Calculating inverse fourier transform and graphing it

Question

Recently I stumbled upon Fourier transforms which means I am very new to it. I've been given a question from my professor to find the inverse Fourier transform of a frequency spectrum which is given by:

I can go through each step to explain what I've done so far.

At first, I wrote the basic formula for inverse Fourier transform which is shown below

I know that omega is always positive, I rewrote the inverse fourier transform like this

After a little algebra and integration, I got to this equation which is shown below

Now the biggest equation is this - I can surely see that the signal seems to be in the complex form. I tried rewriting the exponential part in terms of sine and cosine function and then split the imaginary part and the real part. Is this the correct way to graph it? The equations which contain real part of the function and imaginary part are shown below

Also another thing I thought about is to find the magnitude of the complex function once I have real and imaginary part. Question is if this approach is correct. When I found the module of this function, I got this result (assuming that omega zero is equal to 1)

Answer 1

Complex formulation of Fourier transforms need also negative frequencies! They are there a must. "I know omega is positive" should be written "I do not know why there should be negative frequencies, so I leave them out".

Do again the inverse transform integration, but include the full range i.e.

Qualitatively one should see in 2 seconds that the spectrum belongs to an ideally low pass filtered impulse. That's constant amplitude vs. frequency, only a certain bandwidth taken into the account and linear phase which means constant frequency independent delay.

Do not worry. Like the rest of us, also you will see it soon if you continue with Fourier transforms. And you will if you are going to understand the signal theory.

Not asked, but if we look backwards we see that negative frequency has puzzled people several times. Presenting a plain common sinusoidal voltage as a sum of positive and negative frequency component gives to us a possibility to use complex exponentials and that makes numerous common formulas and equations incredibly simple and symmetric and that's utilized well in communication signal theory books.