ೔ÿ ା௘ ష ೔ÿ ଶ

ൌ

௭ ା௭ష ଶ

|

௭ ା௭ ష

ͳ ܿ‫ߠ͵ݏ݋‬ ‫ݖ‬ ‫ݖ‬ÿ ൅ ͳ ଶ න ර ൌ െ ߠ ൌ ර ‫| ݖ‬ షభ ௭ା௭ ʹ݅ ‫ݖ‬ଷ ሺʹ‫ ݖ‬െ ͳሻሺ‫ ݖ‬െ ʹሻ ଴ ͷ െ Ͷܿ‫ߠݏ݋‬ ௖ ͷെቀ ቁ ݅‫ݖ‬ ௖ ଶ )| | |  |!|
||
 ||| E | 
||‫ ݖ‬ൌ Ͳ| || E |
 | |‫ ݖ‬ൌ ͳ ʹ|| 
 
|| | ଶగ

ଵ ௗమ

1 ‫ ݖ‬ൌ Ͳ| |Ž‹௭՜଴ ଶ

ௗ௭మ

1 ‫ ݖ‬ൌ ͳ ʹ|| |Ž‹௭՜ଵ ൫‫ݖ‬ ଶ ௭ ାଵ െ ଶ௜ 6௖ ௭ ሺଶ௭ିଵሻሺ௭ିଶሻ ‫ݖ‬ ଵ

|

ଶగ

<| J |||||‫׬‬଴

ௗఏ

ሺÿିଷ௦௜ ఏሻమ

ൌ

௭ ାଵ ÿଵ ቃൌ ÿ| ሺଶ௭ିଵሻሺ௭ିଶሻ ௭ ାଵ െ ͳ ʹ
ሺଶ௭ିଵሻሺ௭ିଶሻቃ ൌ ௭ గ ÿଵ ଵ ÿÿ

‫ ݖ‬ଷ ௭

െ

ÿÿ

ଶସ

|||)|

ൌ െ ଶ௜ ሺʹߨ݅ሻ ቀ ÿ െ ଶସቁ ൌ ଵଶ
| |  ||

ÿగ ଷଶ

|

| ஶ ௖௢௦௠௫ గ <| J ||‫׬‬଴ మ ‫ ݔ‬ൌ ݁ ି௠ ݉ ൐ Ͳ| ௫ ାଵ ଶ | |

,|
#|6௖ |

௘೔ ೥

௭ మ ାଵ

 ‫| | | ||)| ݖ‬

‫ ݖ‬ൌ േ݅ |
|‫ ݖ‬ൌ ݅ || 
 
|| |

||!|
||| 
 |

| ௘೔ ೥

1 ‫ ݖ‬ൌ ݅ | |||Ž‹௭՜௜ ሺ‫ ݖ‬െ ݅ሻ ሺ௭ି௜ሻሺ௭ା௜ሻቃ ൌ 
௘೔ ೣ

௖

|,|‫
ି׬‬మ ‫ ݔ‬൅ ݅ ‫୻׬‬ ௫ ାଵ

?
  |

|

௘೔ ೥

௭మ ାଵ

‫ ݖ‬ൌ ߨ݁ ି௠ |

݁ ௜௠௭ ܿ‫ݔ݉ݏ݋‬ න ‫ ݖ‬ൌ ߨ݁ ି௠ | ‫ݔ‬ ൅ ଶ ଶ ୻ ‫ ݖ‬൅ͳ ଴ ‫ ݔ‬൅ͳ
௖௢௦௠௫ గ
||!|
| |1| | 


| |
| |

| 
  ՜ Ͳ|‫׬‬଴ మ ‫ ݔ‬ൌ ݁ ି௠ | ʹන

|

ଶ௜ ି௠

݁ ݁ ቇ ൌ ߨ݁ ି௠ |  ‫ ݖ‬ൌ ʹߨ݅ ቆ ଶ ʹ݅ ‫ ݖ‬൅ͳ ௜௠௭

ර

௘ష




௫ ାଵ

ଶ




c
      |

ஶ

௫௦௜ గ௫

<|   ||‫ି׬‬ஶ ௫ మାଶ௫ାÿ ‫ݔ‬

?
  |




c
      |

CHAPITRE XIII TRANSFORMEE DE LAPLACE   | !| ||! | || 
|| |R3|
| |
|| |
|‫ܨ‬ሺ‫ݐ‬ሻ| | |

||24| ||
|| ஶ


| ‫ܨ‬ሺ‫ݐ‬ሻ| |
|

| |‫ ݐ‬൐ Ͳ| 3Ÿ‫ܨ‬ሺ‫ݐ‬ሻÿ ൌ ݂ሺ‫ݏ‬ሻ ൌ ‫׬‬଴ ݁ ି௦௧ ‫ܨ‬ሺ‫ݐ‬ሻ ‫ ݐ‬ሺͳሻ|)| ‫| | '|  |ݏ‬ |  |!| 
||! ||‫ܨ‬ሺ‫ݐ‬ሻ|
 |
| 
  | ሺͳሻ…‘ ‡”‰‡| | 
|  ||‫|  | 
|

|
 || |&| ݏ‬


| | | 
| 
| 

| | ‫ ݐ‬൐ Ͳ‫ ܨ‬ሺ‫ݐ‬ሻǡ ‫ ܩ‬ሺ‫ݐ‬ሻǡ ÿሺ‫ݐ‬ሻ| | 
|  |  
 | |݂ ሺ‫ݏ‬ሻǡ ݃ሺ‫ݏ‬ሻǡ ‫ݕ‬ሺ‫ݏ‬ሻ | 3  |

|| |
݇ଵ ǡ ݇ଶ|  | |   |  |  | ‫ܨ‬ଵ ሺ‫ݐ‬ሻǡ ‫ܨ‬ଶ ሺ‫ݐ‬ሻ| | 
|  | |

||! | | 
 |݂ଵሺ‫ݏ‬ሻ݁‫݂ݐ‬ଶ ሺ‫ݏ‬ሻ| || 3 Ÿ݇ଵ ‫ܨ‬ଵ ሺ‫ݐ‬ሻ ൅ ݇ ଶ ‫ܨ‬ଶ ሺ‫ݐ‬ሻÿ ൌ ݇ଵ 3‫ܨ‬ଵ ሺ‫ݐ‬ሻ ൅ ݇ଶ 3‫ܨ‬ଶ ሺ‫ݐ‬ሻ ൌ ݇ଵ ݂ଵ ሺ‫ݏ‬ሻ ൅ ݇ଶ ݂ଶሺ‫ݏ‬ሻ

B -8$|&$ |@-J!$ |&$ |,.@D,. |$!$J$.@-D1$ | ûሺ࢚ሻ


3Ÿûሺ࢚ሻÿ ൌ Áሺ࢙ሻ

+|

ͳ ‫ ݏ‬൐ Ͳ| ‫ݏ‬

@|

ͳ ‫ ݏ‬൐ Ͳ| ‫ݏ‬ଶ

‫| ݐ‬

݊ ൌ Ͳǡͳǡʹǡ ǥ|

| . || 


|;݊ ൌ ͳǤʹǤ͵ ǥ ݊| |Ͳ ൌ ͳ|

݁ ௔௧ |

௔

|‫ ݏ‬൐ Ͳ|

௦

|‫ ݏ‬൐ Ͳ|

•‹ ܽ‫| ݐ‬

௦మ ା௔మ

…‘• ܽ‫| ݐ‬

௦మ ା௔మ

‫|ݐ݄ܽݏ‬ ݄ܿܽ‫|ݐ‬

|

d
( "

 |

ͳ ‫ ݏ‬൐ ܽ| ‫ݏ‬െܽ

௔

௦ మ ି௔మ ௦

௦ మ ି௔మ

|‫ ݏ‬൐ ȁܽ ȁ| |‫ ݏ‬൐ ȁܽ ȁ|




c
      |

ଵ


|‫ ܨ‬ሺ‫ݐ‬ሻ ൌ ͳ|3Ÿͳÿ ൌ ௦ ǡ ‫ ݏ‬൐ Ͳ

|

௉ ௘ష ÿ

3 Ÿͳÿ ൌ ‫׬‬଴ ݁ ି௦௧ ሺͳሻ ‫ ݐ‬ൌ Ž‹௉՜ஶ ‫׬‬଴ ݁ ି௦௧ ‫ ݐ‬ൌ Ž‹௉՜ஶ
ஶ

|

௉


|

|

ሺͳሻ
ቀ

௦మ

ቁቚ

ି௦ ଴

ஶ

‫ ܨ‬ሺ‫ݐ‬ሻ ൌ ‫|ݐ‬

ଵି௘ ష ÿ ௦

ൌ

ଵ ௦

௉

௘ష ÿ

3Ÿ‫ݐ‬ÿ ൌ ‫׬‬଴ ݁ ି௦௧ ሺ‫ݐ‬ሻ ‫ ݐ‬ൌ Ž‹௉՜ஶ ‫׬‬଴ ‫ି ݁ݐ‬௦௧ ‫ ݐ‬ൌ Ž‹௉՜ஶ ሺ‫ݐ‬ሻ ቀ

௉ ଴

ି௦

ͳ ݁ ି௦௉ m݁ ି௦௉ ͳ ቇ ൌ ଶ ǡ ‫ ݏ݅ݏ‬൐ Ͳ ൌ Ž‹ ቆ ଶ െ ଶ െ ௉՜ஶ ‫ݏ‬ ‫ݏ‬ ‫ݏ‬ ‫ݏ‬

ቁെ

௉

ஶ ஶ ௘
|‫ ܨ‬ሺ‫ݐ‬ሻ ൌ ݁ ௔௧ ǡ3Ÿ݁ ௔௧ ÿ ൌ ‫׬‬଴ ݁ ି௦௧ ሺ݁ ௔௧ ሻ ‫ ݐ‬ൌ Ž‹௉՜ஶ ‫׬‬଴ ݁ ିሺ௦ି௔ሻ௧  ‫ ݐ‬ൌ
Ž‹௉՜ஶ ିሺ௦ି௔ሻ ÿ | షሺ షೌሻÿ

| |

௘ష ÿ

ቚ ൌ Ž‹௉՜ஶ

|

|

|

ൌ Ž‹௉՜ஶ

|

ஶ

௘ షሺ షೌሻÿ ିሺ௦ି௔ሻ

ൌ

ଵ

௦ି௔

଴

‫ ݏ݅ݏ‬൐ ܽ| ௉


|‫ܨ‬ሺ‫ݐ‬ሻ ൌ •‹ ܽ‫ ݐ‬ǡ3Ÿ•‹ ܽ‫ݐ‬ÿ ൌ ‫׬‬଴ ݁ ି௦௧ •‹ ܽ‫ ݐ  ݐ‬ൌ Ž‹௉՜ஶ ‫׬‬଴ ݁ ି௦௧ •‹ ܽ‫| ݐ  ݐ‬

|

௉

௔

݁ ି௦௧ ሺെܽ‫ ݐܽ݊݅ݏ‬െ ܽ …‘• ܽ‫ݐ‬ሻ ܽ ݁ ି௦௉ ሺ‫ܽ ‹• ݏ‬m ൅ ܽ …‘• ܽmሻ ቤ ÿ| ൌ Ž‹
ൌ Ž‹ െ ௉՜ஶ ௉՜ஶ ‫ ݏ‬ଶ ൅ ܽ ଶ ‫ݏ‬ଶ ൅ ܽଶ ‫ݏ‬ଶ ൅ ܽଶ ଴

ൌ ௦మ ା௔మ ||‫ ݏ‬൐ Ͳ|

ൌ
Ž‹௉՜ஶ

|

ஶ

௉


|‫ ܨ‬ሺ‫ݐ‬ሻ ൌ …‘• ܽ‫ݐ‬ǡ 3Ÿ…‘• ܽ‫ݐ‬ÿ ൌ ‫׬‬଴ ݁ ି௦௧ …‘• ܽ‫ ݐ ݐ‬ൌ Ž‹௉՜ஶ ‫׬‬଴ ݁ ି௦௧ …‘• ܽ‫| ݐ ݐ‬

|

௉ ௘ ష ÿ ሺି௦௖௢௦௔௧ା௔ ୱ୧୬ ௔௧ሻ

|

|

௦

ቚ ;Ž‹௉՜ஶ ÿ

௦ మ ା௔మ

ൌ

௦

௦ మ ା௔మ

௦ మ ା௔మ

଴

െ

௘ ష
ሺ௦ ୡ୭ୱ ௔௉ି௔ ୱ୧୬ ௔௉ሻ ௦ మ ା௔మ

|

||‫ ݏ‬൐ Ͳ|

R

|

න݁

௧

•‹ ÿ‫ ݐ ݐ‬ൌ

න݁

௧

…‘• ÿ‫ ݐ ݐ‬ൌ

݁ ௧ ሺߙ •‹ ÿ‫ ݐ‬െ ÿ …‘• ÿ‫ݐ‬ሻ | ߙ ଶ ൅ ÿଶ

௦ା௜௔

ଵ


|‫ ܨ‬ሺ‫ݐ‬ሻ ൌ ݁ ௜௔௧ ǡ3 ݁௜௔௧ ÿ ൌ ௦ି௜௔ ൌ ௦మ ା௔మ | ݁௜

ஶ

௧

݁ ௧ ሺߙ …‘• ÿ‫ ݐ‬൅ ÿ •‹ ÿ‫ݐ‬ሻ | ߙ ଶ ൅ ÿଶ

ൌ …‘• ߙ‫ ݐ‬൅ ݅‫ݐߙ݊݅ݏ‬ ஶ

ஶ

3 ݁ ௜௔௧ ÿ ൌ ‫׬‬଴ ݁ ି௦௧ ሺ…‘• ܽ‫ ݐ‬൅ ݅‫ݐܽ݊݅ݏ‬ሻ ‫ ݐ‬ൌ ‫׬‬଴ ݁ ି௦௧ …‘• ܽ‫ ݐ ݐ‬൅ ݅ ‫׬‬଴ ݁ ି௦௧ •‹ ܽ‫ݐ ݐ‬ ௦

