what is the difference between transfer function and gain of a circuit?

transfer function:- it is a function that is used to describe the relation between input and output parameters (i.e impedance, admittance, current, voltage, power) of a circuit.

basically how circuit behaves to different parameters....

gain:- it is essentially relate the input voltage, current or power to the output voltage, current or power respectively. (i.e. basically how much you ''gain'' from the circuit, which can be more than 1 or less than 1)

that is what i understand with transfer function and gain...

They are essentially the same thing. Moreover I've seen transfer functions used in regard to frequency response of a circuit. Gain, is simply magnitude change, ie a basic OP-AMP.

--edit--

I sort of insinuated this, but a transfer function can result in more interesting results. Doing the calculations will require differential equations or an understanding of the Leplace transform and its inverse. The latter allows for a lot of simplification.

Gain is just at one specific frequency

A transfer function will tell you the gain of the circuit at all frequencies

hello