Quantum Mechanics algebra - complex analysis

[sxc656]

Hi,

I cannot work out how the working shown in the attached pic is well, er worked out!
Could someone explain the ins and outs of the complex analysis of taking the real or imaginary parts of some formula, for example in the context of the my case.

 

[Pengwuino]

Remember that $$e^{ix}= cos(x) + isin(x)$$. Taking the imaginary part means you're looking at just the sine part. When you combine into that integral form, the solution is simpler.

 

[sxc656]

Remember that $$e^{ix}= cos(x) + isin(x)$$. Taking the imaginary part means you're looking at just the sine part. When you combine into that integral form, the solution is simpler.

Is this what you mean, i am not sure about the last two lines of working.

 

[vela]

You can write

$$sin pr = Im(e^{ipr})$$

Because Im(z+w) = Im(z)+Im(w) and Im(az) = aIm(z) for real a, you can pull the other exponential in as well as reverse the order of I am and the integral.

 

[sxc656]

Thanks to all