௔

ൌ3 Ÿ…‘• ܽ‫ݐ‬ÿ ൅ ݅3 Ÿ•‹ ܽ‫ݐ‬ÿ ൌ ௦మ ା௔మ ൅ ݅ ௦మ ା௔మ |

d
( "

 |




c
      |

௘ ೌÿ ି௘ షೌÿ

 | ‫ ܨ‬ሺ‫ݐ‬ሻ ൌ ‫ݐ݄ܽݏ‬ǡ 3Ÿ‫ܨ‬ሺ‫ݐ‬ሻÿ ൌ3ÿ ଵ

ஶ

ଵ

ଵ

ଵ

ஶ

ଶ

ஶ

ൌ ‫׬‬଴ ݁ ି௦௧ ቀ

௘ ೌÿ ି௘ షೌÿ ଶ

ቁ ‫| ݐ‬

ଵ

ଵ

3 Ÿ‫ܨ‬ሺ‫ݐ‬ሻÿ ൌ ଶ ‫׬‬଴ ݁ ି௦௧ ݁ ௔௧ ‫ ݐ‬െ ଶ ‫׬‬଴ ݁ ି௦௧ ݁ ି௔௧ ‫ ݐ‬ൌ ଶ3 Ÿ݁ ௔௧ ÿ െ ଶ3Ÿ݁ ି௔௧ ÿ ଵ

௔

3 Ÿ‫ܨ‬ሺ‫ݐ‬ሻÿ ൌ ଶ ÿ௦ି௔ െ ௦ା௔ ൌ ௦మ ି௔మ ǡ ‫ ݏ‬൐ ȁܽ ȁ| ௘ ೌÿ ା௘ షೌÿ

 | ‫ܨ‬ሺ‫ݐ‬ሻ ൌ ݄ܿܽ‫ݐ‬ǡ3Ÿ‫ܨ‬ሺ‫ݐ‬ሻÿ ൌ3ÿ

ଶ

ଵ

ଵ

ൌ 3Ÿ݁௔௧ ÿ ൅ 3 Ÿ݁ ି௔௧ ÿ| ଶ

ଶ

ܽ ͳ ͳ ͳ ൠൌ ଶ ൅ ǡ ‫ ݏ‬൐ ȁܽ ȁ| ൌ § ‫ ݏ‬െ ܽଶ ʹ ‫ݏ‬െܽ ‫ݏ‬൅ܽ

&)&)') | ஶ

ஶ

3 Ÿ‫ܨ‬ଵ ሺ‫ݐ‬ሻÿ ൌ ݂ଵሺ‫ݏ‬ሻ ൌ ‫׬‬଴ ݁ ି௦௧ ‫ܨ‬ଵ ሺ‫ݐ‬ሻ ‫
 ݐ‬3Ÿ‫ܨ‬ଶ ሺ‫ݐ‬ሻÿ ൌ ݂ଶሺ‫ݏ‬ሻ ൌ ‫׬‬଴ ݁ ି௦௧ ‫ܨ‬ଶ ሺ‫ݐ‬ሻ ‫݇|
| | ݐ‬ଵ ݁‫݇ݐ‬ଶ | |

|  | ||

ஶ

3 Ÿ݇ଵ ‫ܨ‬ଵ ሺ‫ݐ‬ሻ ൅ ݇ ଶ ‫ܨ‬ଶ ሺ‫ݐ‬ሻÿ ൌ ݇ଵ ‫׬‬଴ ݁ ି௦௧ ‫ܨ‬ଵ ሺ‫ݐ‬ሻ ‫ ݐ‬൅

ൌ

ଵ 3Ÿ‫ܨ‬ଵ ሺ‫ݐ‬ሻÿ

ஶ ି௦௧ ‫ܨ‬ଶ ሺ‫ݐ‬ሻ ‫ݐ‬ ଶ ‫׬‬଴ ݁

൅ ݇ଶ3 Ÿ‫ܨ‬ଶሺ‫ݐ‬ሻÿ ൌ ݇ଵ ݂ଵ ሺ‫ݏ‬ሻ ൅ ݇ଶ ݂ଶ ሺ‫ݏ‬ሻ

Ex  |

‫ ܨ‬ሺ‫ݐ‬ሻ ൌ Ͷ݁ ଷ௧ ൅ ͸‫ ݐ‬ଷ െ ͵ •‹ Ͷ‫ ݐ‬൅ ʹ …‘• ʹ‫|ݐ‬

3 Ÿ‫ܨ‬ሺ‫ݐ‬ሻÿ ൌ Ͷ3Ÿ݁ ଷ௧ ÿ ൅ ͸3 Ÿ‫ ݐ‬ଷ ÿ െ ͵3Ÿ•‹ Ͷ‫ݐ‬ÿ ൅ ʹ3Ÿ…‘• ʹ‫ݐ‬ÿ

‫ݏ‬ ͵ Ͷ ͵͸ ͳʹ ʹ‫ݏ‬ ͳ Ͷ ቁൌ ൰ ൅ ͸ ൬ ସ൰ െ ͵ ൬ ଶ ൰ ൅ʹቀ ଶ ൅ ସെ ଶ ൅ ଶ ൌ Ͷ൬ ‫ ݏ‬൅ Ͷ ‫ݏ‬ ‫ ݏ‬൅ ͳ͸ ‫ ݏ‬൅ ͳ͸ ‫ ݏ‬൅ Ͷ ‫ݏ‬െͷ ‫ݏ‬െͷ ‫ݏ‬

)+'&)&)+ !($- !''  |

Ã)&)ß |


|‫ ܨ‬ሺ‫ݐ‬ሻǡ ‫ ݐ‬൐ Ͳ| |3Ÿ‫ܨ‬ሺ‫ݐ‬ሻÿ ൌ ݂ሺ‫ݏ‬ሻ  3Ÿ݁ ௔௧ ‫ܨ‬ሺ‫ݐ‬ሻÿ ൌ ݂ሺ‫ ݏ‬െ ܽሻ

|

"| 
|

|  
 || ஶ

3 Ÿ‫ܨ‬ሺ‫ݐ‬ሻÿ ൌ ‫׬‬଴ ݁ ି௦௧ ‫ܨ‬ሺ‫ݐ‬ሻ ‫ ݐ‬ൌ ݂ሺ‫ݏ‬ሻ| ஶ

ஶ

3 Ÿ݁ ௔௧ ‫ܨ‬ሺ‫ݐ‬ሻÿ ൌ ‫׬‬଴ ݁ ି௦௧ Ÿ݁௔௧ ‫ܨ‬ሺ‫ݐ‬ሻÿ ‫ ݐ‬ൌ ‫׬‬଴ ݁ ିሺ௦ି௔ሻ௧ ‫ܨ‬ሺ‫ݐ‬ሻ  ‫ ݐ‬ൌ ݂ሺ‫ ݏ‬െ ܽሻ

d
( "

 |




c
      |

Ex 
|‫ ܨ‬ሺ‫ݐ‬ሻ ൌ ‫ ݐ‬ଶ ݁ଷ௧ |

+|

ଶ

ଶ

1  ||3Ÿ‫ ݐ‬ଶÿ ൌ ௦ ൌ ௦ | 3 Ÿ‫ ݐ‬ଶ ݁ ଷ௧ ÿ ൌ

ଶ ሺ௦ିଷሻ

|

6 | |
||‫ܨ‬ሺ‫ݐ‬ሻ ൌ ݁ ିଶ௧ •‹ Ͷ‫|ݐ‬ 1  ||3Ÿ•‹ Ͷ‫ݐ‬ÿ ൌ 3 Ÿ݁ ିଶ௧ •‹ Ͷ‫ݐ‬ÿ ൌ

ସ

௦ మ ାଵÿ

ସ ሺ௦ାଶሻమ ାଵÿ

ସ

ൌ ௦మାସ௦ାଶ଴


|‫ ܨ‬ሺ‫ݐ‬ሻ ൌ ݁ ସ௧ …Š ͷ‫|ݐ‬

3|

௦

1  |||3Ÿ݄ܿͷ‫ݐ‬ÿ ൌ ௦మ ିଶÿ

௦ିସ

௦ିସ

3 Ÿ݁ ସ௧ …Š ͷ‫ݐ‬ÿ ൌ ሺ௦ିସሻమିଶÿ ൌ ௦మ ିÿ௦ିÿ|

S | |
|‫ܨ‬ሺ‫ݐ‬ሻ ൌ ݁ ିଶ௧ ሺ͵ …‘• ͸‫ ݐ‬െ ͷ •‹ ͸‫ݐ‬ሻ|

1  ||||3Ÿ͵ …‘• ͸‫ ݐ‬െ ͷ •‹ ͸‫ݐ‬ÿ ൌ ͵3Ÿ…‘• ͸‫ݐ‬ÿ െ ͷ3Ÿ•‹ ͸‫ݐ‬ÿ ൌ ͵ ቀ 3 Ÿ݁ ିଶ௧ ሺ͵ …‘• ͸‫ ݐ‬െ ͷ •‹ ͸‫ݐ‬ሻÿ ൌ

ଷሺ௦ାଶሻିଷ଴ ሺ௦ାଶሻమ ାଷÿ

ൌ

ଷ௦ିଶସ

௦

௦ మ ାଷÿ

ቁെ ͷቀ

ÿ

ቁൌ

௦ మ ାଷÿ

ଷ௦ିଷ଴ ௦ మ ାଷÿ

௦ మ ାସ௦ାସ଴

ß$&)&))+' |

Ã)&)2  |

‫ ܨ‬ሺ‫ ݐ‬െ ܽሻ‫ ݐ‬൐ ܽ
|
|3‫ ܨ‬ሺ‫ݐ‬ሻ ൌ ݂ሺ‫ݏ‬ሻ ‫ ܩ‬ሺ‫ݐ‬ሻ ൌ § Ͳǡ‫ ݐ‬൏ ܽ

|

3 Ÿ‫ܩ‬ሺ‫ݐ‬ሻÿ ൌ ݁ ି௦௧ ݂ሺ‫ݏ‬ሻሺͶሻ

"| 
|

|  
 ||
#| ஶ ௔ ஶ ஶ ஶ 3 Ÿ‫ܩ‬ሺ‫ݐ‬ሻÿ ‫׬‬଴ ݁ ି௦௧ ‫ ܩ‬ሺ‫ݐ‬ሻ ‫ ݐ‬ൌ ‫׬‬଴ ݁ ି௦௧ ‫ ܩ‬ሺ‫ݐ‬ሻ ‫ ݐ‬൅ ‫׬‬௔ ݁ ି௦௧ ‫ ܩ‬ሺ‫ݐ‬ሻ ‫ ݐ‬ൌ ‫׬‬଴ ݁ ି௦௧ ሺͲሻ ‫ ݐ‬൅ ‫׬‬௔ ݁ ି௦௧ ‫ ܨ‬ሺܽ െ ‫ݐ‬ሻ ‫ݐ‬ |

|

|

,|| |‫ ݐ‬ൌ ‫ ݑ‬൅ ܽ|

ஶ

d
( "

 |

ஶ

ൌ ‫׬‬௔ ݁ ି௦ሺ௨ା௔ሻ ‫ ܨ‬ሺ‫ ݑ‬ሻ ‫ ݑ‬ൌ ݁ ି௔௦ ‫׬‬௔ ݁ ି௦௨ ‫ ܨ‬ሺ‫ ݑ‬ሻ ‫ ݑ‬ൌ ݁ ି௔௦ ݂ሺ‫ݏ‬ሻ|




