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chisigma · #2 July 3rd, 2009, 11:47 PM

We can start remembering the well known identity...

$t^{\omega} = e^{\omega \cdot \ln t} = \sum_{n=0}^{\infty} \frac{(\omega \cdot \ln t)^{n}}{n!}$ (1)

... where both $t$ and $\omega$ are real or even complex variables. Also well known is the Taylor expansion of the fuction $\cos (*)$ that permits us to write...

$\cos (\omega \cdot \ln t) = \sum_{n=0}^{\infty} (-1)^{n} \frac{(\omega \cdot \ln t)^{2n}}{(2n)!}$ (2)

Combining (1) and (2) we arrive to the 'simple' identity...

$\cos (\omega\cdot \ln t) = \frac{t^{i\cdot \omega} + t^{-i\cdot \omega}}{2}$ (3)

An easy question to You before proceeding: is it all right?...

Kind regards

$\chi$ $\sigma$