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Gain is also taken as Laplace transform of output to Laplace transform of Input . for example voltage gain calculation , it is not necessary to make the energy will be zero in L and C ( if present in the given circuit). But in case of Transfer function to avoid the system dynamics , we have to make the inductor and capacitor energy will be zero as initial condition = 0
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an is a built in formula that uses arguments to calculate information. is FUNCTION
With a formula, you know the variable's value, and you have to calculate the value of the function of it. With an equation, you know the function's value, and you have to calculate the value of the variable.
I am not entirely sure what you mean; but basically, you can evaluate (calculate) the function for each value of the variable.