c
      |

Ex    ||3Ÿ‫ܨ‬ሺ‫ݐ‬ሻÿ
‫ ܨ‬ሺ‫ݐ‬ሻ ൌ ÿ మഏ

…‘• ቀ‫ ݐ‬െ

ଶగ ଷ

ቁ ǡ‫ ݐ‬൐

Ͳǡ‫ ݐ‬൏

ஶ

3 Ÿ‫ܨ‬ሺ‫ݐ‬ሻÿ ൌ ‫׬‬଴ ݁ ି௦௧ ሺͲሻ ൅ ‫׬‬మഏ ݁ ି௦௧ …‘• ቀ‫ ݐ‬െ

ൌ݁

ି

మഏ

ଶగ ଷ

ଶగ

ଷ
ଶగ ଷ

ஶ

ቁ  ‫ ݐ‬ൌ ‫׬‬଴ ݁ ି௦ቀ௨ା

ஶ

మഏ

ቁ

…‘• ‫ݑ ݑ‬

మഏ

‫ି ݁ݏ‬ ି௦௨ න݁ …‘• ‫ ݑ ݑ‬ൌ ଶ ‫ ݏ‬൅ͳ ଴

Ã)&) !($- !'' |

Ã)&)3 |

௦

ଵ


|3Ÿ‫ܨ‬ሺ‫ݐ‬ሻÿ ൌ ݂ሺ‫ݏ‬ሻ 3Ÿ‫ܨ‬ሺܽ‫ݐ‬ሻÿ ൌ ݂ ቀ ቁ

|

௔

ஶ

௔

௨

"| 
|

|  
 ||
#|3Ÿ‫ܨ‬ሺ‫ݐ‬ሻÿ ൌ ‫׬‬଴ ݁ ି௦௧ ‫ܨ‬ሺܽ‫ݐ‬ሻ ‫ ݐ| | ݐ‬ൌ | ௔ ଵ

ೠ

ஶ

ଵ

௦

3 Ÿ‫ܨ‬ሺܽ‫ݐ‬ሻÿ ൌ ‫׬‬଴ ݁ ି ೌ ‫ ܨ‬ሺ‫ ݑ‬ሻ ‫ ݑ‬ൌ ݂ ቀ ቁ ௔ ௔

௔

Ex  |

ୱ୧୬ ௧

$  ||3ÿ

ୱ୧୬ ௔௧

ୱ୧୬ ௔௧

ൌ 3ÿ

ୱ୧୬ ௔௧

ൌ ‫ି ݃ݐ‬ଵ ቀ ቁ

3ÿ

|

ଵ

௧

3ÿ

௔௧ ௧

௔

௧

ଵ

ൌ ‫ି ݃ݐ‬ଵ ቀ௦ ቁ   |3 ÿ ଵ

ଵ

ଵ

ୱ୧୬ ௔௧ ௔௧ ௔

ൌ ‫ି݃ݐ‬ଵ § ൠ ൌ ‫ି݃ݐ‬ଵ ቀ ቁ| ௔

ೌ

௔

௦

௔ ௦

)+)$+&' )*+  |

4) 
|‫ܨ‬ሺ‫ݐ‬ሻ|

|‫ ݐ‬൐ Ͳ| |
|3‫ ܨ‬ሺ‫ݐ‬ሻ ൌ ݂ሺ‫ݏ‬ሻ| |3Ÿ‫ ܨ‬ᇱ ሺ‫ݐ‬ሻÿ ൌ ‫ ݂ݏ‬ሺ‫ݏ‬ሻ െ ‫ܨ‬ሺͲሻሺͷሻ


‫ܨ‬ሺ‫ݐ‬ሻ| |
| |Ͳ ൑ ‫ ݐ‬൑ | || 
 | |൐ || |Ͳ ൑ ‫ ݐ‬൑ ||$|
  | | 
|||

d
( "

 |

|| 
||‫ ܨ‬ᇱ ሺ‫ݐ‬ሻ| |
|




c
      | ௣ ஶ ௣ ௣ 3 Ÿ‫ ܨ‬ᇱ ሺ‫ݐ‬ሻÿ ൌ ‫׬‬଴ ݁ ି௦௧ ‫ ܨ‬ᇱ ሺ‫ݐ‬ሻ ‫ ݐ‬ൌ Ž‹௣՜ஶ ‫׬‬଴ ݁ ି௦௧ ‫ ܨ‬ᇱ ሺ‫ݐ‬ሻ ‫ ݐ‬ൌ Ž‹௣՜ஶ
݁ ି௦௧ ‫ܨ‬ሺ‫ݐ‬ሻȁ଴ ൅ ‫׬ ݏ‬଴ ݁ ି௦௧ ‫ ܨ‬ሺ‫ݐ‬ሻ ‫ ݐ‬ÿ ஶ

ൌ ‫ ݏ‬න ݁ ି௦௧ ‫ܨ‬ሺ‫ݐ‬ሻ ‫ ݐ‬െ ‫ ܨ‬ሺͲሻ ൌ ‫݂ݏ‬ሺ‫ݏ‬ሻ െ ‫ܨ‬ሺͲሻ ଴

$|

 | |
||‫ܨ‬ሺ‫ݐ‬ሻ| || 
 |||‫ ݐ‬՜ λ|||| Ž‹௣՜ஶ ݁ ି௦௣ ‫ ܨ‬ሺ‫݌‬ሻ ൌ Ͳ |‫ ݏ‬൐ |

34
| | ‫ܨ‬ሺ‫ݐ‬ሻ|  | | 
| | ‫ ݐ‬ൌ Ͳ| 
| | |

| | 
| 
 | Ž‹௧՜଴ ‫ܨ‬ሺ‫ݐ‬ሻ ൌ ‫ܨ‬ሺͲା ሻ|  |||||| 3 Ÿ‫ܨ‬ሺ‫ݐ‬ሻÿ ൌ ‫݂ݏ‬ሺ‫ݏ‬ሻ െ ‫ ܨ‬ሺͲା ሻ 3,4
| | ‫ܨ‬ሺ‫ݐ‬ሻ|  | | 
| | ‫ ݐ‬ൌ ܽ |  | 3Ÿ‫ܨ‬ÿሺ‫ݐ‬ሻÿ ൌ ‫ ݂ݏ‬ሺ‫ݏ‬ሻ െ ‫ܨ‬ሺͲሻ െ ݁ ି௔௦ Ÿ‫ ܨ‬ሺܽ ା ሻ െ ‫ ܨ‬ሺܽି ሻÿ)| ‫ ܨ‬ሺܽା ሻ െ ‫ ܨ‬ሺܽି ሻ|  | |   | |  | | 
 | | 


| ‫ ݐ‬ൌ ܽ | 
| | | 
| 
 | | 


|
| |

| || |  
 |||  
 5) L     
  ûሺ࢚ሻ
|‫ܨ‬ሺ‫ݐ‬ሻ| |

|
|‫ ݐ‬൐ Ͳ| |
|3Ÿ‫ܨ‬ሺ‫ݐ‬ሻÿ|;݂ሺ‫ݏ‬ሻ| ||3Ÿ‫ܨ‬ÿÿሺ‫ݐ‬ሻÿ|;‫ ݏ‬ଶ ݂ ሺ‫ݏ‬ሻ െ ‫ ܨݏ‬ሺͲሻ െ ‫ ܨ‬ᇱ ሺͲሻǤ|
|‫ܨ‬ሺ‫ݐ‬ሻ|  |‫ ܨ‬ᇱ ሺ‫ݐ‬ሻ| |
| |Ͳ ൑ ‫ ݐ‬൑ | || 
 | |‫ ݐ‬൐ | 
||‫ ܨ‬ᇱᇱ ሺ‫ݐ‬ሻ|| | 
| || |Ͳ ൑ ‫ ݐ‬൑ Ǥ| "| 
|

|  
 || 3 Ÿ‫ ܩ‬ᇱ ሺ‫ݐ‬ሻÿ ൌ ‫ݏ‬3 Ÿ‫ܩ‬ሺ‫ݐ‬ሻÿ െ ‫ ܩ‬ሺͲሻ ൌ ‫݃ݏ‬ሺ‫ݏ‬ሻ െ ‫ܩ‬ሺͲሻ|  |‫ ܩ‬ሺ‫ݐ‬ሻ ൌ ‫ ܨ‬ᇱ ሺ‫ݐ‬ሻ|||| 3 Ÿ‫ܨ‬ÿÿሺ‫ݐ‬ሻÿ ൌ ‫ݏ‬3Ÿ‫ܨ‬ÿሺ‫ݐ‬ሻ െ ‫ܨ‬ÿሺͲሻÿ ൌ ‫ݏ‬ሾ‫ݏ‬3 Ÿ‫ܨ‬ሺ‫ݐ‬ሻÿ െ ‫ܨ‬ሺͲሻሿ െ ‫ ܨ‬ᇱ ሺͲሻ

3 Ÿ‫ܨ‬ÿÿሺ‫ݐ‬ሻÿ ൌ ‫ ݏ‬ଶ 3Ÿ‫ܨ‬ሺ‫ݐ‬ሻÿ െ ‫ ܨݏ‬ሺͲሻ െ ‫ ܨ‬ᇱ ሺͲሻ ൌ ‫ ݏ‬ଶ ݂ ሺ‫ݏ‬ሻ െ ‫ ܨݏ‬ሺͲሻ െ ‫ ܨ‬ᇱ ሺͲሻ

™) D  
  
|‫ܨ‬ሺ‫ݐ‬ሻ|

| |‫ ݐ‬൐ Ͳ| |
|3Ÿ‫ܨ‬ሺ‫ݐ‬ሻÿ ൌ ݂ሺ‫ݏ‬ሻ ||

3 ‫ ܨ‬ሺ ሻ ሺ‫ݐ‬ሻÿ ൌ ‫݂ ݏ‬ሺ‫ݏ‬ሻ െ ‫ݏ‬

ିଵ

‫ ܨ‬ሺͲሻ െ ‫ݏ‬

ିଶ ᇱ ሺ

‫Ͳ ܨ‬ሻ െ ‫ ڮ‬െ ‫ܨݏ‬

ିଶ ሺ

Ͳሻ െ ‫ܨ‬

ିଵ

ሺͲሻ


| ‫ܨ‬ሺ‫ݐ‬ሻǢ ‫ ܨ‬ᇱ ሺ‫ݐ‬ሻǢ ǥ ‫ି ܨ‬ଵ ሺ‫ݐ‬ሻ|  | 
| |Ͳ ൑ ‫ ݐ‬൑ |  |  
 | | | |‫ ݐ‬൐ Ͳ|  ||‫ ܨ‬ሺ ሻ ሺ‫ݐ‬ሻ| |
| || |Ͳ ൑ ‫ ݐ‬൑ Ǥ|

Ex    | |
|
|| ‫ ܨ‬ሺ‫ݐ‬ሻ ൌ •‹ ܽ‫| ݐ‬

‫ ܨ‬ᇱ ሺ‫ݐ‬ሻ ൌ ܽ …‘• ܽ‫|ݐ‬

‫ ܨ‬ሺͲሻ ൌ Ͳ|

‫ ܨ‬ᇱ ሺ Ͳ ሻ ൌ ܽ|

d
( "

 |




c
      | ‫ ܨ‬ᇱᇱሺ௧ሻ ൌ െܽଶ •‹ ܽ‫|ݐ‬

3Ÿ•‹ ܽ‫ݐ‬ÿ ൌ

3 Ÿ‫ ܨ‬ᇱ ሺ‫ݐ‬ሻÿ ൌ ‫ݏ‬3 Ÿ•‹ ܽ‫ݐ‬ÿ െ ‫ܨ‬ሺͲሻ

௔

௦ మ ା௔మ

ൌ‫ݏ‬

‫ܽݏ‬ ܽ െͲൌ ଶ | ଶ ‫ ݏ‬൅ ܽଶ ൅ܽ

‫ݏ‬ଶ

3 Ÿ‫ ܨ‬ᇱᇱ ሺ‫ݐ‬ሻÿ ൌ ‫ ݏ‬ଶ3 Ÿ•‹ ܽ‫ݐ‬ÿ െ ‫ ܨݏ‬ሺͲሻ െ ‫ ܨ‬ᇱ ሺͲሻ

ൌ‫ݏ‬

ܽ ܽଷ െ Ͳ െ ܽ ൌ െ ‫ ݏ‬ଶ ൅ ܽଶ ‫ݏ‬ଶ ൅ ܽଶ

3 Ÿ‫ ܨ‬ᇱᇱᇱ ሺ‫ݐ‬ሻÿ ൌ ‫ ݏ‬ଷ ݂ሺ‫ݏ‬ሻ െ ‫ ݏ‬ଶ ‫ܨ‬ሺͲሻ െ ‫ܨݏ‬ÿሺͲሻ െ ‫ ܨ‬ᇱᇱ ሺͲሻ

3 Ÿ‫ ܨ‬ᇱᇱᇱ ሺ‫ݐ‬ሻÿ ൌ ‫ ݏ‬ଷ

‫ ܨ‬ᇱᇱᇱ ሺ‫ݐ‬ሻ ൌ ܽଷ …‘• ܽ‫ ܨ ݐ‬ᇱᇱ ሺͲሻ ൌ Ͳ

௔

௦ మ ା௔మ

െ ‫ ݏ‬ଶ Ǥ Ͳ െ ‫ݏ‬Ǥ ܽ െ Ͳ ൌ െ

௦௔

௦ మ ା௔మ

)+)$&' -()'+ |

Ã)&)5 
|||‫ܨ‬ሺ‫ݐ‬ሻ|||‫ ݐ‬൐ Ͳ|

௧


||| 3Ÿ‫ܨ‬ሺ‫ݐ‬ሻÿ ൌ ݂ሺ‫ݏ‬ሻ|| ||3ÿ‫׬‬଴ ‫ ܨ‬ሺ‫ݑ‬ሻ ‫ ݑ‬ൌ

௙ሺ௦ሻ ௦

|

௧

"| 
|

|  
 ||| ||‫ ܩ‬ሺ‫ݐ‬ሻ ൌ ‫׬‬଴ ‫ܨ‬ሺ‫ݑ‬ሻ ‫ ܩ| || ݑ‬ᇱ ሺ‫ݐ‬ሻ ൌ ‫ܨ‬ሺ‫ݐ‬ሻ| |‫ ܩ‬ሺͲሻ ൌ Ͳ| 3 Ÿ‫ ܩ‬ᇱ ሺ‫ݐ‬ሻÿ ൌ ‫ݏ‬3 Ÿ‫ܩ‬ሺ‫ݐ‬ሻÿ െ ‫ܩ‬ሺͲሻ ൌ ‫ݏ‬3Ÿ‫ܩ‬ሺ‫ݐ‬ሻÿ ൌ ݂ሺ‫ݏ‬ሻ ||3 Ÿ‫ܩ‬ሺ‫ݐ‬ሻÿ ൌ ௧

