User:VmZH88AZQnCjhT40/Laplace Transforms for Print

Function	Time domain $f(t)={\mathcal {L}}^{-1}\left\{F(s)\right\}$	Laplace s-domain $F(s)={\mathcal {L}}\left\{f(t)\right\}$	Region of convergence	Reference
unit impulse	$\delta (t)\$	$1$	$\mathrm {all} \ s\,$	inspection
delayed impulse	$\delta (t-\tau )\$	$e^{-\tau s}\$		time shift of unit impulse
unit step	$u(t)\$	${1 \over s}$	${\textrm {Re}}\{s\}>0\,$	integrate unit impulse
delayed unit step	$u(t-\tau )\$	${e^{-\tau s} \over s}$	${\textrm {Re}}\{s\}>0\,$	time shift of unit step
ramp	$t\cdot u(t)\$	${\frac {1}{s^{2}}}$	${\textrm {Re}}\{s\}>0\,$	integrate unit impulse twice
delayed nth power with frequency shift	${\frac {(t-\tau )^{n}}{n!}}e^{-\alpha (t-\tau )}\cdot u(t-\tau )$	${\frac {e^{-\tau s}}{(s+\alpha )^{n+1}}}$	${\textrm {Re}}\{s\}>-\alpha \,$	Integrate unit step, apply frequency shift, apply time shift
nth power ( for integer n )	${t^{n} \over n!}\cdot u(t)$	${1 \over s^{n+1}}$	${\textrm {Re}}\{s\}>0\,$ $(n>-1)\,$	Integrate unit step n times
qth power ( for complex q )	${t^{q} \over \Gamma (q+1)}\cdot u(t)$	${1 \over s^{q+1}}$	${\textrm {Re}}\{s\}>0\,$ $({\textrm {Re}}\{q\}>-1)\,$	ref?
nth power with frequency shift	${\frac {t^{n}}{n!}}e^{-\alpha t}\cdot u(t)$	${\frac {1}{(s+\alpha )^{n+1}}}$	${\textrm {Re}}\{s\}>-\alpha \,$	Integrate unit step, apply frequency shift
exponential decay	$e^{-\alpha t}\cdot u(t)\$	${1 \over s+\alpha }$	${\textrm {Re}}\{s\}>-\alpha \$	Frequency shift of unit step
exponential approach	$(1-e^{-\alpha t})\cdot u(t)\$	${\frac {\alpha }{s(s+\alpha )}}$	${\textrm {Re}}\{s\}>0\$	Unit step minus exponential decay
sine	$\sin(\omega t)\cdot u(t)\$	${\omega \over s^{2}+\omega ^{2}}$	${\textrm {Re}}\{s\}>0\$	ref?
cosine	$\cos(\omega t)\cdot u(t)\$	${s \over s^{2}+\omega ^{2}}$	${\textrm {Re}}\{s\}>0\$	ref?
hyperbolic sine	$\sinh(\alpha t)\cdot u(t)\$	${\alpha \over s^{2}-\alpha ^{2}}$	${\textrm {Re}}\{s\}>|\alpha |\$	ref?
hyperbolic cosine	$\cosh(\alpha t)\cdot u(t)\$	${s \over s^{2}-\alpha ^{2}}$	${\textrm {Re}}\{s\}>|\alpha |\$	ref?
Exponentially-decaying sine wave	$e^{-\alpha t}\sin(\omega t)\cdot u(t)\$	${\omega \over (s+\alpha )^{2}+\omega ^{2}}$	${\textrm {Re}}\{s\}>-\alpha \$	ref?
Exponentially-decaying cosine wave	$e^{-\alpha t}\cos(\omega t)\cdot u(t)\$	${s+\alpha \over (s+\alpha )^{2}+\omega ^{2}}$	${\textrm {Re}}\{s\}>-\alpha \$	ref?
nth root	${\sqrt[{n}]{t}}\cdot u(t)$	$s^{-(n+1)/n}\cdot \Gamma \left(1+{\frac {1}{n}}\right)$	${\textrm {Re}}\{s\}>0\,$	ref?
natural logarithm	$\ln \left({t \over t_{0}}\right)\cdot u(t)$	$-{t_{0} \over s}\,\left[\ln(t_{0}s)+\gamma \right]$	${\textrm {Re}}\{s\}>0\,$	ref?
