More of a math problem, but from a physics text

[David92]

Homework Statement

[/B]
So I can't get from line (53) to line (54)

I do not understand how interchanging the variables will cause the imaginary part to disappear

The Attempt at a Solution

I'm not sure where to start
I know exp(ix) = cos(x) +isin(x)
And the only way i see cos(x) coming out is using the exponential identity
exp(ix) + exp(-ix) = 2cos(x) (but this clearly isn't it)
But this 'interchanging' variables really doesn't make sense to me
Please help! The answer is apparently obvious! haha

 

[FactChecker]

Homework Statement

View attachment 113014 [/B]
So I can't get from line (53) to line (54)

From the identity e^iθ = cos(θ)+i*sin(θ), you can split the real and imaginary parts of the integral. You also know that the cosine is an even function, so cos(-θ) = cos(θ) is used to remove the negative exponent.

I do not understand how interchanging the variables will cause the imaginary part to disappear

You are over-thinking it. That is not what makes the imaginary part disappear. If you know that the final answer is real, then you know that the imaginary part is zero without having to do anything. So just ignore the sin terms.

 

[David92]

Oh wow you are right I was overthinking it. Thinking in terms of even and odd functions really helped, thanks!