Sommaire

  La méthode des fractions partielles.
  

      (5s+7)        A      B       C
   -----------  =  --- + ----- + ----- 
   s(s+1)(s-2)      s    (s+1)   (s-2)
         
          
   Multiplions par s(s+1)(s-2)  les numérateurs :
   
    s(s+1)(s-2) (5s+7)      A s(s+1)(s-2)   B s(s+1)(s-2)   C s(s+1)(s-2)
    ----------- ------   =  ------------- + ------------- + ------------- 
    s(s+1)(s-2)   1           s                (s+1)                (s-2)
           
           
   Simplifions :
      
    (5 s+7) = A (s+1)(s-2) + B s(s-2) + C s(s+1)
 
   Si s = (0)  
  
    (5 (0) +7) = A ((0)+1)(0-2) + B (0)(s-2) + C (0)(s+1)
            7  = A    (+1)( -2)    
            A  = -7/2
 
   Si s = (-1)  
       
    (5 (-1)+7) = A ((-1)+1)((-1)-2) + B (-1)((-1)-2) + C (-1)((-1)+1)
            2  = A      (0)    (-3) + B (-1)    (-3) + C (-1)     (0)
            2  = 3 B 
            B  = 2/3 
 
   Si s = (2) 
     
       (5 2+7) = A (s+1)(2-2) + B 2(2-2) + C 2(2+1)
          (17) = A        (0) + B    (0) + C    (6)
          (17) = C 6
             C = 17/6
  
   Donc :
  
      (5 s + 7)         A       B         C
   ---------------  =  --- + ------- + -------  
   s(s + 1)(s - 2)      s    (s + 1)   (s - 2)
   
   
      (5 s + 7)         7/2     2/3       17/6
   ---------------  = - --- + ------- + -------  
   s(s + 1)(s - 2)       s    (s + 1)   (s - 2)