What is G and what is H? What is the difference in the way they are defined?

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Both \(G(s)\,\) and \(H(s)\,\) are notations for transfer functions. Sometimes we use \(G(i\omega)\,\) (or \(H(i\omega)\,\)) to refer to the "frequency response" if the system is asymptotically stable; it can be obtained by evaluating \(G(s)\,\) at \(s=i\omega\,\).