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I saw that when we change time domain into frequency domain, we can use frequency itself (denoted as X(f)) or angular frequency domain like X(w).

I know that w = 2πf. Is there any reason there are two types of frequency domain? What is the adventage of using w-domain instead of f-domain?

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The difference is as you said, only in the constant of multiplication, $2 \pi$. The circular frequency is prefered to the Hertztian frequency when you are working with sines and cosines because you can easily spot the roots of those functions. But other than that, those frequency domains can be equivalently used.

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