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

Answer #1:

Notice, that $forall xinmathbb{R}$:

|exp(xi)|=1

$$left|expleft(xiright)right|=1tag1$$

Because:

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

$$left|expleft(xiright)right|=left|cosleft(xright)+sinleft(xright)iright|=sqrt{underbrace{cos^2left(xright)+sin^2left(xright)}_{=space 1}}=sqrt{1}=1tag2$$

So, for your integral we get:

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

$$int_0^{text{T}_0}left|expleft(omega_0ttext{j}right)right|^2spacetext{d}t=int_0^{text{T}_0}left|cosleft(omega_0tright)+sinleft(omega_0tright)text{j}right|^2spacetext{d}t=$$
?T00(?cos2(?0t)+sin2(?0t)?= 1)2 dt=?T00(?1)2?= 1 dt=

$$int_0^{text{T}_0}left(sqrt{underbrace{cos^2left(omega_0tright)+sin^2left(omega_0tright)}_{=space1}}right)^2spacetext{d}t=int_0^{text{T}_0}underbrace{left(sqrt{1}right)^2}_{=space1}spacetext{d}t=$$
?T001 dt=[t]T00=T0?0=T0

$$int_0^{text{T}_0}1spacetext{d}t=left[tright]_0^{text{T}_0}=text{T}_0-0=text{T}_0tag3$$

Answer #2:

I believe the vertical bars inside the integration denote magnitude. This exponent is a phasor whose magnitude is one and angle is (wt). The square of one is also one.

The magnitude of the phasor (or complex number) is the square root of: the real part squared plus the imaginary part (without j) squared. When you add sin^2 and cos^2, the result is one.

Hope this helps.