֜  ÿන ‫ ܨ‬ሺ‫ ݑ‬ሻ ‫ ݑ‬ൌ ଴

௙ሺ௦ሻ ௦

݂ሺ‫ݏ‬ሻ ‫ݏ‬

Ex  ௧ ୱ୧୬ ௨

+ | @ |3ÿ‫׬‬଴ ௦௜ ௧

3ÿ

௧

௨

௧ ୱ୧୬ ௨

ଵ

ൌ ܿ‫֜ ݃ݐ݋‬3 ÿ‫׬‬଴ ௦

 ‫| ݑ‬

௧

௨

ଵ

 ‫ ݑ‬ൌ ܿ‫݃ݐ݋‬ ௦

6 | @ |3ÿ‫׬‬଴ •‹ ʹ‫| ݑ ݑ‬ ଶ

௧

ଵ ௦

ଶ

3 Ÿ•‹ ʹ‫ݐ‬ÿ ൌ ௦మ ାସ ֜3 ÿ‫׬‬଴ •‹ ʹ‫ ݑ ݑ‬ൌ ௦ሺ௦మ ାସሻ

d
( "

 |




c
      |

'&' &) |

Ã)&)6 
‫ܨ‬ሺ‫ݐ‬ሻ|

| | ൐ Ͳ||
|3 Ÿ‫ܨ‬ሺ‫ݐ‬ሻÿ ൌ ݂ሺ‫ݏ‬ሻǤƒŽ‘”•|

3 Ÿ‫ܨ ݐ‬ሺ‫ݐ‬ሻÿ ൌ ሺെͳሻ

ௗ೙

ௗ௦ ೙

݂ ሺ୬ሻ ሺ•ሻሺͻሻ|

ÿ‘—”±–ƒ„Ž‹”…‡––‡’”‘’”‹±–±ǡ‘—–‹Ž‹•‡Žƒ” ‰Ž‡†‡‡‹„‹–œ’‘—”†±”‹ ‡”†‘…Ž‡•‹‰‡‹–±‰”ƒŽ‡ ௗ௙ ௗ௦

ൌ݂ ᇱ ሺ‫ݏ‬ሻ ൌ

ௗ ஶ ି௦௧ ‫݁ ׬‬ ௗ௦ ଴

ஶ డ

‫ܨ‬ሺ‫ݐ‬ሻ†–ൌ‫׬‬଴

డ௦

ஶ

ஶ

݁ ି௦௧ ‫ܨ‬ሺ‫ݐ‬ሻ ‫ݐ‬ൌെ ‫׬‬଴ െ‫ି ݁ݐ‬௦௧ ‫ܨ‬ሺ‫ݐ‬ሻ ‫ ݐ‬ൌ െ ‫׬‬଴ ݁ ି௦௧ Ÿ‫ܨݐ‬ሺ‫ݐ‬ሻÿ ‫ݐ‬

െ3 Ÿ‫ܨݐ‬ሺ‫ݐ‬ሻÿ†ǯ‘î 3Ÿ‫ ܨݐ‬ሺ‫ݐ‬ሻÿ ൌ െ݂ሺ‫ݏ‬ሻ’‘—”݊ ൌ ͳǤÿ±±”ƒŽ‡‡–’‘—”݊ ൌ ݇

ௗ௙ ௗ௦

ൌ

Ex 
|‫ܨ‬ሺ‫ݐ‬ሻ ൌ ‫ݐ݊݅ݏ‬ǡ—Ž–‹’Ž‹ƒ–’ƒ”–Ǥ‫ܨݐ‬ሺ‫ݐ‬ሻ ൌ ‫ݐ݊݅ݏݐ‬Ǥ

3 Ÿ‫ ܨݐ‬ሺ‫ݐ‬ሻÿ ൌ െ

ௗ

ௗ௦

ሺ3 Ÿ‫ܨ‬ሺ‫ݐ‬ሻÿሻൌ

ௗ

ሺ

௔

ௗ௦ ௦ మ ା௔మ

ሻൌ

ଶ௔௦

ሺ௦ మ ା௔మ ሻమ



ÿ‹‫ܨ‬ሺ‫ݐ‬ሻ ൌ ܿ‫ݐݏ݋‬ǡ—Ž–‹’Ž‹‘•’ƒ”‫ݐ‬ǡ’—‹•’ƒ”‫ܨݐƒ‘ʹݐ‬ሺ‫ݐ‬ሻ ൌ ‫ݐݏ݋ܿݐ‬Ǥ ”‘— ‡”3 Ÿ‫ܨݐ‬ሺ‫ݐ‬ሻÿൌ3Ÿ‫ݐ •‘… ݐ‬ÿൌെ

ௗ

ௗ௦

3Ÿܿ‫ݐݏ݋‬ÿÿൌെ

”‘— ‡”3 Ÿ‫ ݐ‬ଶ ܿ‫ݐݏ݋‬ÿ ൌ

ௗమ

ቀ

௦

ௗ௦ మ ௦ మ ା௔మ

ௗ௦

ሺ

௦

௦ మ ା௔మ

ሻ ൌ െ

‫ ݏ‬ଶ ൅ ܽଶ െ ʹ‫ݏ‬ଶ ቉ ൌ െቈ ሺ‫ ݏ‬ଶ ൅ ܽ ଶ ሻଶ ൌെ

ÿƒ‫ݐ‬ଶ ‫ܨ‬ሺ‫ݐ‬ሻ ൌ ‫ ݐ‬ଶ ܿ‫ݐݏ݋‬Ǥ

ௗ

ଵǤ൫௦ మ ା௔మ
ି௦ሺଶ௦ሻ ሺ௦ଶା௔మ ሻమ

ቃ

‫ݏ‬ଶ െ ܽଶ  ሺ‫ ݏ‬ଶ ൅ ܽ ଶ ሻଶ

ቁ

ଶ௦ ିÿ௔మ ௦

 ൌ ሺ௦మା௔మ ሻ   

*+&) |

d
( "

 |




c
      |

Ã)&)7 ÿ‘‹–‫ܨ‬ሺ‫ݐ‬ሻ†±ˆ‹‹’ƒ”‫ ݐ‬൐ ͲǤÿ‹3Ÿ‫ܨ‬ሺ‫ݐ‬ሻÿ ൌ ݂ሺ‫ݏ‬ሻǡƒŽ‘”• ‫ܨ‬ሺ‫ݐ‬ሻ ൌ ݁ ଶ௧ ǡ3Ÿ݁ଶ௧ ÿൌ



ௗ

ቀ

ଵ

ቁൌሺ௦ିଶሻమ 

ௗమ

ቀ

ଵ

ቁൌሺ௦ିଶሻ 

3 Ÿ‫ ݁ݐ‬ଶ௧ ÿ ൌ െ 3 Ÿ‫ ݐ‬ଶ ݁ ଶ௧ ÿ ൌ

ଵ

௦ିଶ

ௗ௦ ௦ିଶ

ௗ௦మ

௦ିଶ

ௗ

ଵ

ଶ

ଵ

3 Ÿ‫ ݐ‬ଷ ݁ ଶ௧ ÿ ൌ െ ௗ௦ ቀ௦ିଶቁൌ

ÿ  ሺ௦ିଶሻర

D
  ிሺ௧ሻ

3ÿ

௧

ஶ

ൌ‫׬‬௦ ݂ ሺ‫ ݑ‬ሻ ‫ݑ‬Ǥÿ”‘— ‡”“—‡Ž‹௧՜଴

ிሺ௧ሻ ௧

‡š‹•–‡Ǥ

Ex  ଵ

3 Ÿ•‹ ‫ݐ‬ÿൌ ௦௜ ௧

3ÿ

௧

௦ మ ାଵ

‡–Ž‹௧՜଴

ஶ ௗ௨

ൌ‫׬‬௦

௨మ ାଵ

ୱ୧୬ ௧ ௧

ൌ ͳ

ଵ

ൌܽ‫ ݃ݐܿݎ‬ቀ ቁ ௦


 &)/ 

Ã)&)8 ÿ‘‹–‫ܨ‬ሺ‫ݐ‬ሻ†‡’±”‹‘†‡
൐ Ͳǡ–‡ŽŽ‡’‘—”‫ ܨ‬ሺ‫ ݐ‬൅
ሻ ൌ ݂ሺ‫ݐ‬ሻǡƒŽ‘”• 3 Ÿ‫ ܨ‬ሺ‫ݐ‬ሻÿൌ

‫׬‬

ಮ

௘ ష ÿ ிሺ௧ሻௗ௧ ଵି௘ ష ÿ

ሺͻሻ

Ex  ‫ݐ݊݅ݏ‬ǡ Ͳ ൏ ‫ ݐ‬൏ ߨ
ƒሻ| ÿ‘‹–‫ ܨ‬ሺ‫ݐ‬ሻ ൌ ÿ †‡’±”‹‘†‡ʹÿǤ ”‘— ‡” 3Ÿ‫ ܨ‬ሺ‫ݐ‬ሻÿǤ Ͳǡ ߨ ൏ ‫ ݐ‬൏ ʹߨ „ሻ| ‡‰”ƒ’Š‡

d
( "

 |




c
      | |

3 Ÿ‫ ܨ‬ሺ‫ݐ‬ሻÿ ൌ

 ൌ



ଵ

ଵି௘ షమഏ

ଵ

ଵି௘ షమഏ

గ

‫׬‬଴ ݁ ି௦௧ ‫ܨ‬ሺ‫ݐ‬ሻ ‫ ݐ‬ గ ͳ න ݁ ି௦௧ •‹ ‫ ݐ ݐ‬ ൌ ͳ െ ݁ ିଶగ௦ ଴ గ ݁
ି௦௧ ሺെ‫ ݐ ‹• ݏ‬െ …‘• ‫ݐ‬ሻ ͳ ቤ  ൌ ‫ݏ‬ଶ ൅ ͳ ͳ െ ݁ ିଶగ௦ ଴

ÿ

ଵା௘ షഏ ௦ మ ାଵ

ଵ  ሻሺ௦ మ ାଵሻ

ൌ ሺଵି௘ షഏ

&)) %ûሺ࢚ሻ | | |



‹‰—”‡ ͳǤ ”‡’”±•‡–‡ —‡ ˆ‘…–‹‘ †‹•…‘–‹—‡ ƒ—š ’‘‹–• – ͳǡ –ʹǡ –͵ǡ ƒ ‡… †‡• Ž‹‹–‡•  ‰ƒ—…Š‡ Ž‹‹–‡•†”‘‹–‡‡š‹•–ƒ–Ǥ ሺ– ͳ ሻ‡– ሺ–ͳ൅ሻǤ

ÿ‡ˆ‘…–‹‘‡•–…‘–‹—‡’ƒ”‘”…‡ƒ—š†ƒ•—‹–‡” ƒŽŽ‡ˆ‡”±ሾܽǡ ܾሿǡ•‹…‡–‹–‡” ƒŽŽ‡’‡—–
–”‡ †‹ ‹•± ‡ — ‘„”‡ ˆ‹‹ †‡ •‘—• ‹–‡” ƒŽŽ‡ †ƒ• Ž‡•“—‡Ž• Žƒ ˆ‘…–‹‘ ‡•– …‘–‹—‡ …̵‡•–  †‹”‡ ’‘•• †‡Ž‡•Ž‹‹–‡•‰ƒ—…Š‡•‡–†”‘‹–‡•ˆ‹‹‡•Ǥ 


 -))%&' 

ÿǯ‹Ž‡š‹•–‡†‡•…‘•–ƒ–‡•”±‡ŽŽ‡• ൐ Ͳ‡–ÿǡ–‡ŽŽ‡“—‡’‘—”–‘—–‫ ݐ‬൐ ȁ ݁ ି௦௧ ‫ܨ‬ሺ‫ݐ‬ሻȁ ൏ ‫—‘ܯ‬ȁ‫ ܨ‬ሺ‫ݐ‬ሻȁ ൏ ‫ ݁ܯ‬௦௧ 

d
( "

 |

Ǣ‘ƒ 




c
      | ÿ †‹– “—‡ ‫ܨ‬ሺ‫ݐ‬ሻ ‡•– —‡ ˆ‘…–‹‘ †ǯ‘”†”‡ ‡š’‘‡–‹‡Ž ÿ “—ƒ† ‫ ݐ‬՜ λ ‘— •‹’Ž‡‡– †ǯ‘”†”‡ ‡š’‘‡–‹‡ŽǤ

Ex  ‫ ܨ‬ሺ‫ݐ‬ሻ ൌ ‫ݐ‬ଶ ‡•–‡š’‘‡–‹‡Ž†ǯ‘”†”‡͵…ƒ”ȁ‫ ݐ‬ଶ ȁ ൌ ‫ ݐ‬ଶ ൏ ݁ ଷ௧ ’‘—”‫ ݐ‬൐ Ͳ

   

$  | | 
|
 || 3 Ÿ‫ ݐ‬ଶ ݁ ି௦௧ ÿ
3 Ÿ݁ ି௧ …‘• ʹ‫ݐ‬ÿ
3Ÿʹ݁ ଷ௧ •‹ Ͷ‫ݐ‬ÿ
3 Ÿሺ‫ ݐ‬൅ ʹሻଶ ݁ ௧ ÿ
3Ÿ݁ ଶ௧ ሺ͵ •‹ Ͷ‫ ݐ‬െ Ͷ …‘• Ͷ‫ݐ‬ሻÿ

3 Ÿ݁ ିସ௧ ݄ܿʹ‫ݐ‬ÿ
3 Ÿ݁ ି௧ ሺ͵‫ ݐʹ݄ݏ‬െ ͷ݄ܿʹ‫ݐ‬ሻÿ
3Ÿ݁ ି௧ ‫݊݅ݏ‬ଶ ‫ݐ‬ÿ
3 Ÿሺͳ ൅ ‫ି ݁ݐ‬௧ ሻଷ ÿ

