Chercher alpha beta et phi

t²/(1-t²)= alpha + (beta/(1-t)) + (phi/(1+t))

:'( je crois que alpha = -1

start by putting everything on the same denominator. Then multiply by (1-t^2). This gives:

t^2 = alpha (1-t^2) + beta(1+t) + phi(1-t)
t^2 = -alpha t^2 + (beta - phi).t + alpha + beta + phi
now the coefficients of t^2, t and t^0 in the RHS and LHS must be equal.
so:
alpha=-1 (as you said)
beta = phi
-1 + 2phi = 0 or
beta and phi are both equal to 1/2.

t^2 / (1-t^2) = A + B/ (1-t) + C/(1+t)
t^2 = A(1-t^2)+B(1+t) +C (1-t)

= A -A t^2 +B +Bt +C -Ct
=(A+B+C) +(B-C)t -At^2
-A = 1 ==> A=-1 ...correct

A+B+C =0, B-C =0
B+C =1
B-C=0
B=C=1/2
t^2 /(1-t^2) = -1 +1/2 /(1-t) +1/2/(1+t)