Differential Equations - Laplace Transforms: Table

	$f (t) = L^{- 1} {F (s)}$	$F (s) = L {f (t)}$
	1	$\frac{1}{s}$
	$e^{a t}$	$\frac{1}{s - a}$
	$t^{n}, n = 1, 2, 3, \dots$	$\frac{n!}{s^{n + 1}}$
	$t^{p}$ , $p > - 1$	$\frac{Γ (p + 1)}{s^{p + 1}}$
	$\sqrt{t}$	$\frac{\sqrt{π}}{2 s^{\frac{3}{2}}}$
	$t^{n - \frac{1}{2}}, n = 1, 2, 3, \dots$	$\frac{1 \cdot 3 \cdot 5 \dots (2 n - 1) \sqrt{π}}{2^{n} s^{n + \frac{1}{2}}}$
	$\sin (a t)$	$\frac{a}{s^{2} + a^{2}}$
	$\cos (a t)$	$\frac{s}{s^{2} + a^{2}}$
	$t \sin (a t)$	$\frac{2 a s}{{(s^{2} + a^{2})}^{2}}$
	$t \cos (a t)$	$\frac{s^{2} - a^{2}}{{(s^{2} + a^{2})}^{2}}$
	$\sin (a t) - a t \cos (a t)$	$\frac{2 a^{3}}{{(s^{2} + a^{2})}^{2}}$
	$\sin (a t) + a t \cos (a t)$	$\frac{2 a s^{2}}{{(s^{2} + a^{2})}^{2}}$
	$\cos (a t) - a t \sin (a t)$	$\frac{s (s^{2} - a^{2})}{{(s^{2} + a^{2})}^{2}}$
	$\cos (a t) + a t \sin (a t)$	$\frac{s (s^{2} + 3 a^{2})}{{(s^{2} + a^{2})}^{2}}$
	$\sin (a t + b)$	$\frac{s \sin (b) + a \cos (b)}{s^{2} + a^{2}}$
	$\cos (a t + b)$	$\frac{s \cos (b) - a \sin (b)}{s^{2} + a^{2}}$
	$\sinh (a t)$	$\frac{a}{s^{2} - a^{2}}$
	$\cosh (a t)$	$\frac{s}{s^{2} - a^{2}}$
	$e^{a t} \sin (b t)$	$\frac{b}{{(s - a)}^{2} + b^{2}}$
	$e^{a t} \cos (b t)$	$\frac{s - a}{{(s - a)}^{2} + b^{2}}$
	$e^{a t} \sinh (b t)$	$\frac{b}{{(s - a)}^{2} - b^{2}}$
	$e^{a t} \cosh (b t)$	$\frac{s - a}{{(s - a)}^{2} - b^{2}}$
	$t^{n} e^{a t}, n = 1, 2, 3, \dots$	$\frac{n!}{{(s - a)}^{n + 1}}$
	$f (c t)$	$\frac{1}{c} F (\frac{s}{c})$
	$u_{c} (t) = u (t - c)$ Heaviside Function	$\frac{e^{- c s}}{s}$
	$δ (t - c)$ Dirac Delta Function	$e^{- c s}$
	$u_{c} (t) f (t - c)$	$e^{- c s} F (s)$
	$u_{c} (t) g (t)$	$e^{- c s} L {g (t + c)}$
	$e^{c t} f (t)$	$F (s - c)$
	$t^{n} f (t), n = 1, 2, 3, \dots$	${(- 1)}^{n} F^{(n)} (s)$
	$\frac{1}{t} f (t)$	$\int_{s}^{\infty} F (u) d u$
	$\int_{0}^{t} f (v) d v$	$\frac{F (s)}{s}$
	$\int_{0}^{t} f (t - τ) g (τ) d τ$	$F (s) G (s)$
	$f (t + T) = f (t)$	$\frac{\int_{0}^{T} e^{- s t} f (t) d t}{1 - e^{- s T}}$
	$f^{'} (t)$	$s F (s) - f (0)$
	$f^{″} (t)$	$s^{2} F (s) - s f (0) - f^{'} (0)$
	$f^{(n)} (t)$	$s^{n} F (s) - s^{n - 1} f (0) - s^{n - 2} f^{'} (0) \dots - s f^{(n - 2)} (0) - f^{(n - 1)} (0)$

Table Notes

This list is not a complete listing of Laplace transforms and only contains some of the more commonly used Laplace transforms and formulas.
Recall the definition of hyperbolic functions. $\cosh (t) = \frac{e^{t} + e^{- t}}{2} \sinh (t) = \frac{e^{t} - e^{- t}}{2}$
Be careful when using “normal” trig function vs. hyperbolic functions. The only difference in the formulas is the “ $+ a^{2}$ ” for the “normal” trig functions becomes a “ $- a^{2}$ ” for the hyperbolic functions!
Formula #4 uses the Gamma function which is defined as $Γ (t) = \int_{0}^{\infty} e^{- x} x^{t - 1} d x$ If $n$ is a positive integer then,
$Γ (n + 1) = n!$ The Gamma function is an extension of the normal factorial function. Here are a couple of quick facts for the Gamma function
$\begin{matrix} Γ (p + 1) = p Γ (p) p (p + 1) (p + 2) \dots (p + n - 1) = \frac{Γ (p + n)}{Γ (p)} Γ (\frac{1}{2}) = \sqrt{π} \end{matrix}$

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