σ = 0 corresponds to the case of sinusoidal steady state. If you allow σ > 0 then you are choosing signals to represent both the stimulus and response that are growing exponentials. If σ < 0 then your signals are all decaying exponentials. When σ = 0 the signals are all simple sinusoids, and so the response you compute is the sinusoidal steady-state response. The case where σ = 0 is also referred to as phasor analysis.

-Ken

The interesting thing (provided you assume nice functions that are absolutely integrable) is that both the representations are entire. That is, you could use either the Fourier kernel exp(jwt) or the Laplace kernel exp(st) for unique linear decompositions with arbitrarily bounded LMS error. In some sense it's like how a function can be expanded either as a Fourier series (sine and cosine) or as a Chebyshev series (T_n) with the latter converging much faster within a certain zone (what would be called the 'passband' in Chebyshev filters). Also gives some idea of why wavelet representations are more dense than Fourier or Laplace, and why the Karhunen-Loeve expansion of an AWGN contains the least number of eigenvalues.

Basis functions bear out George Orwell's inunction that "All animals are equal, but some animals are more equal than others"

M.G.Rajan
www.eecalc.com