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)
