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Olá,
z = cis(r) = cos(r) + isen(r)
z^n = cis(nr) = cos(nr) + isen(nr)
Sum(cis(kr)) = Sum(z^k) , k = 1 ... n
Sum(z^k) = z + z^2 + z^3 + ... + z^n = z(z^n - 1) /
(z - 1) [somatorio de PG]
Sum(z^k) = cis(r) [ cis(nr) - 1 ] / [ cis(r) - 1
]
Sum(cis(kr)) = cis(r) [ cis(nr) - 1 ] / [ cis(r) -
1 ] = cis(r) [ cis(nr) - 1 ] * [ cis(-r) - 1 ] / [ 1 - cis(-r) - cis(r) + 1
]
cis(r) + cis(-r) = cos(r) + isen(r) + cos(r) -
isen(r) = 2cos(r)
Sum(cis(kr)) = [ cis(r)cis(nr)cis(-r) -
cis(r)cis(-r) - cis(r)cis(nr) + cis(r) ] / [ 2 + 2cos(r) ]
Re(Sum(cis(kr))) = Sum(cos(kr))
Im(Sum(cis(kr))) = Sum(sen(kr))
Sum(cis(kr)) = [ cis(nr) - 1 - cis[(n+1)r] + cis(r)
] / 2[1 + cos(r)]
Re(Sum(cis(kr))) = [ cos(nr) - 1 + cos[(n+1)r] +
cos(r) ] / 2[1+cos(r)]
Im(Sum(cis(kr))) = [ sen(nr) - 1 - sen[(n+1)r] +
sen(r) ] / 2[1+cos(r)]
Sum(cos(kr)) = [ cos(nr) - 1 + cos[(n+1)r] +
cos(r) ] / 2[1+cos(r)]
Sum(sen(kr)) = [ sen(nr) - 1 - sen[(n+1)r] +
sen(r) ] / 2[1+cos(r)]
Sum(sen(a+kr)) = Sum(sen(a)cos(kr) + cos(a)sen(kr))
= sen(a)*Sum[cos(kr)] + cos(a)*Sum[sen(kr)]
Sum(cos(a+kr)) = Sum(cos(a)cos(kr) - sen(a)sen(kr))
= cos(a)*Sum[cos(kr)] - sen(a)*Sum[sen(kr)]
abraços,
Salhab
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