  
    Ã     |


| | ||! ||
|‫ܨ‬ሺ‫ݐ‬ሻ| |݂ሺ‫ݏ‬ሻ| 
|
|3Ÿ‫ܨ‬ሺ‫ݐ‬ሻÿ ൌ ݂ሺ‫ݏ‬ሻ  ‫ܨ‬ሺ‫ݐ‬ሻ|  |  | |
 ||݂ሺ‫ݏ‬ሻ| ||
|‫ ܨ‬ሺ‫ݐ‬ሻ ൌ 3+݂ሺ‫ݏ‬ሻ|)|3+| |   ||


|
 ||!  |
|3Ÿ݁ ିଷ௧ ÿ ൌ

ଵ |

௦ାଷ

 |3+ÿ

ଵ

௦ାଷ

ൌ ݁ ିଷ௧ |

3 
 





| ݇ଵǡ ݇ଶ| |  | |   |  |  | ݂ଵ ሺ‫ݏ‬ሻǡ ݂ଶ ሺ‫ݏ‬ሻ| | 
| | ! | | ‫ܨ‬ଵ ሺ‫ݐ‬ሻ݁‫ܨݐ‬ଶሺ‫ݐ‬ሻ| 
 | | 3 +Ÿ݇௔ଵ ݂ଵ ሺ‫ݏ‬ሻ ൅ ݇ଶ ݂ଶ ሺ‫ݏ‬ሻÿ ൌ ݇ଵ 3 Ÿ݂ଵ ሺ‫ݏ‬ሻÿ ൅ ݇ଶ 3 Ÿ݂ଶ ሺ‫ݏ‬ሻÿ ൌ ݇ଵ ‫ܨ‬ଵ ሺ‫ݐ‬ሻ ൅ ݇ଶ ‫ܨ‬ଶሺ‫ݐ‬ሻ|





|

!+) +&)''+ | ௉ሺ௦ሻ

@ |
|


|

ொሺ௦ሻ

||)|mሺ‫ݏ‬ሻ| ||ܳሺ‫ݏ‬ሻ| ||  E| |||mሺ‫ݏ‬ሻ|  |

|

| 
||ܳሺ‫ݏ‬ሻ5||  | 
|| |||
|


|| ||

d
( "

 |




c
      |

஺௦ାÿ ஺ ǡ )‫ݎ‬ ሺ௔௦ା௕ሻÿ ሺ௔௦మ ା௕௦ା௖ሻÿ

ൌ ͳǡʹǡ͵ ǥ| $|   | | 
|
 | | | | 
| ௉ሺ௦ሻ |




|
| | 
 ||  |3+ÿ

ொሺ௦ሻ

Ex  +| 6|

ଶ௦ିÿ ሺଷ௦ିସሻሺଶ௦ାଵሻ

ଷ௦ మ ିସ௦ାଶ

஺

ൌ

ଷ௦ିସ

ሺ௦ మ ାଶ௦ାସሻ మ ሺ௦ିÿሻ

൅ ሺଶ௦ାଵሻ

஼ ሺଶ௦ାଵሻ మ

ÿ

ൌ ሺ௦మ

஺௦ାÿ

ାଶ௦ାସሻ మ

൅

൅

஼௦ା஽

௦ మ ାଶ௦ାସ

!|  |‫ܣ‬ǡ ǡ ‫ ܥ‬ǥ| || 
|

஽

|

ଶ௦ାଵ ா

൅

௦ିÿ

|

ଷ௦ା

3 | @ ||3+ÿ௦మ ିଶ௦ିଷ | ,|
||

ଷ௦ା

௦ మ ିଶ௦ିଷ

ଷ௦ା

ൌ ሺ௦ିଷሻሺ௦ାଵሻ ൌ

஺

௦ିଷ

൅

ÿ |

௦ାଵ

$||



 | |ሺ‫ ݏ‬െ ͵ሻሺ‫ ݏ‬൅ ͳሻ||||͵‫ ݏ‬൅ ͹ ൌ ‫ܣ‬ሺ‫ ݏ‬൅ ͳሻ ൅ ሺ‫ ݏ‬െ ͵ሻ ൌ ሺ‫ ܣ‬൅ ሻ‫ ݏ‬൅ ‫ ܣ‬െ ͵ | | ‫ܣ‬൅ ൌ͵
|)||
|‫ ܣ‬ൌ Ͷǡ ൌ െͳǤ| |
 | |

 |||ÿ ‫ܣ‬െ͵ ൌ ͹

3 +ÿ

ଷ௦ା

௦ మ ିଶ௦ିଷ

ൌ Ͷ3 ͳ ÿ

ଵ

௦ିଷ

െ3 ͳÿ

ଵ

‫ݏ‬ଶ

௦ାଵ

͵‫ ݏ‬൅ ͹ Ͷ ͳ | ൌ െ െ ʹ‫ ݏ‬െ ͵ ‫ ݏ‬െ ͵ ‫ ݏ‬൅ ͳ

ൌ Ͷ݁ ଷ௧ െ ݁ ି௧ |

ଶ௦ మ ିସ

S | |@ ||3+ÿሺ௦ାଵሻሺ௦ିଶሻሺ௦ିଷሻ | ଶ௦ మ ିସ

஺

஼

,|
|ሺ௦ାଵሻሺ௦ିଶሻሺ௦ିଷሻ ൌ ሺ௦ାଵሻ ൅ ሺ௦ିଶሻ ൅ ሺ௦ିଷሻ ሺͳሻ| ÿ

,|



| ||||2+4| |ሺ‫ ݏ‬൅ ͳሻ| ||
| |‫| ݏ‬െͳ|||| ʹ‫ ݏ‬ଶ െ Ͷ ͳ ൌെ | ௦՜ିଵ ሺ‫ ݏ‬െ ʹሻሺ‫ ݏ‬െ ͵ሻ ͸

‫ ܣ‬ൌ Ž‹

,|



| ||||ሺͳሻ| |ሺ‫ ݏ‬െ ʹሻ| ||
| |‫| |ݏ‬6|||| Ͷ ʹ‫ ݏ‬ଶ െ Ͷ ൌെ | ௦՜ଶ ሺ‫ ݏ‬൅ ͳሻሺ‫ ݏ‬െ ͵ ሻ ͵

ൌ Ž‹

,|



| ||||ሺͳሻ| |ሺ‫ ݏ‬െ ͵ሻ| ||
| |‫| |ݏ‬3|||| ʹ‫ݏ‬ଶ െ Ͷ ͹ ൌ | ௦՜ଷ ሺ‫ ݏ‬൅ ͳ ሻሺ‫ ݏ‬െ ʹሻ ʹ

‫ ܥ‬ൌ Ž‹ ଶ௦ మ ିସ

- |3+ÿሺ௦ାଵሻሺ௦ିଶሻሺ௦ିଷሻ ൌ

ିଵ ÿ

ଵ

ସ

ଵ

ଵ

ଵ

ସ

3 ͳ ÿ௦ାଵ െ ଷ 3 ͳ ÿ௦ିଶ ൅ ଶ 3 ͳ ÿ௦ିଷ ൌ െ ÿ ݁ ି௧ െ ଷ ݁ଶ௧ ൅ ଶ ݁ ଷ௧ |

ÿ௦ మ ିଵÿ௦ିଵଵ

9 | @ |3+ÿሺ௦ାଵሻሺ௦ିଶሻ |

d
( "

 |




c
      |

‫ܣ‬ ‫ܥ‬ ‫| ܦ‬ ͷ‫ ݏ‬ଶ െ ͳͷ‫ ݏ‬െ ͳͳ ൌ ൅ ൅ ൅ ሺ‫ ݏ‬൅ ͳሻሺ‫ ݏ‬െ ʹሻଷ ሺ‫ ݏ‬൅ ͳሻ ሺ‫ ݏ‬െ ʹሻଷ ሺ‫ ݏ‬െ ʹሻଶ ‫ ݏ‬െ ʹ

"|  |‫ܣ‬ǡ ݁‫| |

|| ݐ‬6|||ሺͳሻ| |ሺ‫ ݏ‬൅ ͳሻ| ||
| |‫| ݏ‬െͳ|||| ͳ ͷ‫ ݏ‬ଶ െ ͳͷ െ ͳͳ ൌെ | ଷ ௦՜ିଵ ሺ‫ ݏ‬െ ʹ ሻ ͵

‫ ܣ‬ൌ Ž‹

,|



| ||||ሺͳሻ| |ሺ‫ ݏ‬െ ʹሻଷ| ||
| |‫| ݏ‬െʹ|||| ͷ‫ ݏ‬ଶ െ ͳͷ െ ͳͳ ൌ െ͹| ௦՜ିଶ ‫ݏ‬൅ͳ

ൌ Ž‹

"| 
|‫|
|
 |ܦݐ݁ܥ‬-| |J|||
||ሺͳሻ|| ÿ௦ మ ିଵÿ௦ିଵଵ ሺ௦ାଵሻሺ௦ିଶሻ

ିଵ

ି

஼

ଷ ൌ ሺ௦ାଵሻ ൅ ሺ௦ିଶሻ ൅ ሺ௦ିଶሻమ ൅

஽ ||||||||||||||||||||||||

௦ିଶ

"| 
|| |&|||
| |

 ||||
 |‫ ݏ‬ൌ Ͳ݁‫ ݏݐ‬ൌ ͳ| ||  || ଵଵ

ÿଶଵÿ ଶ

ଵ

஼

஽

ൌ െଷ൅ÿ ൅ସ െ ଶ ଵ

ൌ െÿ ൅ ͹ ൅ ‫ ܥ‬െ ‫ܦ‬


||&|| #||
|‫ ܥ‬ൌ Ͷ| |‫ ܦ‬ൌ ଵ| ଷ

ିଵ

ଵ

ÿ௦ మ ିଵÿ௦ିଵଵ

ଷ ଷ | ൠ ൌ 3 ͳ §ሺ௦ାଵሻ ൅ ሺ௦ିଶሻ ൅ ሺ௦ିଶሻమ ൅ ௦ିଶ

ÿ௦ మ ିଵÿ௦ିଵଵ

ൌ ൫െͳ ͵
݁ି௧ െ ൫͹ ʹ
‫ ݐ‬ଶ݁ ଶ௧ ൅ Ͷ‫݁ݐ‬ଶ௧ ൅ ൫ͳ ͵
݁ ଶ௧ |

3 +ÿ ሺ௦ାଵሻሺ௦ିଶሻ 3 +ÿ ሺ௦ାଵሻሺ௦ିଶሻ

ି

ସ

Ã)&)+')+)$*)+

L 

Ã)&)
|3+Ÿ݂ሺ‫ݏ‬ሻÿ ൌ ‫ ܨ‬ሺ‫ݐ‬ሻ|  |֜ 3 ͳ Ÿ݂ሺ‫ ݏ‬െ ܽሻÿ ൌ ݁ ௔௧ ‫ܨ‬ሺ‫ݐ‬ሻ|

%$&'

ଵ


|3 ͳ ÿ௦మ ାସ

ଵ ଶ

ଵ

ଶ

ଵ

•‹ ʹ‫| |ݐ‬3 ͳ ÿ௦మିଶ௦ାÿ ൌ 3 ͳ ÿሺ௦ିଵሻమାସ ൌ ଶ ݁ ௧ •‹ ʹ‫|ݐ‬

ሺ ሻ  |
|3+Ÿ݂ሺ‫ݏ‬ሻÿ ൌ ‫ ܨ‬ሺ‫ݐ‬ሻ ֜ 3 ͳ Ÿ݁ ି௔௦ ݂ ሺ‫ݏ‬ሻÿ ൌ ÿ‫ ݐ ܨ‬െ ܽ ‫ ݐ‬൐ ܽ
| Ͳ‫ ݐ‬൏ ܽ

