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拉普拉斯变换简表

Reflect314数学公式
f(t)f(t)F(s)F(s)
111s\frac{1}{s}
eate^{at}1s−a\frac{1}{s-a}
tm (m>−1)t^{m} \ (m>-1)Γ(m+1)sm+1\frac{\Gamma (m+1)}{s^{m+1}}
tmeat (m>−1)t^{m}e^{at} \ (m>-1)Γ(m+1)(s−a)m+1\frac{\Gamma (m+1)}{(s-a)^{m+1}}
sin⁡at\sin atas2+a2\frac{a}{s^2+a^2}
cos⁡at\cos atss2+a2\frac{s}{s^2+a^2}
sinh⁡at\sinh atas2−a2\frac{a}{s^2-a^2}
cosh⁡at\cosh atss2−a2\frac{s}{s^2-a^2}
tsin⁡att\sin at2as(s2+a2)2\frac{2as}{(s^2+a^2)^2}
tcos⁡att\cos ats2−a2(s2+a2)2\frac{s^2-a^2}{(s^2+a^2)^2}
tsinh⁡att\sinh at2as(s2−a2)2\frac{2as}{(s^2-a^2)^2}
tcosh⁡att\cosh ats2+a2(s2−a2)2\frac{s^2+a^2}{(s^2-a^2)^2}
tmsin⁡at (m>−1)t^{m}\sin at \ (m>-1)Γ(m+1)2j(s2+a2)m+1[(s+ja)m+1−(s−ja)m+1]\frac{\Gamma (m+1)}{2\mathrm{j} (s^2+a^2)^{m+1}}[(s+\mathrm{j}a)^{m+1}-(s-\mathrm{j}a)^{m+1}]
tmcos⁡at (m>−1)t^{m}\cos at \ (m>-1)Γ(m+1)2(s2+a2)m+1[(s+ja)m+1+(s−ja)m+1]\frac{\Gamma (m+1)}{2 (s^{2}+a^{2})^{m+1}}[(s+\mathrm{j}a)^{m+1}+(s-\mathrm{j}a)^{m+1}]
e−btsin⁡ate^{-bt}\sin ata(s+b)2+a2\frac{a}{(s+b)^{2}+a^{2}}
e−btcos⁡ate^{-bt}\cos ats+b(s+b)2+a2\frac{s+b}{(s+b)^{2}+a^{2}}
e−btsin⁡(at+c)e^{-bt}\sin (at+c)(s+b)sin⁡c+acos⁡c(s+b)2+a2\frac{(s+b)\sin c+a\cos c}{(s+b)^{2}+a^{2}}
sin⁡2t\sin^{2}t12(1s−ss2+4)\frac{1}{2}\left (\frac{1}{s}-\frac{s}{s^{2}+4}\right)
cos⁡2t\cos^{2}t12(1s+ss2+4)\frac{1}{2}\left (\frac{1}{s}+\frac{s}{s^{2}+4}\right)
sin⁡atsin⁡bt\sin at \sin bt2abs[s2+(a+b)2][s2+(a−b)2]\frac{2abs}{[s^{2}+(a+b)^{2}][s^{2}+(a-b)^{2}]}
eat−ebte^{at}-e^{bt}a−b(s−a)(s−b)\frac{a-b}{(s-a)(s-b)}
aeat−bebtae^{at}-be^{bt}(a−b)s(s−a)(s−b)\frac{(a-b) s}{(s-a)(s-b)}
1asin⁡at−1bsin⁡bt\frac{1}{a}\sin at-\frac{1}{b}\sin btb2−a2(s2+a2)(s2+b2)\frac{b^{2}-a^{2}}{(s^{2}+a^{2})(s^{2}+b^{2})}
cos⁡at−cos⁡bt\cos at-\cos bt(b2−a2)s(s2+a2)(s2+b2)\frac{(b^{2}-a^{2}) s}{(s^{2}+a^{2})(s^{2}+b^{2})}
1a2(1−cos⁡at)\frac{1}{a^{2}}(1-\cos at)1s(s2+a2)\frac{1}{s (s^{2}+a^{2})}
1a3(at−sin⁡at)\frac{1}{a^{3}}(at-\sin at)1s2(s2+a2)\frac{1}{s^{2}(s^{2}+a^{2})}
1a4(cos⁡at−1)+12a2t2\frac{1}{a^{4}}(\cos at-1)+\frac{1}{2a^{2}}t^{2}1s3(s2+a2)\frac{1}{s^{3}(s^{2}+a^{2})}
1a4(cosh⁡at−1)−12a2t2\frac{1}{a^4}(\cosh at-1)-\frac{1}{2a^2}t^21s3(s2−a2)\frac{1}{s^3 (s^2-a^2)}
