《电路》（英文版）15-4 Laplace transform of some simple time function

8 15-4 Laplace transform of some simple time function 1. The unit step function () 1t>0 u(t) 0t<0 0 L{u()} e-stu(t)dt ∫ oO st SO

§15-4 Laplace transform of some simple time function 1. The unit step function   − = 0 L{u(t)} e u(t)dt st      = 0 0 1 0 ( ) t t u t s u t 1 ( )  u(t) 1 0 t s s e st 1 0 =  − = −   − = 0 e dt st

2. The unit impulse function 6(t-t0)=0t≠t +0 and 8(t-to)dt=1 or 8(t-to )dt= fo L{a(-o)2=0c-3(-10)=e-3 δ(t-t0)e Jto=0()分1

2. The unit impulse function     − = − = − =    + − + − 0 0 ( ) 1 ( ) 1 ( ) 0 0 0 0 0 t t and t t dt or t t dt t t t t      0 0 0 0 ( ) ( ) st st L t t e t t dt e  − −   − =  − = 0 ( ) 0 s t t t e  −  − If t 0 = 0  (t) 1 t ( ) 0  t − t 0 1 0 t

3. The exponential function (a+s)t Le-au( d=j -la+s)t (a+s)0 s+a 1 s-+Oc 4. The ramp function tu(t {u(t)} e at 2 tl(1)<>-2

3. The exponential function    + =  − + = − + s s e s t 1 ( ) 0 ( )   + −  s e t 1 4. The ramp function t u(t)   2 0 1 ( ) s L t u t t e dt st   − = = 2 1 ( ) s tu t  Le u(t) −t   − − = 0 e e dt st  t   − + = 0 ( ) e dt  s t