Proving Complex Conjugate is Real: Euler's Identity

[JPBenowitz]

Homework Statement

So we are given $\alpha$exp(i$\varpi$t) +$\alpha$*exp(-i$\varpi$t) and are asked to prove the resulting equation is real.

Homework Equations

$\alpha$ + $\alpha$* = 2Re($\alpha$) and Euler's Identity

The Attempt at a Solution

I tried expanding out the exp's to cosines and isines but couldn't reach the solution.

 

[Mute]

What if you also write $\alpha = a + i b$, where a and b are both real numbers? Can you do the problem then?

 

[JPBenowitz]

What if you also write $\alpha = a + i b$, where a and b are both real numbers? Can you do the problem then?

I figured it out but I didn't do it that way. That seems awfully more tedious than usual.

 

[Mute]

I figured it out but I didn't do it that way. That seems awfully more tedious than usual.

It's more tedious than the quick solution, yes, but it was more along the lines of the approach you had tried to take by expanding the exponentials into sines and cosines, so I opted to guide you along that direction, in case the problem wanted you to show it explicitly.

 

[Dick]

Why are you writing a bar over the omega? If omega is real then it's sort of obviously true. Because you are adding two complex conjugates. If omega isn't real then it's not even true.