(adsbygoogle = window.adsbygoogle || []).push({}); [SOLVED] Representing a wave as a complex number.

I'm just a bit confused as to the validity of representing the equation of a wave or oscillatory motion as a complex number. As is my understanding the argument for doing so goes thus:

Assuming our amplitude is 1, our equation is:

[tex]y(t) = cos ( \omega t)[/tex]

Which we can write as

[tex] cos ( \omega t) = [Re] exp(i \omega t)[/tex]

Which it certainly is. but then people seem to go on to manipulate [tex][Re] exp(i \omega t)[/tex] as if it were simply [tex]exp(i \omega t)[/tex] and then consider the answers correct. For example my lecturers appear to be squaring [tex]exp(i \omega t)[/tex] as if it were [tex] = cos( \omega t)[/tex]. Which, as far as I can tell, reduces to absurdity quite quickly:

[tex] cos( \omega t) = exp(i \omega t)

\Rightarrow

cos ( \omega t) = cos ( \omega t) + iSin( \omega t)

\Rightarrow

cos^2 ( \omega t) = cos^2 ( \omega t) - sin^2 ( \omega t) +2iCos( \omega t)Sin( \omega t)

[/tex]

The real part of the right side is clearly not equal to the real part of the left side. And so, I don't understand how [tex]exp(i \omega t)[/tex] can be used, usefully, to describe a wave.

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# Homework Help: Representing a wave as a complex number.

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