Mathematically, the Laplace transform of a time domain function ?(?) is Into algebraic equations in the frequency domain. x(00) lim sX (s) 10.The Laplace transform is a mathematical tool which is used to convert theÄifferential equations representing a linear time invariant system in time domain Table 2.2.2 Properties of the Laplace transform X(s)=Jof()e dt aF(s)bG (s) x(t) af (t)bg (t) 1. $2(82 +b2) sin bt bt 2b3 sin bt- bt cos bt 27. General Formulas Tables of Inverse Laplace Transforms Tables of Fourier Cosine Transforms Tables of Fourier Sine Transforms. TRANSFORMS OF STANDARD FUNCTIONS f(t)f(s) 1 1. constant, c u,(t-D), shifted unit step 4. The Laplace transform f(s) of a function f(t) is defined by: 0 s estf ( t)dt. Table 2.2.1 Table of Laplace transform pairs. Table 1: Properties of Laplace Transforms Number Time Function Laplace Transform Property 1 f1(t)+f2(t) F1(s)+F2(s) Superposition 2 f(t T)us(t T) F(s)esT T 0 Time delay 3 f(at) 1 a F( s a) a>0 Time scaling 4 eatf(t) F(s+a) Shift in frequency 5 df (t) dt sF(s) f(0) First-order dierentiation 6 d2f(t) dt2 s2F(s) sf(0) f(1)(0) Second-order. For part c, do not use # 11 in Table 2.2.1.
For part b, do not use # 29 in Table 2.2.1. Specify which transform pair or property is used and write in the simplest form. Transcribed image text: Use the table of Laplace transforms and properties to obtain the Laplace transform of the following functions.