# Euler’s relation and the energy of a complex exponential signal

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### Question :

Euler’s relation and the energy of a complex exponential signal

I’m having a little hard time understanding the equation:

Why Euler’s relation’s square is equal to 1?

I know how to write Euler’s equation in terms of sin and cos:

But when I do the math, the square of this equality is nothing but 1. I can’t get rid of from the sin and cos in the equation:
$$(ejwt)2=cos2(jwt)+2jcos(jwt)sin(jwt)+j2sin2(jwt)$$

$$=cos2(jwt)−sin2(jwt)+2jcos(jwt)sin(jwt)$$

I’m probably missing something with the Euler’s relation here so I would like to hear what am I missing.

Notice, that $$?x?Rforall xinmathbb{R}$$:

$$|exp(xi)|=1$$

Because:

$$|exp(xi)|=|cos(x)+sin(x)i|=?cos2(x)+sin2(x)?= 1=?1=1$$

So, for your integral we get:

$$?T00|exp(?0tj)|2 dt=?T00|cos(?0t)+sin(?0t)j|2 dt=$$

$$?T00(?cos2(?0t)+sin2(?0t)?= 1)2 dt=?T00(?1)2?= 1 dt=$$

$$?T001 dt=[t]T00=T0?0=T0$$