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For any system to be stable, poles can not lie in ROC.Thus, there should not be any poles on the (imaginary axis) Re(s)=0.

Suppose α and β are the poles of the system function H(s) where Re(α)<0 and Re( β)>0.

As Re{s}=0 is contained in the ROC and

, the only possible option is

( to have a non-empty ROC).

Looking at inverse transform of

As Re{s} = 0 lies in ROC , we will have to take to be the inverse.

Thus, in a rational system, with ROC of the system function including Re(s)=0, the poles to the left of imaginary axis contribute right-sided exponentially decaying term and poles to the right of the imaginary axis contribute left-sided exponentially decaying term.

α contributes a right-sided decaying exponential	β contributes a left handed decaying exponent.

Poles to the right of imaginary axis contribute -P_β(t)e^βtu(-t), where P_β(t) is a polynomial of degree k-1

k = order of pole at in H(s)

Similarly poles to the left of imaginary axis contribute P_α(t)e^αtu(t)

The absolute integral

Thus the absolute integral sum of the absolute integrals of these terms (finite number because the system function is rational)

Therefore, the system is stable.

Later, we shall prove the theorem ,that irrespective of the polynomial p(t), converges if and only if converges, in order to justify the convergence of each absolute integral.