9.1. For each of the following integrals, specify the values of the real parameter o which ensure that the integral conv

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9 1 For Each Of The Following Integrals Specify The Values Of The Real Parameter O Which Ensure That The Integral Conv 1 (457.08 KiB) Viewed 20 times

9.1. For each of the following integrals, specify the values of the real parameter o which ensure that the integral converges: (a) √e¯5¹¸¯(σ+jw)t dt е (b) [e-5te-(o+jw)t dt -(o+jw)t dt 5 -5t-(o+jw)t dt (c) se¯ e (d) e e (e) e-5¹e-(o+jw)t dt (f) [e-5\t\e-(o+jw)t dt 9.2. Consider the signal x(t) = e-5¹u(t = 1), - and denote its Laplace transform by X(s). (a) Using eq. (9.3), evaluate X(s) and specify its region of convergence. (b) Determine the values of the finite numbers A and to such that the Laplace trans- form G(s) of g(t) = Ae-5¹u(-t-to) has the same algebraic form as X(s). What is the region of convergence corre- sponding to G(s)? 9.3. Consider the signal x(t) = e¯5¹u(t) + e¯ßt u(t), and denote its Laplace transform by X(s). What are the constraints placed on the real and imaginary parts of ß if the region of convergence of X(s) is Re{s} > −3?