da luca.barletta » 09/11/2006, 19:39
Consider the polar form $z=rhoe^(itheta)$.
Let's find the non-trivial solutions of the equation (trivial: $rho=0$):
$rhoe^(i2theta)+costhetasintheta-rhoe^(itheta)=0$ (1)
Immaginary part of (1):
$rhosin(2theta)-rhosintheta=0$
discarding the trivial solution:
$sin(2theta)-sintheta=0 rarr {(theta_1=kpi),(theta_2=pi/3+2kpi),(theta_3=-pi/3+2kpi):}, k in ZZ$
Real part of (1):
$rhocos(2theta)+sinthetacostheta-rhocostheta=0$ (2)
Make the substitution $theta=theta_1$ in (2):
$rho-rho(-1)^k=rho(1-(-1)^k)=0$
discarding the trivial solution, $k$ must be even, so $theta_1=2kpi, k in ZZ$.
Make the substitution $theta=theta_2$ in (2):
$-rho/2+1/2sqrt(3)/2-rho/2=0 rarr rho=sqrt(3)/4$
Make the substitution $theta=theta_3$ in (2):
$-rho/2-1/2sqrt(3)/2-rho/2=0 rarr rho=-sqrt(3)/4<0$
Finally, solutions are:
$(rho=0) vv (theta=2kpi, k in ZZ) vv (rho=+sqrt(3)/4, theta=+pi/3+2kpi, k in ZZ)$
Ultima modifica di
luca.barletta il 09/11/2006, 21:46, modificato 1 volta in totale.
Frivolous Theorem of Arithmetic:
Almost all natural numbers are very, very, very large.