let 0<x<1 and Γ(x) =∫_0 ^∞ t^(x−1) e^(−t) dt
1) prove that Γ(x).Γ(1−x) =(π/(sin(πx))) (compliments formulae)
2) calculate Γ(n) and Γ(n+(1/2)) with n from N.
solving
ax^4 +bx^3 +cx^2 +dx+e=0
(a≠0, b, c, d, e)∈Q
special cases (easy to solve)
ax^4 +e=0 solve at^2 +e=0 ⇒ x=±(√t_(1, 2) )
ax^4 +cx^2 +e=0 solve at^2 +ct+e=0 ⇒ x=±(√t_(1, 2) )
always try all factors of ±e
because a(x−α)(x−β)(x−γ)(x−δ)=ax^4 +...+αβγδ
⇒ e=αβγδ
next we must find the nature of the solutions
4 real solutions
2 real & 2 complex solutions
4 complex solutions
a, b, c, d, e ∈Q ⇒ complex solutions always in
conjugated pairs
draw the function or calculate some values
to find the number of real solutions
divide by a
x^4 +px^3 +qx^2 +rx+s=0
[p=(b/a) q=(c/a) r=(d/a) s=(e/a)]
I′ll soon post some cases I′ve been able to solve
as comments