# Calculating inverse fourier transform and graphing it

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) • What is "the module" of the function?
– JRE
Mar 7 at 16:49
• Magnitude of the complex number (we call it ''modulo") Mar 7 at 16:51
• Fourier transform works in complex signals, but we often use real signals anyway (imaginary part is 0). When you Fourier-transform a real signal, the output is a frequency spectrum and the magnitude/modulo tells you the amplitude and the argument/angle tells you the phase. It's harder to say what the imaginary part means for inverse Fourier transform, but consider that forward and inverse are almost identical, it could still be interpreted the same way. Mar 7 at 16:52
• @JRE Slovak language - Modul môže byť: absolútna hodnota (reálneho či komplexného čísla alebo prvku v čiastočne usporiadanom vektorovom priestore alebo vektora). I know, it's quite weird. Should have told you that the magnitude is the concern here. Mar 7 at 16:58
• "I know that omega is always positive" No, it's not. Redo your integration from $-\omega_0$ to $+\omega_0$ Mar 7 at 17:26

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