quintica polinomi y serie exponencial

x^{5}+ax^{3}+(a^{2}/5)·x+c = 0

u^{5}+v^{5} = c

5uv( u^{3}+v^{3} ) = a·( u^{3}+v^{3} )

10·(uv)^{2}( u+v ) = (2a)·uv·(u+v)

a·uv(u+v) = (a^{2}/5)·(u+v)

x^{5}+5x^{3}+5x+c = 0

x^{5}+10x^{3}+20x+c = 0

f(x) = f(a)+c_{1}·(e^{x}+(-1)·e^{a})+c_{2}·(e^{x}+(-1)·e^{a})^{2}+...

... c_{n}·(e^{x}+(-1)·e^{a})^{n}+...

d_{x}[f(a)] = c_{1}·e^{a}

c_{1} = d_{x}[f(a)]·e^{(-a)}

d_{xx}^{2}[f(a)] = c_{1}·e^{a}+c_{2}e^{2a}

c_{2} = ( d_{xx}^{2}[f(a)]+(-1)·d_{x}[f(a)] )·e^{(-2)a} 

d_{xxx}^{3}[f(a)] = c_{1}e^{a}+c_{2}e^{2a}+c_{3}e^{3a}

c_{3} = ( d_{xxx}^{3}[f(a)]+(-1)·d_{xx}^{2}[f(a)] )·e^{(-3)a}

e^{x} = e^{a}+(e^{x}+(-1)·e^{a})

e^{(-x)} = ( 2+(-1)·e^{x} )+2·sum[ k = 2 --> oo ][ (-1)^{k}·(e^{x}+(-1))^{k} ]

cosh(x) = 1+sum[ k = 2 --> oo ][ (-1)^{k}·(e^{x}+(-1))^{k} ]

sinh(x) = ( e^{x}+(-1) )+(-1)·sum[ k = 2 --> oo ][ (-1)^{k}·(e^{x}+(-1))^{k} ]