%$&'
|3 ͳ ÿ

ଵ

௦ మ ାଵ

ൌ •‹ ‫ݐ‬ǡ||||3 ͳ

గ •‹ ቀ‫ ݐ‬െ ቁ ǡ‫ ݐ݅ݏ‬൐ ߨ ͵|
ଷ ÿ ÿ ൌ ௦ మ ାଵ Ͳǡ‫ ݐ݅ݏ‬൏ ߨ ͵ ௘

ష

ഏ

|

d
( "

 |




c
      |

Ch
 
 ǯ h ௧

ଵ


|3 ͳŸ݂ሺ‫ݏ‬ሻÿ ൌ ‫ܨ‬ሺ‫ݐ‬ሻ 3 ͳŸ݂ሺ݇‫ݏ‬ሻÿ ൌ ௞ ‫ ܨ‬ቀ௞ቁ|

%$&' |

|

௦


|3 ͳ ÿ

ଶ௦

ൌ …‘• Ͷ‫||ݐ‬3 ͳ ÿሺଶ௦ሻమ

௦ మ ାଵÿ

ାଵÿ

ଵ

ൌ …‘• ଶ

ସ௧ ଶ

ଵ

ൌ …‘• ʹ‫|ݐ‬ ଶ

T
 

    
|3 ͳŸ݂ሺ‫ݏ‬ሻÿ ൌ ‫ܨ‬ሺ‫ݐ‬ሻ| 3 ͳ ݂ ሺ ሻ ሺ‫ݏ‬ሻÿ ൌ 3 ͳ ÿ

ௗ೙

ௗ௦ ೙

%$&' |

|

|

|

ଵ

݂ሺ‫ݏ‬ሻ ൌ ሺെͳሻ ‫ܨ ݐ‬ሺ‫ݐ‬ሻ|

ଵ

ௗ

ିଶ௦

ିଶ௦


||3 ͳ ÿ௦మାଵ ൌ •‹ ‫| |ݐ‬ௗ௦ ቀ௦మ ାଵቁ ൌ ሺ௦మାଵሻమ|||3 ͳ ÿሺ௦మାଵሻమ ൌ െ‫|||ݐ ‹• ݐ‬

3 ͳ ÿሺ௦మ

௦

ାଵሻమ

ଵ

ൌ ‫| ݐ ‹• ݐ‬ ଶ

T
 

 ǯ
  ஶ

ிሺ௧ሻ|


|3 ͳŸ݂ሺ‫ݏ‬ሻÿ ൌ ‫ܨ‬ሺ‫ݐ‬ሻ 3 ͳ ‫׬‬଴ ݂ ሺ‫ݑ‬ሻ ‫ ݑ‬ÿ ൌ

%$&' |

|

|

|


||3 ͳ ÿ

ଵ

ଵ

ൌ 3 ͳÿ െ

௦ሺ௦ାଵሻ

ஶ ଵ

௦

ଵ

3 ͳ ÿ‫׬‬଴ ቀ െ ௨

ଵା௨

௧

ଵ

௦ାଵ

ൌ ͳ െ ݁ ି௧ | ଵ

ቁ  ‫ ݑ‬ൌ 3 ͳ ÿŽ ቀͳ ൅ ቁ ൌ ௦

ଵି௘ షÿ | ௧

M 
 

|3 ͳŸ݂ሺ‫ݏ‬ሻÿ ൌ ‫ܨ‬ሺ‫ݐ‬ሻ |‫ ܨ‬ሺͲሻ ൌ Ͳ| ||3 ͳŸ‫݂ݏ‬ሺ‫ݏ‬ሻÿ ൌ ‫ ܨ‬ᇱ ሺ‫ݐ‬ሻ|||!|




| ||| | || 
|‫ܨ‬ሺ‫ݐ‬ሻ |||
|‫ܨ‬ሺͲሻ ് Ͳ| |3 ͳŸ‫ ݂ݏ‬ሺ‫ݏ‬ሻ െ ‫ܨ‬ሺͲሻÿ ൌ ‫ ܨ‬ᇱ ሺ‫ݐ‬ሻ|

%$&' |

|

|

|

ଵ


||3 ͳ ÿሺ௦మ

3 ͳ ÿሺ௦మ

௦

ାଵሻ

ାଵሻమ

ൌ •‹ ‫ ݐ‬ǡ ݁‫ݐ‬ ଵ

ൌ •‹ ‫| ݐ‬

ௗ

ቀ

ଵ

ௗ௦ ௦ మ ାଵ

ିଶ௦

ቁ ൌ ሺ௦మ

ାଵሻమ

ିଶ௦

||||3 ͳ ÿሺ௦మ

ାଵሻమ

ൌ െ‫| ݐ ‹• ݐ‬

ଶ

|

D
  ௙ሺ௦ሻ

x
3 ͳŸ݂ሺ‫ݏ‬ሻÿ ൌ ‫ܨ‬ሺ‫ݐ‬ሻ0

3 ͳ ÿ 
 
‫ܨ‬ሺ‫ݐ‬ሻ



A


%$&' |

|

ଵ

ଵ

௧

ଵ

‫׬‬଴ ‫ ܨ‬ሺ‫ ݑ‬ሻ ‫ ݑ‬A


  




   

௦

 


ଵ

௧ଵ

ଵ


||3 ͳ ÿ௦మାସ ൌ ଶ •‹ ʹ‫||ݐ‬3 ͳ ÿ௦ሺ௦మ ାସሻ ൌ ‫׬‬଴ ଶ •‹ ʹ‫ ݑ ݑ‬ൌ ସ ሺͳ െ …‘• ʹ‫ݐ‬ሻ|

d
( "

 |

௦




c
      |

௙ሺ௦ሻ



|  
 |  |' |
| |3 ͳ ÿ ௦೙ ǡ ݊ ൌ ͳǡʹǡ͵ ǥ|

P   

௧

x
3 ͳŸ݂ሺ‫ݏ‬ሻÿ ൌ ‫ܨ‬ሺ‫ݐ‬ሻ3 ͳŸ݃ሺ‫ݏ‬ሻÿ ൌ ‫ܩ‬ሺ‫ݐ‬ሻ
3 ͳŸ݂ ሺ‫ݏ‬ሻ݃ሺ‫ݏ‬ሻÿ ൌ ‫׬‬଴ ‫ ܨ‬ሺ‫ݑ‬ሻ‫ܩ‬ሺ‫ ݐ‬െ ‫ݑ‬ሻ ‫ ݑ‬ൌ ‫
ܩ ܨ‬

%$&' |

|

|

|

ଵ

ଵ


||3 ͳ ÿ௦మ ൌ ‫|||ݐ‬3 ͳ ÿሺ௦ାଵሻమ ൌ ‫ି ݁ݐ‬௧ |||||| | #||  
| |

3ͳÿ

ଵ

ଵ

௦ మ ሺ௦ାଵሻమ

௧

ൌ ‫׬‬଴ ሺ‫ି ݁ݑ‬௨ ሻሺ‫ ݐ‬െ ‫ ݑ‬ሻ ‫ ݑ‬ൌ ‫ି ݁ݐ‬௧ ൅ ʹ݁ ି௧ ൅ ‫ ݐ‬െ ʹ|

|

d
( "

 |




c
      |

A 
 |

,) | 
|||| ݉|
||| |

||| 
 |<||| 
|
| || |
|||  
||

 ||| |
  |"I | |

| 
| ÿሺ‫ݐ‬ሻ| |  |
  ||݉|||
  |‫| ||

|

 |||
 |ݐ‬

||݉| |
|| |||  |ሺȂ ݇ÿሺ‫ݐ‬ሻሻ|)|݇| ||  |  || |  |
|
|| |!

|| | 
|  || |
||| 
||   ||݉

ௗమ

ௗ௧ మ

ൌ െ݇ÿ||݉ÿ ᇱᇱ ൅ ݇ÿ ൌ Ͳ|


| | |
| | | | | | 

 |  


| | |
|
  | | | |5| ݉

ௗమ

ௗ௧ మ

ൌ െ݇ÿ െ ÿ

ௗ

ௗ௧

| | | ݉ÿ ᇱᇱ ൅ ÿÿ ൅ ݇ÿ ൌ ͲǤ| B|  | 


|
 
 | | 
|

|
|
|| || #||| 
| 
 ||
|||  |‫ܨ‬ሺ‫ݐ‬ሻ |&|| ௗమ

| 
||  |||݉ ௗ௧ మ ൌ െ݇ÿ െ ÿ ௗ௧ ൅ ‫ ܨ‬ሺ‫ݐ‬ሻǤ| ௗ

|

) ' )/ | B| 

|   
| 
 |  | |   | | 

| 
 |  | | 
|  | |
  |݇|

d
( "

 |




c
      | |

| +| 6| 3| S|

B| ||


|
 | ||   
|‫||  =|||ܧ‬ |B|
 |ܴ||,| |B|
|

|
 |‫|||ܮ‬K| B| || 
|‫|||| ܥ‬

I| 
  |/| |||0|||| |

|
||||I|||  || || | ||  |!| || 

|| |||
| ௗொ

|  || | 
| | ௗ௧ ൌ ƒ | |  | | ||
 
| |||- #| |

 ||  ||| |

"| |||

| | ||  
 |||  ||

|  |  ||  
 |||
 ||ܴƒ ൌ ܴ

 |  ||  
 |||
 |‫ܮ‬

ொ

ௗÿ

ௗ௧

 |  ||  
 ||| 
||ൌ ௖ |

ௗொ

|

ௗ௧ ௗమ ொ

ൌ‫ܮ‬

ௗ௧ మ

|

 |  ||  
 ||| |ൌ െܽ‫ ݈݁݅ݐ݊݁ݐ݋݌݁ ݊݋݅ݐܽݐ݊݁݉݃ݑ‬ൌ െ‫| ܧ‬ |
|



|  || | 
||+irkoffA

+9): | + | !|| 
||
 
|| |
|
 |||
|


 ||T| |

| |

|2 
||T4 | 6 | !|| 
|| ||  
 | | ||

|| |

 | &| | | | 

| 
 | 

| | | 
|  | 

# | 
 | !
| |  

||I| ||‫ܮ‬

ௗమ ொ ௗ௧ మ

൅ܴ

ௗఏ ௗ௧

൅

ொ ௖

ൌ ‫|ܧ‬

|

d
( "

 |




c
      |

Ã),'.$+&)+ | 
||  | | | 
|ÿ ൌ Ͳ| |||ÿ ൌ ‫| ܮ‬%
 | | |||ÿ | |

| 

|  |  |||
ሺÿሻ| |
|| |
|
| 
  |D ||

| | | | 

|  |  | |  #| ÿሺÿሻ| | 
 | ÿ| 
| 
 | 
| 



|||

ௗర

ௗ ర

ൌ

ÿሺ ሻ ாÿ

Ͳ ൏ ÿ ൏ ‫ܮ‬ሺͻሻ|



| 
|   |  |   |   | | | 
| |  

| | |   | !| 
| ‫ | 
  | | |  | | |


 | |  | ƒܧ‬2‫ | |  | | ܧ‬U| |  |  

|| |  | | ƒ| | |

|| |  | |  || 4 |!|| ‫ƒܧ‬ÿ ᇱᇱ ሺÿሻ݁‫ƒܧݐ‬ÿ ᇱᇱᇱ ሺÿሻ|  | 
# | | |  | | 



 | 
 | | 
 | |ÿ | . || |ÿ| |
 | | || || | | |
| | 

 ||| !|

|


 |
|| 
|



|ሺͻሻ|  || |0|| | |  |  |   |!|

| | || | |
 || + | $ 
|
|

||
|||ÿ ᇱ ൌ ÿ ൌ Ͳ| 6 | $ 
|
  || |ÿ ᇱᇱ ൌ ÿ ൌ Ͳ| ͵Ǥ| $ 
|
||ÿ ᇱᇱ ൌ ÿ ᇱᇱᇱ ൌ Ͳ

d
( "

 |




c
      | |

CHAPITRE XIV. TRANSFORMATIONS CONFORMES |

ñ |  | ‫ ݑ‬ൌ ‫ ݑ‬ሺ‫ݔ‬ǡ ‫ݕ‬ሻ
ሺͳሻ| | 

| |  | | 
| | ‫ ݒ‬ൌ ‫ ݒ‬ሺ‫ݔ‬ǡ ‫ݕ‬ሻ |  
|
| 
|| | | | 
 || ||‫|| 
 | | | ݒݑ‬ | ‫|
|!| ݕݔ‬ሺͳሻ| |  |
|| | 
 |
||| 
 || | | ‫| 

|
 ||  ||ݕݔ|| || || | 
 || | ݒݑ‬ &||  ||| || 
 || ||‫| ||  ||||||  |ݕݔ‬ 
||| || 
 || ||‫| 
| ||  ||| ݒݑ‬

! | | 
| §

|

ß | ; ,-)+)$  B|
|| ࣬|| ||| || ||||||2&4|||
||࣬|| ||‫| ݒݑ‬

&| | 

| 
| ο-|  | | ο- | 
 |  
 | | 
| | | 
||  | ||| |||| | | |
V |

  | ο஺௨௩

డሺ௨ǡ௩ሻ

݈݅݉ ο஺௨௩ ൌ ቚడሺ௫ǡ௬ሻቚ|

|

,)|݈݅݉| || | |W‫||ݕݔܣ‬ο‫| 
 | |)| || |  | ݒݑܣ‬ డሺ௨ǡ௩ሻ

డሺ௫ǡ௬ሻ

ൌ

డ௨ డ௫ డ௩ డ௫

డ௨ డ௬ డ௩

డ௬

ൌ

డ௨ డ௩

డ௫ డ௬

െ

డ௨ డ௩

 |  | |A
|| | 
|ሺͳሻ|

డ௬ డ௫


| | | ሺͳሻ|
||| #|
|

 || |||
||| | | | 
 | | 
| ‫ ݔ‬ൌ ‫ ݔ‬ሺ‫ݑ‬ǡ ‫ݒ‬ሻǡ ‫ ݕ‬ൌ ‫ݕ‬ሺ‫ݑ‬ǡ ‫ݒ‬ሻ|   |   | 
|
 | | ሺͳሻ |
|| || |
| |
 |

 | |A
||

| 
| | డሺ௫ǡ௬ሻ డሺ௨ǡ௩ሻ

డሺ௨ǡ௩ሻ

||  | 
|
| | || 
 ||డሺ௫ǡ௬ሻ  |
| |||A
| |
 |

||||
|
||||'||   |1
 ||  | ||
||  | | |
 |

 |||  ||࣬| |
| |A
|

||࣬ǡ| | | 
|ሺͳሻ| |
 |

డሺ௨ǡ௩ሻ

డሺ௫ǡ௬ሻ

|| |

|

d
(
  (  |




c
      |

2 | 
)$ $&'%-)+)$ | D | | 

# |
  ||
| ||)|| | |
 | || 
|

| | | 
|


||
| 
|| | 
 | |‫ ݖ‬ൌ ‫ ݔ‬൅ ݅‫|
 |ݕ‬ ‫ ݓ‬ൌ ‫ ݑ‬൅ ݅‫ ݒ‬ൌ ݂ ሺ‫ݖ‬ሻ ൌ ݂ ሺ‫ ݔ‬൅ ݅‫ݕ‬ሻǤ|