12a3(sin⁡at−atcos⁡at)\frac{1}{2a^3}(\sin at-at \cos at)1(s2+a2)2\frac{1}{(s^2+a^2)^2}
12a(sin⁡at+atcos⁡at)\frac{1}{2a}(\sin at+at \cos at)s2(s2+a2)2\frac{s^2}{(s^2+a^2)^2}
1a4(1−cos⁡at)−12a3tsin⁡at\frac{1}{a^4}(1-\cos at)-\frac{1}{2a^3}t \sin at1s(s2+a2)2\frac{1}{s (s^2+a^2)^2}
(1−at)e−at(1-at) e^{-at}s(s+a)2\frac{s}{(s+a)^2}
t(1−a2t)e−att\left (1-\frac{a}{2}t\right) e^{-at}s(s+a)3\frac{s}{(s+a)^3}
1a(1−e−at)\frac{1}{a}(1-e^{-at})1s(s+a)\frac{1}{s (s+a)}
1ab+1b−a(e−btb−e−ata)\frac{1}{ab}+\frac{1}{b-a}\left (\frac{e^{-bt}}{b}-\frac{e^{-at}}{a}\right)1s(s+a)(s+b)\frac{1}{s (s+a)(s+b)}
e−at−eat2(cos⁡3at2−3sin⁡3at2)e^{-at}-e^{\frac{at}{2}}\left (\cos\frac{\sqrt{3}at}{2}-\sqrt{3}\sin\frac{\sqrt{3}at}{2}\right)3a2s3+a3\frac{3a^2}{s^3+a^3}
sin⁡atcosh⁡at−cos⁡atsinh⁡at\sin at \cosh at-\cos at \sinh at4a3s4+4a4\frac{4a^3}{s^4+4a^4}
12a2sin⁡atsinh⁡at\frac{1}{2a^2}\sin at \sinh atss4+4a4\frac{s}{s^4+4a^4}
12a3(sinh⁡at−sin⁡at)\frac{1}{2a^3}(\sinh at-\sin at)1s4−a4\frac{1}{s^4-a^4}
12a2(cosh⁡at−cos⁡at)\frac{1}{2a^2}(\cosh at-\cos at)ss4−a4\frac{s}{s^4-a^4}
1πt\frac{1}{\sqrt{\pi t}}1s\frac{1}{\sqrt{s}}
2tπ2\sqrt{\frac{t}{\pi}}1ss\frac{1}{s\sqrt{s}}
1πteat(1+2at)\frac{1}{\sqrt{\pi t}}e^{at}(1+2at)s(s−a)s−a\frac{s}{(s-a)\sqrt{s-a}}
12πt3(ebt−eat)\frac{1}{2\sqrt{\pi t^3}}(e^{bt}-e^{at})s−a−s−b\sqrt{s-a}-\sqrt{s-b}
1πtcos⁡2at\frac{1}{\sqrt{\pi t}}\cos 2\sqrt{at}1se−as\frac{1}{\sqrt{s}}e^{-\frac{a}{s}}
1πtcosh⁡2at\frac{1}{\sqrt{\pi t}}\cosh 2\sqrt{at}1seas\frac{1}{\sqrt{s}}e^{\frac{a}{s}}
1πtsin⁡2at\frac{1}{\sqrt{\pi t}}\sin 2\sqrt{at}1sse−as\frac{1}{s\sqrt{s}}e^{-\frac{a}{s}}
1πtsinh⁡2at\frac{1}{\sqrt{\pi t}}\sinh 2\sqrt{at}1sseas\frac{1}{s\sqrt{s}}e^{\frac{a}{s}}
1t(ebt−eat)\frac{1}{t}(e^{bt}-e^{at})ln⁡s−as−b\ln \frac{s-a}{s-b}
2tsinh⁡at\frac{2}{t}\sinh atln⁡s+as−a=2Artanh⁡as\ln \frac{s+a}{s-a}=2\operatorname{Artanh}\frac{a}{s}
2t(1−cos⁡at)\frac{2}{t}(1-\cos at)ln⁡s2+a2s2\ln \frac{s^2+a^2}{s^2}
2t(1−cosh⁡at)\frac{2}{t}(1-\cosh at)ln⁡s2−a2s2\ln \frac{s^2-a^2}{s^2}
1tsin⁡at\frac{1}{t}\sin atarctan⁡as\arctan \frac{a}{s}
1t(cosh⁡at−cos⁡bt)\frac{1}{t}(\cosh at-\cos bt)ln⁡s2+b2s2−a2\ln \sqrt{\frac{s^2+b^2}{s^2-a^2}}
