1拉氏变换的定义若时间函数f(t)在t>0有定义，则f(t)的拉普拉斯变换(简称拉氏变换)为L[f(t)]=F(s)=Jsf(t)-e-tsdtjF(s)像0f(t)原像2拉普拉斯反变换f(t)=二产小F(s)estds，可
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示为：f(t)心F(s)]2no-js1.表A-1拉氏变换的基本性质1线性定理齐次性L[af(t)]=aF(s)叠加性L[f(t)土f(t)]=F(s)土F(s)1212df(t)L[]=sF(s)-f(0)dtL[d；f(t)]=s2F(s)-sf(0)-f'(0)dt22微分定理一般形式L[dnf(t)]=s”F(s)-Xsn-kf(k-1)(0)dt”k=1f(t)dk-1/(t)f(k-1)(t)=7dtk-1初始条件为0时d”f(t)L[—=s”F(s)dt”L[Jf(t)dt]=加+L[fff(t)(力)2]F(s)+[Jf(t)dtLo+JJf(t)(dt)2Lo一般形式L[JJf(/)(dt)2]—+t0+t0s252s3积分定理L[f•jf(t)(dt)n]—d+X1[f(t)(dt)n]s”s”一k+1t—0k1初始条件为0时L[t'Jf(t)(dt)”]—皿Sn4延迟定理(或称t域平移定理)L[f(t-T)1(t-T)]—e-tsF(s)5衰减定理(或称s域平移定理)L[f(t)e-at]—F(s+a)6终值定理limf(t)—limsF(s)tTssT07初值定理limf(t)—limsF(s)tT0sTs8卷积定理L[Jf(t-T)f(T)dT]—L[Jf(t)f(t-T)dT]—F(s)F(s)012012122.表A-2常用函数的拉氏变换和z变换表序号时间函数e(t)拉氏变换E(s)Z变换E(z)<5(t)8(t-kT)6(t)=为6(t-nT)Tn二01—e-Ts1(t)Tz(z-1)2T2z(z+1)2(z-1)3tnn!Sn+1恤血竺—)a—on!Qa”z—e-aTz-e-aTte—atTze—aT(s+a)2(z-e-aT)2101—e—at(1—e-aT)z(z—1)(z—e-aT)11e—at—e—bt(s+a)(s+b)z—e—aTz—e—bT12sinwts2+w2zsinwTz2-2zcoswT+113coswtz(z-coswT)z2-2zcoswT+114e—atsinwtze-aTsinwT(s+a)2+w2z2—2ze-aTcoswT+e—2aT15e-atcoswts+a(s+a)2+w2z2—ze-aTcoswTz2—2ze-aTcoswT+e-2aTs—(1/T)lna
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