|

3 | %$&'+)+)$+ | ,|
#| |
|  
|࣬|
 #|| |||

| |‫ ݔ‬ൌ Ͳǡ ‫ ݕ‬ൌ Ͳǡ ‫ ݔ‬ൌ ʹǡ ‫ ݕ‬ൌ ͳ| & 
| |
|࣬ÿ| ||‫|| | | |࣬|||
| | | ݒݑ‬  | ‫ ݓ‬ൌ ‫ ݖ‬൅ ሺͳ െ ʹ݅ሻ| |

| ‫ ݑ‬ൌ ͳ| ‫ ݒ‬ൌ െʹ| ‫ ݑ‬ൌ ͵| ‫ ݒ‬ൌ െͳ| ‫ ݑ‬ൌ ‫ ݔ‬൅ ͳ
‫ ݓ‬ൌ ሺ‫ ݔ‬൅ ͳሻ ൅ ݅ሺ‫ ݕ‬െ ʹሻ ฺ § | ‫ ݒ‬ൌ ‫ݕ‬െʹ ,|  | ||  | 
 ||࣬| | ||| 
 | || ||࣬ÿ| | 
 | !| 
|
|
||  
||   |$| |‫ ݓ‬ൌ ‫ ݖ‬൅ ͳ|
| 
|| 
||  |
 | !|
|| | | | | | |

 | ‫ ݓ‬ൌ ξʹ݁

గ௜

‫ݔ‬ ‫ݕ‬ ‫ݔ‬ ‫ݕ‬

ൌ Ͳ| ൌ Ͳ| ൌ ʹ| ൌ ͳ|

 | |||||  | |||||  | |||||  | |||||

ସ ‫|ݖ‬

‫ݑ‬ൌ‫ݔ‬െ‫ݕ‬ ‫ ݑ‬൅ ݅‫ ݒ‬ൌ ሺͳ ൅ ݅ ሻሺ‫ ݔ‬൅ ݅‫ݕ‬ሻ ൌ ‫ ݔ‬െ ‫ ݕ‬൅ ݅ሺ‫ ݔ‬൅ ‫ݕ‬ሻ ÿ‫ ݒ‬ൌ ‫ ݔ‬൅ ‫
|ݕ‬

d
(
  (  |




c
      | |

| ‫ݑ‬
ൌ െ‫ ݑݑ݋ ݕ‬ൌ െ‫| ݒ‬ ‫ݒ‬ൌ‫ݕ‬ ‫ݑ‬ൌ‫ݔ‬
| | ‫ ݕ‬ൌ Ͳ|  | ||| ‫ ݑݑ݋‬ൌ ‫| ݒ‬ ‫ݒ‬ൌ‫ݔ‬ ‫ݑ‬ൌʹെ‫ݕ‬ | | ‫ ݔ‬ൌ ʹ|  | |||
ൠ ‫ ݑݑ݋‬൅ ‫ ݒ‬ൌ Ͷ| ‫ ݒ‬ൌʹ൅‫ݕ‬ ‫ݑ‬ൌ‫ݔ‬െͳ ‫ ݒݑ݋‬െ ‫ ݑ‬ൌ ʹ| | | ‫ ݕ‬ൌ ͳ|  | |||
‫ݒ‬ൌ‫ݔ‬൅ͳ $| | | 
|‫ ݓ‬ൌ ߙ‫|

|
 || |
||| |ݖ‬ | 
 | | !|
||

 | ‫ ݓ‬ൌ ξʹ݁

గ௜

ଶ ‫ݖ‬

‫ ݔ‬ൌ Ͳ|  | |||

൅ ሺͳ െ ʹ݅ሻ|

‫ ݑ‬൅ ݅‫ ݒ‬ൌ ሺͳ ൅ ݅ ሻሺ‫ ݔ‬൅ ݅‫ݕ‬ሻ ൅ ͳ െ ʹ݅ ฺ §

|

‫ ݑ‬ൌ ‫ ݔ‬െ ‫ ݕ‬൅ ͳ
| ‫ ݒ‬ൌ ‫ݔ‬൅‫ݕ‬െʹ

| !|
|| ‫ ݔ‬ൌ Ͳ|  | ||||| ‫ ݑ‬൅ ‫ ݒ‬ൌ െͳ| | | ‫ ݕ‬ൌ Ͳ|  | ||||| ‫ ݑ‬െ ‫ ݒ‬ൌ ͵| | | ‫ ݔ‬ൌ ʹ|  | ||||| ‫ ݑ‬൅ ‫ ݒ‬ൌ ͵| | | ‫ ݕ‬ൌ ͳ|  | ||||| ‫ ݑ‬െ ‫ ݒ‬ൌ ͳ| !| 
| | | 
|| 
| || 
|||4| 

| |  
 | $|  | | 
|‫ ݓ‬ൌ ߙ‫ ݖ‬൅ ÿ | |   | | 
|| 
| 
| |  
||,|  |
|| 

| 
| | 

 | | 
| 
| ‫ ݓ‬ൌ ܽ‫ݖ‬ଵ | 2 
|  |  
4|  | ÿ ‫ݖ‬ଵ ൌ ‫ ݖ‬൅ ߙ|2  
4 | |

‚ | )+)$ )$ |

d
(
  (  |




c
      |

 || |‫ ݑ‬ൌ ‫ݑ‬ሺ‫ݔ‬ǡ ‫ݕ‬ሻ| ||‫ ݒ‬ൌ ‫ݒ‬ሺ‫ݔ‬ǡ ‫ݕ‬ሻ| | 
 |ሺ‫ݔ‬଴ ǡ ‫ݕ‬଴ ሻ|| ||‫| || |
|ݕݔ‬ 
 |ሺ‫ݑ‬଴ ǡ ‫ݒ‬଴ ሻ|| ||‫| ݒݑ‬ |

|   | | | |‫ܥ‬ଵ ݁‫ܥݐ‬ଶ | |   | | ሺ‫ݔ‬଴ǡ ‫ݕ‬଴ሻ|  |  
 | | | | ‫ܥ‬ଵ ÿ݁‫ܥݐ‬ଶ ÿ| ||  ||ሺ‫ݑ‬଴ ǡ ‫ݒ‬଴ ሻ |B| 
| 

||  | |||‫ܥ‬ଵ ݁‫ܥݐ‬ଶ ||ሺ‫ݔ‬଴ ǡ ‫ݕ‬଴ ሻ|  | |||| ||||  | ||‫ܥ‬ଵ ÿ݁‫ܥݐ‬ଶ ÿ||ሺ‫ݑ‬଴ ǡ ‫ݒ‬଴ ሻ| |
|conforme ||ሺ‫ݔ‬଴ ǡ ‫ݕ‬଴ሻ | B| 
|
| | | |||
| |
 |||  |
| isogonale |!| #|
 | |  |

Th  
| ݂ሺ‫ݖ‬ሻ| |  |  
|  | 
| ݂ ᇱ ሺ‫ݖ‬ሻ ് Ͳ| | | | 
 | |   | | | 
| ‫ ݓ‬ൌ ݂ሺ‫ݖ‬ሻ| ||| | | 
 ||࣬ | &| ||| 
||| 
|
|| | 

|| 
 |‫ݔ‬଴ || | || | |||
|  || | ||‫| |

| |
| |ݓ‬ |  | |  
   | | ȁ݂ ᇱ ሺ‫ݖ‬ሻȁଶ Ǥ| !| 
| 
 | | | | | | | 

||‫ݖ‬଴ ||'|



| 

 | |ȁ݂ÿሺ‫ݖ‬଴ ሻȁǤ| &||
|| | ||| |


 |  | ||
|| ||| 
| | 
|' |   || |

5 | !).$ $ | 
|‫
| | 
||| 
|||| ܥ‬#||  |࣬ |

d
(
  (  |




c
      | |

| 
| |  | ‫ܥ‬ÿ|| |||| || 

| | |
|
 | |
#| |   | | ࣬ÿ| | | |  | !| 
|࣬ÿ|  |  | 
|   | | 
| 
 | !|

#| | 1
| | | 
| |  
| 
| 
 | | 
|‫ ݓ‬ൌ ݂ሺ‫ݖ‬ሻ|  
||࣬|
||  | 
 ||࣬ÿ ||  | 
 ||‫ܥ|| 
 || ܥ‬ÿ |!| |  | 

| |
| 
| || 
|   |

|

| 
|   | ' |  
||
  || ||‫ܥ‬ÿ|| ||| 
 |||࣬| ||| 
 || ‫ܥ‬ÿ| | | | | 
 | | |‫ | ܥ‬,| | | | | #| | 1
|  
|   | 
 ||݂ሺ‫ݖ‬ሻ|
||| |

|
 | D | | 
 | | | #|| 1
|||)| ||
|  || || ||
 | |
 
||  | |
#|
|  ||
 
 | |

6 | Ã+%+&+*)+ |  || ||%
| | ||| | | || |,|  | |  ||


|| || 
'| | 
 | |  |‫ ݖ‬ൌ ݂ሺ‫ݖ‬ሻ|| | | 
|| 

| 
| |

|
|| | 

| 
 |
|| 
 |
 
 ||| |


|| 
| 
 |  |

Ex  : | | !| 
 |
||
 
 || | 
|‫ ݓ‬ൌ ‫ݖ‬ଶ | | 
||‫ ݖ‬ଶ ൌ ‫|ݖ‬  
|‫ ݖ‬ൌ Ͳǡ ͳ || |

7 | '/+"&+)+)$+ 1.| T

‫ ݓ‬ൌ ‫ ݖ‬൅ ÿ )ÿ | ||  |  |"|

| 
| |
|| ||| |

  || |

||  |ÿ |

d
(
  (  |




c
      |

2.| R
‫ ݓ‬ൌ ݁௜ఏ௭ )ߠ| ||  |  |"|

| 
| |
|| ||| 
 || 
| |ߠ | 3.| H h  ‫ ݓ‬ൌ ܽ‫ ܽ|
|  || 
| |
| |
 |

|"| ݖ‬൐ ͳ‫ Ͳ݅ݏݑ݋‬൏ ܽ ൏ ͳ | ,|
#| | 
|||| 

|| |
 
 | 4.| I



ͳ ‫ݓ‬ൌ | ‫ݖ‬

8 | ; ,-)+)$ +|


|‫ ݓ‬ൌ ݂ ሺ‫ݖ‬ሻ ൌ ‫ ݑ‬൅ ݅‫|| |࣬||
 | |ݒ‬

െ

డ௨ డ௬

డሺ௨ǡ௩ሻ

డሺ௫ǡ௬ሻ

ൌ ȁ݂ ᇱ ሺ‫ݖ‬ሻȁଶ |

డ௨


|݂ሺ‫ݖ‬ሻ| | 
|||࣬| | |
||1
|

డ௫

డ௩ డ௩

ൌ

డ௬ డ௫

ǡ

ഏ೔

ଶ

| | 

||࣬|||||| ߲‫ݑ߲ ݑ‬ ߲‫ݑ߲ ݑ‬ ߲‫ ݑ‬ଶ ߲‫ ݑ‬ଶ ߲‫ݑ‬ ߲‫ ݑ‬ଶ ߲ሺ‫ݑ‬ǡ ‫ݒ‬ሻ ߲‫ ݕ߲ ݔ‬ÿ ÿ ߲‫ ݕ߲ ݔ‬ÿ ൌ ÿÿ ൌÿ ൌ ൬ ൰ ൅ ൬ ൰ ൌ ൬ ൅ ݅ ൰ ൌ ȁ݂ ᇱ ሺ‫ݖ‬ሻȁଶ | ÿ ÿ ߲‫ݒ߲ ݒ‬ ߲‫ݑ߲ ݑ‬ ߲‫ݔ‬ ߲‫ݕ‬ ߲‫ݔ‬ ߲‫ݕ‬ ߲ሺ‫ݔ‬ǡ ‫ݕ‬ሻ െ ߲‫ݕ߲ ݔ‬ ߲‫ݔ߲ ݕ‬

ൌ

Ex  : ഏ೔

+|


|‫ ݓ‬ൌ ݂ሺ‫ݖ‬ሻ ൌ ξʹ݁ ర ‫ ݖ‬൅ ሺͳ െ ʹ݅ሻ|!|A
||‫ ܬ‬ൌ

3|

‫ ݑ‬ൌ‫ݔ‬െ‫ݕ‬ డሺ௨ǡ௩ሻ ൌ
|
 ÿ‫ ݒ‬ൌ ‫ ݔ‬൅ ‫ ܬ
ݕ‬ൌ డሺ௫ǡ௬ሻ

డሺ௨ǡ௩ሻ

డሺ௫ǡ௬ሻ

ൌ ȁ݂ ᇱ ሺ‫ݖ‬ሻȁଶ ൌ ÿξʹ݁ ర ÿ ൌ ʹ|

6 | ‫ ݓ‬ൌ ݂ ሺ‫ݖ‬ሻ ൌ ‫ݖ‬ଶ ‫ ܬ‬ൌ ȁ݂ ᇱ ሺ‫ݖ‬ሻȁଶ ൌ ȁʹ‫ݖ‬ȁଶ ൌ ሺʹ‫ ݔ‬൅ ʹ݅‫ݕ‬ሻଶ ൌ Ͷሺ‫ݔ‬ଶ ൅ ‫ ݕ‬ଶ ሻ|

