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[tex]\beta = \frac{v}{c}= \tanh \psi= - i \tan i \psi= \frac{( e^{\psi} - e^{-\psi})}{( e^{\psi} + e^{-\psi})} [/tex],

the rotation of [tex](x',t')[/tex]-axes being defined as imaginary (considering x and t real, and c=1).

However, I have found books where this rotation is considered as real, while the rotation in Figure_1(a) is shown as imaginary (considering x real, t imaginary, and c=1).

Could you explain which rotation is real and which imaginary?