X(s)=9[x(t) where the operator notation s means to multiply the signal x) being operated upon by the complex expo- nential e-st and then to integrate that product over the time interval (oo, too) Region of Absolute Convergence In evaluating the Laplace transform integral that corresponds to a given signal, it is generally found that this integral will exist(that is, the integral has finite magnitude) for only a restricted set of s values. The definition of region of absolute convergence is as follows. The set of complex numbers s for which the magnitude of the Laplace transform integral is finite is said to constitute the region of absolute convergence for that integral transform. This region of convergence is always expressible as σ+<Re(s)<o where o, and o denote real parameters that are related to the causal and anticausal components, respectively, of the sigr Laplace transform is being sought Laplace Transform Pair Tables It is convenient to display the Laplace transforms of standard signals in one table. Table 6.1 displays the time ignal x( t) and its corresponding Laplace transform and region of absolute convergence and is sufficient for our needs Example. To find the Laplace transform of the first-order causal exponential signal x,(t)=e-atu(t where the constant a can in general be a complex number. The Laplace transform of this general exponential signal is determined upon evaluating the associated Laplace x1(s) eu(t)e dt In order for X,(s) to exist, it must follow that the real part of the exponential argument be positive, that is, Re(s +a)=Re(s)+ re(a)>0 If this were not the case, the evaluation of expression(6. 1)at the upper limit t= too would either be unbounded if Re(s)+ Re(a)<0 or undefined when Re(s)+ Re(a)=0. On the other hand, the upper limit evaluation is zero when Re(s)+ Re(a)>0, as is already apparent. The lower limit evaluation at t=0 is equal to 1/(5+ a) for all choices of the variable s The Laplace transform of exponential signal e-at u(t) has therefore been found and is given by ∠[e-"u(t) Rels)© 2000 by CRC Press LLC X(s) = +[x(t)] where the operator notation + means to multiply the signal x(t) being operated upon by the complex expo￾nential e–st and then to integrate that product over the time interval (–•, +•). Region of Absolute Convergence In evaluating the Laplace transform integral that corresponds to a given signal, it is generally found that this integral will exist (that is, the integral has finite magnitude) for only a restricted set of s values. The definition of region of absolute convergence is as follows. The set of complex numbers s for which the magnitude of the Laplace transform integral is finite is said to constitute the region of absolute convergence for that integral transform. This region of convergence is always expressible as s+ < Re(s) < s– where s+ and s– denote real parameters that are related to the causal and anticausal components, respectively, of the signal whose Laplace transform is being sought. Laplace Transform Pair Tables It is convenient to display the Laplace transforms of standard signals in one table. Table 6.1 displays the time signal x(t) and its corresponding Laplace transform and region of absolute convergence and is sufficient for our needs. Example. To find the Laplace transform of the first-order causal exponential signal x1(t) = e –at u(t) where the constant a can in general be a complex number. The Laplace transform of this general exponential signal is determined upon evaluating the associated Laplace transform integral (6.1) In order for X1(s) to exist, it must follow that the real part of the exponential argument be positive, that is, Re(s + a) = Re(s) + Re(a) > 0 If this were not the case, the evaluation of expression (6.1) at the upper limit t= +• would either be unbounded if Re(s) + Re(a) < 0 or undefined when Re(s) + Re(a) = 0. On the other hand, the upper limit evaluation is zero when Re(s) + Re(a) > 0, as is already apparent. The lower limit evaluation at t = 0 is equal to 1/(s + a) for all choices of the variable s. The Laplace transform of exponential signal e–at u(t) has therefore been found and is given by X s e u t e dt e dt e s a at st s a t s a t 1 0 0 ( ) ( ) ( ) ( ) ( ) = = = - + - - - + +• -• +• - + +• Ú Ú L [e u(t)] Re( ) Re( ) s a s a -at = + > - 1 for