డሺ௨ǡ௩ሻ డሺ௫ǡ௬ሻ

S | J ||డሺ௫ǡ௬ሻ Ǥ డሺ௨ǡ௩ሻ ൌ ͳ| ߲‫ݑ‬ ߲ሺ‫ݑ‬ǡ ‫ݒ‬ሻ ߲ሺ‫ݔ‬ǡ ‫ݕ‬ሻ ߲‫ݔ‬ Ǥ ൌ ÿÿ ߲‫ݒ‬ ߲ሺ‫ݔ‬ǡ ‫ݕ‬ሻ ߲ሺ‫ݑ‬ǡ ‫ݒ‬ሻ ߲‫ݔ‬ ൌ ͳ| |

ñ< |

డ௨

డ௨

డ௫

డ௬

డ௫ డ௩

߲‫ݔ߲ ݑ‬ ߲‫ݕ‬ÿ ߲‫ݑ‬ ÿ ߲‫ ݒ‬ÿ ߲‫ݕ‬ ߲‫ݑ߲ ݕ‬

డ௬ డ௩

ൌቚ

ͳ ͳ

െͳ ቚ ൌ ʹ| ͳ

߲‫ݕ߲ ݑ߲ ݔ߲ ݑ‬ ߲‫ݔ‬ ൅ ߲‫ ݒ‬ÿ ൌ ÿ߲‫ݑ߲ ݕ߲ ݑ߲ ݔ‬ ÿ ߲‫ݕ‬ ߲‫ݕ߲ ݒ߲ ݔ߲ ݒ‬ ൅ ߲‫ݒ‬ ߲‫ݑ߲ ݕ߲ ݑ߲ ݔ‬

߲‫ݕ߲ ݑ߲ ݔ߲ ݑ‬ ൅ ߲‫ ݒ߲ ݕ߲ ݒ߲ ݔ‬ÿ ͳ Ͳ ቚ ൌቚ ߲‫ݕ߲ ݒ߲ ݔ߲ ݒ‬ÿ Ͳ ͳ ൅ ߲‫ݒ߲ ݕ߲ ݑ߲ ݔ‬

)+)$')

| !| 
| ‫ ݓ‬ൌ ߙ‫ ݖ‬൅ ÿሺͳሻ| )| ߙ݁‫ݐ‬ÿ|  | |   |  |  |   | |


|

 | $  | | | |  | 
| ሺͳሻ| | | | | 
| 
| ‫ ݓ‬ൌ ÿ ൅ ÿ |)|ÿ ൌ ݁ ௜ఏబ ‫ ݎ |)|ݎ‬ൌ ܽ‫ ߙ | ݖ‬ൌ ܽ݁௜ ఏబ ||| 
|| | 
|

|

| | | 
|||| 
|| 
| 

||  
| 
|  
 |

d
(
  (  |




c
      | |

ññ |

)+)$!$()&!/

| !| 
| ‫ ݓ‬ൌ

௭ାÿ

௭ା

ሺʹሻ| |  | ߙÿ െ ÿ ് Ͳ| |  |   | 
|  
 |



| 
|  | ' | 
| | | 
| | | 
| 

| |

 
| 
| 
|  |
 
 |!| 
|ሺʹሻ|| |  
 || |

|  | | | | | |  | | | | | | 
 ||| |
| 
|  | |  | 

|||


|

!| 
| |

| | ||| 
| 
 |

 || ||| || | 
| 
 | || | ||| | ||   |' || 


| ሺ௭ ି௭భ ሻሺ௭మ ି௭ ሻ ሺ͵ሻ| |  |
  || ‫ݖ‬ଵ ǡ ‫ݖ‬ଶ ǡ ‫ݖ‬ଷ ǡ ‫ݖ‬ସ మ ି௭భ ሻሺ௭ర ି௭ ሻ


| ‫ݖ‬ଵǡ ‫ݖ‬ଶ ǡ ‫ݖ‬ଷ ǡ ‫ݖ‬ସ  |

 | |
| ሺ௭ర

||


  | |

 | |  | 
| 
| |

|  
 |  |' |

| | 
|| 
|| 
 | 

#|  | 
| 
 || 
| | 
  |

Ex  1 |

| 
|‫ݖ‬଴ | || |

|||| 
 | |"||
 | 
|࣬||| |
| ௭ି௭


 |!| 
‫ ݓ‬ൌ ݁ ௜ఏబ ቀ௭ି௭തതതబ ቁሺͶሻ|
| |
 | 
||0|

| బ

| 
 
| ࣬ÿ| |  | 
| ȁ‫ݒݑ‬ȁ ൌ ͳ|  | 
 | | 
 | | |  |  |

| | | 
 | |   | !-|   |ߠ଴ |  | ' |  
| |
  | | | 
 | 

|| |||' | ||| 
 |||  | &| |
|  | ||

|
|
 |||
| | | 

||-|J||&|$|| | 
 || ||| | | |'| 

| |-|J||&|$|| | || | || |&| ||)|| 
 | || 


|| | |  | || #|
|| | 

| | -|  | | | | 
| 
7
 || | 
 |   |  
 | | -| | 24|| |
| |||
| |
#|| ࣬| 
| | || െλ| 2 
 |-4||൅λ|2 
 |4||
| | |
||-||-|| ||
 |

d
(
  (  |




c
      |

Ex  2 @ | | 
|  
| 
| |  |  
 | | 
 | ‫ ݖ‬ൌ Ͳǡ െ݅ǡ െͳ| | 
 | ‫ ݓ‬ൌ ݅ǡ ͳǡͲ | ,| | ሺ௪ି௪భ ሻሺ௪మ ି௪ ሻ ሺ௪మ ି௪భ ሻሺ௪ି௪ ሻ

ൌ

ሺ௭ି௭భ ሻሺ௭మ ି௭ ሻ | ሺ௭ି௭ ሻሺ௭ି௭ ሻ

ሺ௪ି௜ሻሺଵି଴ሻ

ሺ௪ି଴ሻሺଵି௜ሻ

ሺ௭ି଴ሻሺି௜ାଵሻ

ൌ ሺ௭ାଵሻሺିଵି଴ሻ ฺ ‫ ݓ‬ൌ െ݅ ቀ

௭ାଵ ௭ିଵ

ቁ| ||

Ex  3 @ || 
| 
|
|
| |

|| | 
|| || | 
 ||| | |
|| 

|0|| 
 |‫ ݖ‬ൌ ݅ | | | 
 |‫ ݓ‬ൌ Ͳ| || 
 || |R


| | 
 |‫ ݓ‬ൌ െͳ| |

| -| ‫ ݖ‬ൌ ݅ |  | ‫ ݓ‬ൌ Ͳ|  | | ‫ ݖ‬ൌ λ | ‫ ݓ‬ൌ െͳ | &| ‫ ݓ‬ൌ ݁ ௜ఏబ ቀ Ͳ ൌ ݁ ௜ఏబ ቀ

௜ି௭బ

௭ି௭బ

௭ି௭ തതത బ

ቁ| |
|||||||||||||||||||||||

ቁ|
|
||‫ݖ‬଴ ൌ ݅ |& ||| |  |‫ ݖ‬ൌ λ|||
|‫ ݓ‬ൌ ݁ ௜ఏబ ൌ െͳ ||

തതത ௜ି௭ బ

௭ି௜

!| 
|| ||‫ ݓ‬ൌ ሺെͳሻ ቀ

௭ା௜

ቁൌ

௜ି௭ ௜ି௭ҧ

|

|

ñß |

Ã)&)++)'+)+)$+!$()&!/+ 

 <| !| 
|  
| | |  | | | | | |  | | | || |
|  |
|||| 

|| |||


|

| 
| 
|  | |  | | | | | |‫ݖݖܣ‬ҧ ൅ ‫ ݖ‬൅ ത ‫ݖ‬ҧ ൅ ‫ ܥ‬ൌ Ͳ| )| ‫ ܣ‬൐ Ͳǡ ‫ ܥ‬൐ Ͳ| |)| | |  |
|‫ ܣ‬ൌ Ͳ|| | ||
|||
 | ଵ ଵ ഥ ൅ ത‫ ݓ‬൅ ‫ݓ‬ ഥ ൅ ‫ ܣ‬ൌ Ͳ|
|  | Ã



‫ ݓ‬ൌ ௭ ||‫ ݖ‬ൌ ௪ᇱ|

|
|
 |‫ݓݓܥ‬

  || || || || ௪  | Ã



‫ ݓ‬ൌ ܽ‫ ݖ||ݖ‬ൌ |

|
|
 | ௔

d
(
  (  |




c
      |

‫ݓݓܣ‬ ഥ ൅ ሺ ‫ݖ‬ҧሻ‫ ݓ‬൅ ሺ ത ‫ݖ‬ሻ‫ݓ‬ ഥ ൅ ‫ݓܽܥ‬ ഥ ൌ Ͳ
| ||  |

&| |'|
#||  | | 
 || 
 |||  | |  
| | | || ||  | <| !| 
|  
|  | ' | 
| | | 
| |


| 

||  
| 
| |
 
 |


| |


||‫ ݓ‬ൌ

௭ିÿ

௭ା

ൌ  ൅ ሺ௭ା ÿି

ሻ

ൌ ߣ ൅ ௭ାఔ||)| ߣ ൌ  ǡ ÿ ൌ ÿ

ଵ

ሺÿି

మ

ሻ

| | ൌ  |||

 ||   |!| 
| |
  ||
ൌ ‫ ݖ‬൅ ǡ ‫ ݎ‬ൌ | |‫ ݓ‬ൌ ߣ ൅ ÿ‫|
|ݎ‬  | | 
|| |  
| 
| 
| |
 
 |

<| "| 
|| 
| 
|
|| 
 |‫ݖ‬ଵ ǡ ‫ݖ‬ଶ ǡ ‫ݖ‬ଷ|| |||| 
|  | | ‫ݓ‬ଵ ǡ ‫ݓ‬ଶ ǡ ‫ݓ‬ଷ | | | |  | ,|  | |   | | |  | |  | ሺ௪ି௪భ ሻሺ௪మ ି௪ ሻ మ ି௪భ ሻሺ௪ି௪ ሻ


|   | | 
  | | |‫ݖ‬ଵ ǡ ‫ݖ‬ଶ ǡ ‫ݖ‬ଷ ݁‫ ݖݐ‬ሺ௪


| |

<|

ൌ

ሺ௭ି௭భ ሻሺ௭మ ି௭ ሻ | | |  | | ሺ௭ି௭ ሻሺ௭ି௭ ሻ ௭ି௭


| |  | | | 
 |  
|  | | | 
| ‫ ݓ‬ൌ ݁ ௜ఏబ ቀ௭ି௭തതതబ ቁ| బ

| |

| 
|| ||| | 
 ||| 
 
|| |
|| | | | 
 || 
|ȁ‫ݓ‬ȁ ൑ ͳ | ,||ȁ‫ݓ‬ȁ ൌ ቚ݁ ௜ఏబ ቀ

& #| |
|

௭ି௭బ

௭ି௭ തതത బ

ቁቚ ൌ ቚ

௭ି௭బ

ቚ|

௭ି௭ തതത బ

|

|

ñ2 |


|| || |
 | 
|ȁ‫ ݖ‬െ ‫ݖ‬଴ȁ ൑ ȁ‫ ݖ‬െ ‫ݖ‬ഥ଴ ȁ|

|

)+)$ !=)>#!)+'

| 
| |  E| | | | | |  | |  | | |‫ݓ‬ଵ ǡ ‫ݓ‬ଶ ǡ ǥ ǡ ‫| 
 | | ݓ‬   | 
 ||‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ǥ ǡ ‫| || || | || ݔ‬

d
(
  (  |




c
      | |

| B| 
| 
|   | 
 
| ࣬| |  | 
| | | 
 |  
| | ||| | |
#||  || | | || | ሺͳሻ ÿభ

ÿభ ÿమ ÿ೙ ‫ݓ‬ ൌ ‫ܣ‬ሺ‫ ݖ‬െ ‫ݔ‬ଵ ሻഏషభ ሺ‫ ݖ‬െ ‫ݔ‬ଶ ሻഏషభ ǥ ሺ‫ ݖ‬െ ‫ ݔ‬ሻഏషభ| ‫ݖ‬ ÿమ

ÿ೙

ሺʹሻ‘—‫ ݓ‬ൌ ‫׬ ܣ‬ሺ‫ ݖ‬െ ‫ݔ‬ଵ ሻഏషభ ሺ‫ ݖ‬െ ‫ݔ‬ଶ ሻഏషభ ǥ ሺ‫ ݖ‬െ ‫ ݔ‬ሻഏషభ ‫ ݖ‬൅ )| -|  | J|  | |   |   |

,| || + | "
| | 
 ||‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ǥ ǡ ‫| 

|
 |

||  || ݔ‬ 6 | |!|  |-| |J| 
 | | 


| 
 
| | | 

||   | 3 | |D |  | | | 

| | 
 | |  |‫|
 | |  | | ||


 | | ݔ‬  |||ሺͳሻ݁‫ݐ‬ሺʹሻ|
 |  | S | &|  |


|||   |' |
||| |

||  |  | |

ñ3 | )+)$).)%&)$+++)$ &)$)/ |  | | | | | | | | | | | | | 
|   |  
 | ‫ ݔ‬ൌ ‫ ܨ‬ሺ‫ݐ‬ሻǡ ‫ ݕ‬ൌ ‫ ܩ‬ሺ‫ݐ‬ሻ | ሺͳሻ| ,)| |  | | |  | 8|  | 
 | 

  |- | | 
|‫ ݖ‬ൌ ‫ ܨ‬ሺ‫ݓ‬ሻ ൅ ݅‫ ܩ‬ሺ‫ݓ‬ሻሺʹሻ|  | |||| | | ||| || |

d
(
  (  |




c
      | |

| |

d
(
  (  |




c
      | |

d
(
  (  |