1πtsin⁡(2at)\frac{1}{\pi t}\sin (2a\sqrt{t})erf⁡(as)\operatorname{erf}\left (\frac{a}{\sqrt{s}}\right)
1πte−2at\frac{1}{\sqrt{\pi t}}e^{-2a\sqrt{t}}1sea2serfc⁡(as)\frac{1}{\sqrt{s}}e^{\frac{a^{2}}{s}}\operatorname{erfc}\left (\frac{a}{\sqrt{s}}\right)
erfc⁡(a2t)\operatorname{erfc}\left (\frac{a}{2\sqrt{t}}\right)1se−as\frac{1}{s}e^{-a\sqrt{s}}
erf⁡(t2a)\operatorname{erf}\left (\frac{t}{2a}\right)1seaˉ2s2erfc⁡(as)\frac{1}{s}e^{\bar{a}^{2}s^{2}}\operatorname{erfc}(as)
1πte−2at\frac{1}{\sqrt{\pi t}}e^{-2\sqrt{at}}1seaserfc⁡(as)\frac{1}{\sqrt{s}}e^{\frac{a}{s}}\operatorname{erfc}\left (\sqrt{\frac{a}{s}}\right)
1π(t+a)\frac{1}{\sqrt{\pi (t+a)}}1seaserfc⁡(as)\frac{1}{\sqrt{s}}e^{as}\operatorname{erfc}(\sqrt{as})
1aerf⁡(at)\frac{1}{\sqrt{a}}\operatorname{erf}(\sqrt{at})1ss+a\frac{1}{s\sqrt{s+a}}
1aeaterf⁡(at)\frac{1}{\sqrt{a}}e^{at}\operatorname{erf}(\sqrt{at})1s(s−a)\frac{1}{\sqrt{s}(s-a)}
u(t)u (t)1s\frac{1}{s}
tu(t)tu (t)1s2\frac{1}{s^{2}}
tmu(t)(m>−1)t^{m}u (t)\quad (m>-1)1sm+1Γ(m+1)\frac{1}{s^{m+1}}\Gamma (m+1)
δ(t)\delta (t)11
δ(n)(t)\delta^{(n)}(t)sns^{n}
sgn⁡t\operatorname{sgn}t1s\frac{1}{s}
J0(at)\mathrm{J}_{0}(at)1s2+a2\frac{1}{\sqrt{s^{2}+a^{2}}}
I0(at)\mathrm{I}_0 (at)1s2−a2\frac{1}{\sqrt{s^2 - a^2}}
J0(2at)\mathrm{J}_0 (2\sqrt{at})1se−as\frac{1}{s}e^{-\frac{a}{s}}
e−btI0(at)e^{-b t}\mathrm{I}_0 (at)1(s+b)2−a2\frac{1}{\sqrt{(s + b)^2 - a^2}}
tJ0(at)t\mathrm{J}_0 (at)s(s2+a2)3/2\frac{s}{(s^2 + a^2)^{3/2}}
tI0(at)t\mathrm{I}_0 (at)s(s2−a2)3/2\frac{s}{(s^2 - a^2)^{3/2}}
J0(at(t+2b))\mathrm{J}_0 (a\sqrt{t (t + 2b)})1s2+a2eb(ss2+a2)\frac{1}{\sqrt{s^2 + a^2}}e^{b (s\sqrt{s^2 + a^2})}
  1. erf⁡(x)=2π∫0xe−t2dt\operatorname{erf}(x) = \frac{2}{\sqrt{\pi}}\int_{0}^{x}\mathrm{e}^{-t^2}\mathrm{d}t ，称为误差函数； erfc⁡(x)=1−erf⁡(x)=2π∫x+∞e−t2dt,\operatorname{erfc}(x) = 1 - \operatorname{erf}(x) = \frac{2}{\sqrt{\pi}}\int_{x}^{+\infty}\mathrm{e}^{-t^2}\mathrm{d}t, 称为余误差函数.
  2. In(x)=j−nJn(jx)I_{n}(x)=\mathrm{j}^{-n}J_{n}(\mathrm{j}x) . JnJ_{n} 称为第一类 n 阶贝塞尔 (Bessel) 函数; InI_{n} 称为第一类 n 阶变形的贝塞尔函数, 或称为虚变量的贝塞尔函数.
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