Bessel function of the first kind, of order n	$J_{n}(\omega t)\cdot u(t)$	${\frac {\omega ^{n}\left(s+{\sqrt {s^{2}+\omega ^{2}}}\right)^{-n}}{\sqrt {s^{2}+\omega ^{2}}}}$	${\textrm {Re}}\{s\}>0\,$ $(n>-1)\,$	ref?
Modified Bessel function of the first kind, of order n	$I_{n}(\omega t)\cdot u(t)$	${\frac {\omega ^{n}\left(s+{\sqrt {s^{2}-\omega ^{2}}}\right)^{-n}}{\sqrt {s^{2}-\omega ^{2}}}}$	${\textrm {Re}}\{s\}>|\omega |\,$	ref?
Bessel function of the second kind, of order 0	$Y_{0}(\alpha t)\cdot u(t)$	$-{2\sinh ^{-1}(s/\alpha ) \over \pi {\sqrt {s^{2}+\alpha ^{2}}}}$	${\textrm {Re}}\{s\}>0\,$	ref?
Modified Bessel function of the second kind, of order 0	$K_{0}(\alpha t)\cdot u(t)$			ref?
Confluent hypergeometric function	$e^{-t^{2}}t^{2c-2}M(a,c,t^{2})$	$2^{1-2c}\Gamma (2c-1)U\left(c-{\frac {1}{2}},a+{\frac {1}{2}},{\frac {s^{2}}{4}}\right)$	${\textrm {Re}}\,c>{\frac {1}{2}},\,{\textrm {Re}}\,s>0$	ref?
Incomplete Gamma function	$e^{-t^{2}}{\frac {\gamma (\alpha ,-t^{2})}{(-1)^{\alpha }}}$	${\frac {\Gamma (2\alpha )}{4^{\alpha }}}e^{\frac {s^{2}}{4}}\Gamma \left({\frac {1}{2}}-\alpha ,{\frac {s^{2}}{4}}\right)$	${\textrm {Re}}\,c>{\frac {1}{2}},\,{\textrm {Re}}\,s>0$	ref?
Error function	$\mathrm {erf} (t)\cdot u(t)$	${e^{s^{2}/4}\left(1-\operatorname {erf} \left(s/2\right)\right) \over s}$	${\textrm {Re}}\{s\}>0\,$	ref?
Rational function in s	$\sum _{i=1}^{l}\sum _{j=1}^{k_{i}}{\frac {a_{ij}}{(j-1)!}}t^{j-1}e^{\lambda _{i}t}$	$F(s)={\frac {N(s)}{(s-\lambda _{1})^{k_{1}}\cdots (s-\lambda _{l})^{k_{l}}}}=\sum _{i=1}^{l}\sum _{j=1}^{k_{i}}{\frac {a_{ij}}{(s-\lambda _{i})^{j}}}$ $a_{ij}={\frac {1}{(k_{i}-j)!}}\lim _{s\to \lambda _{i}}{\frac {d^{k_{i}-j}}{ds^{k_{i}-j}}}\left(\left(s-\lambda _{i}\right)^{k_{i}}F(s)\right)$	${\textrm {Re}}\{s\}>max({\textrm {Re}}\{\lambda \})\,$	Partial fraction expansion and nth power with frequency shift
Explanatory notes: $u(t)\,$ represents the Heaviside step function. $\delta (t)\,$ represents the Dirac delta function. $\Gamma (z)\,$ represents the Gamma function. $\gamma \,$ is the Euler–Mascheroni constant. $t\,$, a real number, typically represents time, although it can represent any independent dimension. $s\,$ is the complex angular frequency, and ${\textrm {Re}}\{s\}$ is its real part. $\alpha \,$, $\beta \,$, $\tau \,$, and $\omega \,$ are real numbers. $n$ is an integer.