How Do I Determine the Fourier Transform of f(x) = sin(x)?

[ZeroScope]

Determine the Fourier transform of f(x) = sin (x), pi/2 > x > -pi/2 ; 0 otherwise

To do this,
i) express f(x) as a complex exponential,
ii) write down the Fourier integral,
iii) solve the integral, and
iv) replace the complex exponentials by simple trigonometric functions.

Attempt at answer

To start with i get stuck when deciding if i substitute the complex exponential term of sin (x);
see attachment

then substitute this into the Fourier integral and try and evaluate. I am not sure what to do from this point, presuming of course its correct up to this.

 

[HallsofIvy]

The formulas you give are correct, of course. After "replace the complex exponentials by simple trigonometric functions", can either, depending upon the problem, take only the real part of the solution (the way you set up the problem in the first place should make it clear if you should do that) or use the entire solution. If your original problem had only real numbers in its initial or boundary values, you should be able to "incorporate" the "i" in the constants involved in the general solution.

 

[ZeroScope]

What I am having trouble with is, is solving the integral. I end up with an equation with a lot of fractions and exponentials.

 

[coomast]

Hello ZeroScope,

Sorry for this late reply, but can you show what you have obtained as integral?
My result is a function with only an imaginary part. I would like to check if this is the one you got.