Re: Inverse function solution
- To: mathgroup at smc.vnet.net
- Subject: [mg132652] Re: Inverse function solution
- From: Roland Franzius <roland.franzius at uos.de>
- Date: Tue, 29 Apr 2014 01:33:11 -0400 (EDT)
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Am 28.04.2014 03:45, schrieb Narasimham:
> Solve[ {x == Cos[u], y == Cos[u + v] }, {u, v} ]
>
> Its closed/analytic solution is not possible, even numerically.
>
> The known solutions are ellipses from sine waves with a phase difference, having x^2, x y and y^2 terms, as also sketched in Lissajous curves:
>
> ParametricPlot[{Cos[u], Cos[u + v]}, {u, -Pi, Pi}, {v, -Pi, Pi}]
>
> Can there be a work around?
>
Yes, replace trigonometric functions by rationals of exponentials.
{x == Cos[u], y == Cos[u + v]} // TrigExpand // TrigToExp
{x == E^(-I u)/2 + E^(I u)/2,
y == 1/2 E^(-I u - I v) + 1/2 E^(I u + I v)}
Solve[ {2 x == q + 1/q, 2 y == p q + 1/(p q)}, {q, p}]
q-> Exp[I u], p-> Exp[I v]
--
Roland Franzius