拉普拉斯變換公式

1、附錄A1.表A-1拉氏變換的基本性質(zhì)1齊次性線性定理疊加性2微分定理一般形式初始條件為0時(shí)一般形式積分定理初始條件為0時(shí)延緩定理（或稱t域平移定理）衰減定理（或稱s域平移定理）終值定理初值定理卷積定理拉普拉斯變換及反變換Laf(t)aF(s)Lf1(t)f2(t)F1(s)F2(s)Ldf(t)sF(s)f(0)dtLd2f(t)s2F(s)sf(0)f（）dt20Ldnf(t)nsnF(s)snkf(k1)(0)dtnk1f(k1)(t)dk1f(t)dtk1Ldnf(t)snF(s)dtnLf(t)dtF(s)f(t)dtt0ss2F(s)f(t)dtt0f(t)(dt)2t0Lf(t)(

2、dt)s2s2s共n個(gè)n共n個(gè)nF(s)1nLf(t)(dt)1f(t)(dt)t0snk1snk共n個(gè)Lf(t)(dt)nF(s)snLf(tT)1(tT)eTsF(s)Lf(t)eatF(sa)limf(t)limsF(s)ts0limf(t)limsF(s)t0sttL0f1(t)f2()dL0f1(t)f2(t)dF1(s)F2(s)2表A-2常用函數(shù)的拉氏變換和z變換表序拉氏變換E(s)時(shí)間函數(shù)e(t)號(hào)Z變換E(z)1234567891011121311eTs1s12s13s1sn11sa1(sa)2as(sa)ba(sa)(sb)s22ss22(sa)22sa(t)T(t)(tn

3、T)n01(t)tt22tnn!eatteateateatebtsintcosteatsintat1zz1zz1Tz(z1)2T2z(z1)2(z1)3lim(1)nnzn(aT)a0n!azezaTzeTzeaT(zeaT)2(1eaT)z(z1)(zaT)ezzzeaTzebTzsinTz22zcosT1z2z(zcosT)2zcosT1zeaTsinTz22zeaTcosTe2aTz2zeaTcosT14(sa)22ecostz22zeaTcosTe2aT151s(1/T)lnaat/Tzza3用查表法進(jìn)行拉氏反變換用查表法進(jìn)行拉氏反變換的要點(diǎn)在于將變換式進(jìn)行部分分式張開，此后逐項(xiàng)查表進(jìn)

4、行反變換。設(shè)F(s)是s的有理真分式B(s)bmsmbm1sm1b1sb0（nm）F(s)ansnan1sn1a1sa0A(s)式中系數(shù)a0,a1,.,an1,an，b0,b1,bm1,bm都是實(shí)常數(shù)；m,n是正整數(shù)。按代數(shù)定理可將F(s)張開為部分分式。分以下兩種情況談?wù)?。A(s)0無重根這時(shí)，F(xiàn)(s)可張開為n個(gè)簡(jiǎn)單的部分分式之和的形式。c1c2cinciF(s)cn（F-1）ss1ss2ssissni1ssi式中，s1,s2,sn是特色方程A(s)0的根。ci為待定常數(shù)，稱為F(s)在si處的留數(shù)，可按下式計(jì)算：cilim(ssi)F(s)（F-2）ssi或ciB(s)A(s)ssi（F

5、-3）式中，A(s)為A(s)對(duì)s的一階導(dǎo)數(shù)。依照拉氏變換的性質(zhì)，從式（F-1）可求得原函數(shù)f(t)L1F(s)L1ncinciesit(F-4)i1ssii1A(s)0有重根設(shè)A(s)0有r重根s1，F(xiàn)(s)可寫為FsB(s)(ss1)r(ssr1)(ssn)=crcr1c1cr1cicn(ss1)r1(ss1)ssr1ssissn(ss1)r式中，s1為F(s)的r重根，sr1，,sn為F(s)的n-r個(gè)單根；其中，cr1，,cn仍按式(F-2)或(F-3)計(jì)算，cr，cr1，,c1則按下式計(jì)算：crcr1crjc1lim(ss1)rF(s)ss1limd(ss1)rdsss11d(j)lim(j)(sj!ss1ds1limd(r1)(r1)!ss1ds(r1)F(s)r(F-5)s1)F(s)(ss1)rF(s)原函數(shù)f(t)為f(t)L1F(s)L1crcr