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Transformadas de Laplace

Fecha de publicación: 18/10/2026

f(s)f(s)
F(t′)F(t’)
ks\frac{k}{s}
k con k=constantek \text{ con }k= \text{constante}
1s2\frac{1}{s^{2}}
tt
1sn\frac{1}{s^{n}}
tn−1(n−1)! con n=0,1,2…\frac{t^{n-1}}{(n-1)!}\text{ con }n=0,1,2\dots
1sn\frac{1}{s^{n}}
tn−1Γ(n) con n>0\frac{t^{n-1}}{\Gamma(n)}\text{ con }n>0
1s−a\frac{1}{s-a}
eate^{at}
1(s−a)n\frac{1}{(s-a)^{n}}
tn−1eat(n−1)! con n=0,1,2,…\frac{t^{n-1}e^{at}}{(n-1)!}\text{ con }n=0,1,2,\dots
1(s−a)n\frac{1}{(s-a)^{n}}
tn−1eatΓ(n) con n=>0\frac{t^{n-1}e^{at}}{\Gamma(n)}\text{ con }n=>0
1s2+a2\frac{1}{s^{2}+a^{2}}
sen(at)a\frac{sen(at)}{a}
ss2+a2\frac{s}{s^{2}+a^{2}}
cos(at)a\frac{cos(at)}{a}
1(s−b)2+a2\frac{1}{(s-b)^{2}+a^{2}}
ebtsen(at)a\frac{e^{bt}sen(at)}{a}
s−b(s−b)2+a2\frac{s-b}{(s-b)^{2}+a^{2}}
ebtcos(at)e^{bt}cos(at)
1s2−a2\frac{1}{s^{2}-a^{2}}
senh(at)a\frac{senh(at)}{a}
ss2−a2\frac{s}{s^{2}-a^{2}}
cosh(at)cosh(at)
1(s−b)2−a2\frac{1}{(s-b)^{2}-a^{2}}
ebtsenh(at)a\frac{e^{bt}senh(at)}{a}
s−b(s−b)2−a2\frac{s-b}{(s-b)^{2}-a^{2}}
ebtcosh(at)e^{bt}cosh(at)
1(s−a)(s−b)\frac{1}{(s-a)(s-b)}
ebt−eatb−a con a≠b\frac{e^{bt}-e^{at}}{b-a} \text{ con }a\neq b
s(s−a)(s−b)\frac{s}{(s-a)(s-b)}
bebt−aeatb−a con a≠b\frac{be^{bt}-ae^{at}}{b-a} \text{ con }a\neq b
1(s2+a2)2\frac{1}{(s^{2}+a^{2})^{2}}
sen(at)−at cos(at)2a3\frac{sen(at)-at\text{ }cos(at)}{2a^{3}}
s(s2+a2)2\frac{s}{(s^{2}+a^{2})^{2}}
tsen(at)2a\frac{tsen(at)}{2a}
s2(s2+a2)2\frac{s^{2}}{(s^{2}+a^{2})^{2}}
sen(at)+at cos(at)2a\frac{sen(at)+at\text{ }cos(at)}{2a}
s3(s2+a2)2\frac{s^{3}}{(s^{2}+a^{2})^{2}}
cos(at)−12at sen(at)cos(at)-\frac{1}{2}at\text{ }sen(at)
s2−a2(s2+a2)2\frac{s^{2}-a^{2}}{(s^{2}+a^{2})^{2}}
tcos(at)tcos(at)
1(s2−a2)2\frac{1}{(s^{2}-a^{2})^{2}}
at cosh(at)−senh(at)2a3\frac{at\text{ }cosh(at)-senh(at)}{2a^{3}}
s(s2−a2)2\frac{s}{(s^{2}-a^{2})^{2}}
tsenh(at)2a\frac{tsenh(at)}{2a}
s2(s2−a2)2\frac{s^{2}}{(s^{2}-a^{2})^{2}}
