A transfer function is also known as the network function is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a (linear time invariant) system. The transfer function is the ratio of the output Laplace Transform to the input Laplace Transform assuming zero initial conditions. Many important characteristics of dynamic or control systems can be determined from the transfer function. The transfer function is commonly used in the analysis of single-input single-output electronic system for instance. It is mainly used in signal processing, communication theory, and control theory. The term is often used exclusively to refer to linear time-invariant systems (LTI). In its simplest form for continuous time input signal x(t) and output y(t), the transfer function is the linear mapping of the Laplace transform of the input, X(s), to the output Y(s).Zeros are the value(s) for z where numerator of the transfer function equals zero. The complex frequencies that make the overall gain of the filter transfer function zero.Poles are the value(s) for z where the denominator of the transfer function equals zero. The complex frequencies that make the overall gain of the filter transfer function infinite.

The general procedure to find the transfer function of a linear differential equation from input to output is to take the Laplace Transform of both sides assuming zero conditions, and to solve for the ratio of the output Laplace over the input Laplace.The transfer function provides a basis for determining important system response characteristics without solving the complete differential equation. As defined, the transfer function is a rational function in the complex variable ‘s’ that is It is often convenient to factor the polynomials in the numerator and the denominator, and to write the transfer function in terms of those factors:

$${G(s) = \frac{N(s)}{D(s)} = k\frac{(s-z_1 )(s-z_2 )... ... ...(s-z_n)}{(s-p_1 )(s-p_2 )... ... ...(s-p_m)} }$$ Where, the numerator and denominator polynomials, N(s) and D(s).
The values of s for which N(s) =0, are known as zeros of the system. i.e; at: $${s = z_1, z_2……….. z_n}$$ The values of s for which D(s) =0, are known as poles of the system. i.e; at: $${s = p_1, p_2……….. p_m}$$ Obtain Pole, zero, gain values of any transfer functions and transfer function from pole, zero and gain and plot them. Also verify your result theoretically.