![]() On the convergence of some classes of Dirichlet series. inverse Laplace transform representation of a real or complex function f(t) of the real variable t by the integral transformation. We investigate the exponential decay of the tail probability P(X > x) of a continuous type random variable X.Let (s) be the LaplaceStieltjes transform of the probability distribution function F(x) P(X x) of X, and 0 be the abscissa of convergence of (s).We will prove that if < 0 < 0 and the singularities of (s) on the axis of convergence are only a finite number of. On the absolute convergence of Dirichlet series. Once we know the method of obtaining the Laplace transform, however, it is. Analogously, the two-sided transform converges absolutely in a strip of the form a Re(s) b, and possibly including the lines Re(s) a or Re(s) b. This is equivalent to the absolute convergence of the Laplace transform of the impulse response function in the region Re(s) 0. The Laplace transform of any Laplace transformable function f(t) can be found by multiplying f(t) by e-st and then integrating the product from t 0 to t 00. The constant a is known as the abscissa of absolute convergence, and depends on the growth behavior of f(t). Next, we have LEMMA 2 If 0 <0, then for s with
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