senh(at)+at cosh(at)2a\frac{senh(at)+at\text{ }cosh(at)}{2a}
s3(s2−a2)2\frac{s^{3}}{(s^{2}-a^{2})^{2}}
cosh(at)+12at senh(at)cosh(at)+\frac{1}{2}at\text{ }senh(at)
s2+a2(s2−a2)2\frac{s^{2}+a^{2}}{(s^{2}-a^{2})^{2}}
tcosh(at)tcosh(at)
1(s2+a2)3\frac{1}{(s^{2}+a^{2})^{3}}
(3−a2t2)sen(at)−3at cos(at)8a5\frac{(3-a^{2}t^{2})sen(at)-3at\text{ }cos(at)}{8a^{5}}
s(s2+a2)3\frac{s}{(s^{2}+a^{2})^{3}}
tsen(at)−at2cos(at)8a3\frac{tsen(at)-at^{2}cos(at)}{8a^{3}}
s2(s2+a2)3\frac{s^{2}}{(s^{2}+a^{2})^{3}}
(1+a2t2)sen(at)−at cos(at)8a3\frac{(1+a^{2}t^{2})sen(at)-at\text{ }cos(at)}{8a^{3}}
s3(s2+a2)3\frac{s^{3}}{(s^{2}+a^{2})^{3}}
3tsen(at)+at2cos(at)8a\frac{3tsen(at)+at^{2}cos(at)}{8a}
s4(s2+a2)a3\frac{s^{4}}{(s^{2}+a^{2})a^{3}}
(3−a2t2)sen(at)+5at cos(at)8a\frac{(3-a^{2}t^{2})sen(at)+5at\text{ }cos(at)}{8a}
s5(s2+a2)3\frac{s^{5}}{(s^{2}+a^{2})^{3}}
(8−a2t2)cos(at)−7at sen(at)8\frac{(8-a^{2}t^{2})cos(at)-7at\text{ }sen(at)}{8}
3s2−3a2s(s2+a2)4\frac{3s^{2}-3a^{2}s}{(s^{2}+a^{2})^{4}}
12t2cos(at)\frac{1}{2}t^{2}cos(at)
s4−6a2s+a4(s2+a2)4\frac{s^{4}-6a^{2}s+a^{4}}{(s^{2}+a^{2})^{4}}
16t3cos(at)\frac{1}{6}t^{3}cos(at)
s3−a2s(s2+a2)4\frac{s^{3}-a^{2}s}{(s^{2}+a^{2})^{4}}
t3sen(at)24a\frac{t^{3}sen(at)}{24a}
1(s2−a2)3\frac{1}{(s^{2}-a^{2})^{3}}
(3+a2t2)senh(at)−3at cosh(at)8a5\frac{(3+a^{2}t^{2})senh(at)-3at\text{ }cosh(at)}{8a^{5}}
s(s2−a2)3\frac{s}{(s^{2}-a^{2})^{3}}
at2cosh(at)−tsenh(at)8a3\frac{at^{2}cosh(at)-tsenh(at)}{8a^{3}}
s2(s2−a2)3\frac{s^{2}}{(s^{2}-a^{2})^{3}}
at cosh(at)+(a2t2−1)senh(at)8a3\frac{at\text{ }cosh(at)+(a^{2}t^{2}-1)senh(at)}{8a^{3}}
ss3+a3\frac{s}{s^{3}+a^{3}}
eat23a(cos(3at2)+3sen(3at2)−e−3at2)\frac{e^{\frac{at}{2}}}{3a}\left(cos\left(\frac{\sqrt{3at}}{2} \right)+\sqrt{3}sen \left(\frac{\sqrt{3at}}{2}\right)-e^{-\frac{3at}{2}} \right)
s2s3+a3\frac{s^{2}}{s^{3}+a^{3}}
13(e−at+2eat2cos(3at2))\frac{1}{3} \left(e^{-at}+2e^{\frac{at}{2}}cos\left(\frac{\sqrt{3at}}{2} \right) \right)
1s3−a3\frac{1}{s^{3}-a^{3}}
