Basic complex number math -- what am I doing wrong?

[Isaac0427]

For this, f and g are real functions, and k is a real constant.

I have $\psi = fe^{ikx}+ge^{ikx}$ and I want to find $\left|\psi \right|^2$. I went about this two different ways, and got two different answers, meaning I must be doing something wrong.

Method 1:
$\psi =(f+g)e^{ikx}$
$\left|\psi \right|^2=(f+g)^2\left|e^{ikx}\right|^2=(f+g)^2=f^2+g^2+2fg$

Method 2:
Since $|a+b|^2=|a|^2+|b|^2+2\Re (ab)$,
$|\psi |^2=f^2+g^2+2\Re \left( fge^{2ikx} \right)=f^2+g^2+2\cos(2kx)fg$

Method 2 gives me that extra cosine term. Where did I go wrong?

 

[mjc123]

Since |a+b|2=|a|2+|b|2+2R(ab)|a+b|2=|a|2+|b|2+2ℜ(ab)|a+b|^2=|a|^2+|b|^2+2\Re (ab),

Edit: Why isn't formatting preserved in quotes?
Where does this come from? If we write a = a + ic and b = b + id
|a + b|² = |a+b+i(c+d)|² = (a+b)² + (c+d)² = a² + c² + b² + d² + 2(ab + cd)
ab = ab - cd + i(ad +bc)
Hence |a + b|² is not equal to |a|² + |b|² + 2Re(ab)

 

[TeethWhitener]

Since $|a+b|^2=|a|^2+|b|^2+2\Re (ab)$,

Where are you getting this from?
Edit: mjc beat me to it. BTW: You can format the TeX in the quotes directly.

 

[Isaac0427]

Where are you getting this from?

Oh man... I messed up my identities... It should be 2Re(ab*) which would give me where I went wrong. Thank you for pointing this out!