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A) Use the Fourier transform to find $i_{a}$ in the circuit in Fig. P17.28 if $v_{g}$ equals $30 e^{-||5 |}$ A. b) Find $i_{o}(0)$. c) Find $i_{o}\left(0^{+}\right)...

Question

A) Use the Fourier transform to find $i_{a}$ in the circuit in Fig. P17.28 if $v_{g}$ equals $30 e^{-||5 |}$ A. b) Find $i_{o}(0)$. c) Find $i_{o}\left(0^{+}\right)$. d) Use the Laplace transform method to find $i_{0}$ for $t \geq 0$. e) Does the solution obtained in (d) agree with $l_{0}$ for $t>0^{+}$ from $(a) ?$

a) Use the Fourier transform to find $i_{a}$ in the circuit in Fig. P17.28 if $v_{g}$ equals $30 e^{-||5 |}$ A. b) Find $i_{o}(0)$. c) Find $i_{o}\left(0^{+}\right)$. d) Use the Laplace transform method to find $i_{0}$ for $t \geq 0$. e) Does the solution obtained in (d) agree with $l_{0}$ for $t>0^{+}$ from $(a) ?$



Answers

If $ f(t) $ is continuous for $ t \ge 0 $, the Laplace transform of $ f $ is the function $ F $ defined by
$$ F(s) = \int_0^\infty f(t) e^{-st}\ dt $$
and the domain of $ F $ is the set consisting of all numbers s for which the integral converges. Find the Laplace transforms of the following functions.
(a) $ f(t) = 1 $










(b) $ f(t) = e^t $






(c) $ f(t) = t $


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