Let a network be described by a transfer function H(s).
Let us denote the input signal as r(t) and the output signal
as y(t). I am interested in finding the output y(t) when the
input is a sinusoidal function, i.e., 
.
The Laplace transform of the input is:
and hence the output can be written as:
Let
Let
This implies:
Then:
Noting that
we can write:
In the above we haven't paid any attention to the third part of the
expression on the left hand side of the equation (1). Let us
look at that expression now.
Note that both 
and 
can be either real or complex.
When is real then is also real. For complex
there will be another 
and in general
we can write:
It is easy to see that when (Real part of 
) 
then
.
For all passive RLC networks 
is always less than 0. The
condition also means that all the roots
of the denominator of the transfer function H(s) are in the left half of
the complex plane. Roots of the denominator of the transfer function are
also known as the system poles; zeros are the roots of the transfer
function numerator.
In other words for any system with the poles in the left half complex plane
the steady-state response to a sinusoid of frequency 
\
can be worked out by evaluating the magnitude and the phase of the complex number
and then noting that the magnitude of the output is
given by the magnitude of the input sinusoid times the magnitude of
and the phase shift between the input sinusoid and the
output is given by the phase of .