Hi again!
I'm sorry; I was too busy these last days, so I could not answer before now.
I would like to justify some statements I wrote in previous posts.
Imagine a LTI system for which the relationship between x(t) and y(t) is described by an ordinary linear differential equation with constant coefficients. It is known from the theory of differential equatons that the general solution for y(t) can be expressed by the sum of two parts, i.e.:
a) the solution for x(t)=0 [i.e. for the homogeneous equation]. It is the transient response, and it is a superposition of the "natural modes" of the system, each mode multiplied by some arbitrary constant.
b) a solution for the particular x(t) applied. If it does not contain any natural mode, it is called the forced or steady-state response.
Let me explain with an example.
Suppose a system with lumped elements, input x(t) and output y(t), described by the following linear differential equation:
a2*y''(t) + a1*y'(t) + a0*y(t) = b1*x'(t) + b0*x(t)
a) Any "natural response" is the solution of the homogeneous equation
a2*y''(t) + a1*y'(t) + a0*y(t) = 0
For this kind of linear equation, the solution is a sum of "natural modes" (as many as the order of the equation in y), that have the form k*exp(si*t). In the current case (second order) there are two modes (s1 and s2) that are the roots of the characteristic equation, i.e.:
a2*s2 + a1*s + a0 = 0.
(Of course, these are the poles of the circuit. Each natural mode is associated with a pole of the transfer function.)
The two constants that multiply the natural modes in the solution for a given problem can be determined by two conditions (frequently called "initial conditions").
b) Now we will find a particular solution (forced response) for the non-homogeneous equation for the special case of exponential input.
Suppose that we have an input x(t)=exp(s*t) where s is some complex value. In the general case t is ranging from -∞ to +∞, although any real interval can be considered. Frequently, 0 to ∞ is assumed. Te advantage of taking -∞<t<∞ is that no natural modes are excited by the transient at t=0.
For this exponential case, one particular solution of the differential equation (the forced response) is y(t) with the same shape of x(t), i.e. y(t)=H*exp(s*t) where H is some complex constant. (The theory of differential equtions teaches this, but alternatively, instead of invoke the theory we could assume that shape and later we would find that this was the right guess.)
In this case, the derivatives are: y'(t)=s*H*exp(s*t) and y''(t)=s2*H*exp(s*t).
Replacing in the differential equation, we get
a2*s2*H*exp(s*t) + a1*s*H*exp(s*t) + a0*H*exp(s*t) = b1*s*exp(s*t) + b0*exp(s*t)
Then:
H = ( b1*s + b0 ) / ( a2*s2 + a1*s + a0 )
Remember that s can be any complex value. As the "gain" H depends of the value of s, we shall write
H(s) = ( b1*s + b0 ) / ( a2*s2 + a1*s + a0 ) for any complex s
This is our well-known transfer function.
We arrived at it solving the differential equation of the system for a complex exponential input. The result is that H(s) is the "gain" (output/input) for the forced response of the system when the exponential input is applied.
As LvW pointed out, H(s) is, for each value of s, an eigenvalue. And the complex expontial functions are the "eigenfunctions" of the linear, time-invariant transformation described by the differential equation.
I hope it is clear.
Best regards
Z