e−at23a(e3at2−cos(3at2)−3sen(3at2))\frac{e^{-\frac{at}{2}}}{3a}\left(e^{\frac{3at}{2}}-cos\left(\frac{\sqrt{3at}}{2} \right)-\sqrt{3}sen\left(\frac{\sqrt{3at}}{2} \right) \right)
ss3−a3\frac{s}{s^{3}-a^{3}}
e−at23a(3sen(3at2)−cos(3at2)+e3at2)\frac{e^{-\frac{at}{2}}}{3a}\left(\sqrt{3}sen\left(\frac{\sqrt{3at}}{2} \right)-cos\left(\frac{\sqrt{3at}}{2} \right)+e^{\frac{3at}{2}} \right)
s2s3−a3\frac{s^{2}}{s^{3}-a^{3}}
13(eat+2e−at2cos(3at2))\frac{1}{3}\left(e^{at}+2e^{-\frac{at}{2}}cos\left(\frac{\sqrt{3at}}{2} \right) \right)
1s4+4a4\frac{1}{s^{4}+4a^{4}}
14a3(sen(at)cosh(at)−cos(at)senh(at))\frac{1}{4a^{3}}\left(sen(at)cosh(at)-cos(at)senh(at)\right)
ss4+4a4\frac{s}{s^{4}+4a^{4}}
sen(at)senh(at)2a2\frac{sen(at)senh(at)}{2a^{2}}
s3s4+4a4\frac{s^{3}}{s^{4}+4a^{4}}
cos(at)cosh(at)cos(at)cosh(at)
1s4−a4\frac{1}{s^{4}-a^{4}}
12a3(senh(at)−sen(at))\frac{1}{2a^{3}} \left(senh(at)-sen(at) \right)
ss4−a4\frac{s}{s^{4}-a^{4}}
12a2(cosh(at)−cos(at))\frac{1}{2a^{2}}\left(cosh(at)-cos(at) \right)
s2s4−a4\frac{s^{2}}{s^{4}-a^{4}}
12a(cosh(at)−cos(at))\frac{1}{2a}\left(cosh(at)-cos(at) \right)
s3s4−a4\frac{s^{3}}{s^{4}-a^{4}}
12(cosh(at)+cos(at))\frac{1}{2} \left(cosh(at)+cos(at) \right)
1s+a+s+b\frac{1}{\sqrt{s+a}+\sqrt{s+b}}
e−bt−e−at2(b−a)πt3\frac{e^{-bt}-e^{-at}}{2(b-a)\sqrt{\pi t^{3}}}
1ss+a\frac{1}{s\sqrt{s+a}}
erf(at)a\frac{\text{erf}(\sqrt{at})}{\sqrt{a}}
1s(s−a)\frac{1}{\sqrt{s}(s-a)}
eaterf(at)a\frac{e^{at}\text{erf}(\sqrt{at})}{\sqrt{a}}
1s−a+b\frac{1}{\sqrt{s-a}+b}
eat(1πt−beb2terfc(bt))e^{at}\left(\frac{1}{\sqrt{\pi t}}-be^{b^{2}t}\text{erfc}(b\sqrt{t}) \right)
1s2+a2\frac{1}{\sqrt{s^{2}+a^{2}}}
J0(at)J_{0}(at)
1s2−a2\frac{1}{\sqrt{s^{2}-a^{2}}}
I0(at)I_{0}(at)

Nota: La función «erf()\text{erf()}» y «erfc()\text{erfc()}» son la función error y la función de error complementaria. En otros sitios como el formulario de donde procede esta página se puede encontrar como «fer()\text{fer()}» y «fcer()\text{fcer()}«.