Laplace Transform Table - University Of Marylandhck/Laplace.pdfLaplace Transform Table Function...

Laplace Transform Table Function Laplace Transform t n , n =0, 1,... n!/s n+1 t n , n> -1 Γ(n + 1)/s n+1 e at 1/(s - a) t n e at n!/(s - a) n+1 e at cos(ωt) (s - a)/((s - a) 2 + ω 2 ) e at sin(ωt) ω/((s - a) 2 + ω 2 ) e at (sin(ωt) - ωt cos(ωt)) 2ω 3 /((s - a) 2 + ω 2 ) 2 te at sin(ωt) 2ω(s - a)/((s - a) 2 + ω 2 ) 2 f ′ (t) sL(f ) - f (0) f ′′ (t) s 2 L(f ) - sf (0) - f ′ (0) f (n) (t) s n L(f ) - s n−1 f (0) -···- sf (n−2) (0) - f (n−1) (0) e at f (t) L(f )(s - a), i.e., plug in s - a for s. tf (t) -L(f ) ′ t n f (t) (-1) n L(f ) (n) t 0 g (t - u)f (u) du = f * g L(f )L(g ) H c (t)f (t - c) e −cs L(f ) δ (t - c)f (t) e −cs f (c) f (t) ∞ 0 e −st f (t) dt Note that the variable s and the function f can be complex valued. Thus the results for trig functions actually follow from those for exponential functions. The gamma function above is Γ(x)= ∞ 0 x n−1 e −x dx. It satisﬁes Γ(x + 1) = xΓ(x) and Γ(n + 1) = n! for n a positive integer. Also Γ(1/2) = √ π.

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Transcript of Laplace Transform Table - University Of Marylandhck/Laplace.pdfLaplace Transform Table Function...

Laplace Transform Table

Function Laplace Transform

tn, n = 0, 1, . . . n!/sn+1

tn, n > −1 Γ(n + 1)/sn+1

eat 1/(s − a)

tneat n!/(s− a)n+1

eat cos(ωt) (s − a)/((s− a)2 + ω2)

eat sin(ωt) ω/((s− a)2 + ω2)

eat(sin(ωt) − ωt cos(ωt)) 2ω3/((s − a)2 + ω2)2

teat sin(ωt) 2ω(s− a)/((s − a)2 + ω2)2

f ′(t) sL(f)− f(0)

f ′′(t) s2L(f) − sf(0) − f ′(0)

f (n)(t) snL(f) − sn−1f(0) − · · · − sf (n−2)(0) − f (n−1)(0)

eatf(t) L(f)(s− a), i.e., plug in s − a for s.

tf(t) −L(f)′

tnf(t) (−1)nL(f)(n)

∫t

0g(t− u)f(u) du = f ∗ g L(f)L(g)

Hc(t)f(t − c) e−csL(f)

δ(t − c)f(t) e−csf(c)

f(t)∫∞

0e−stf(t) dt

Note that the variable s and the function f can be complex valued. Thus the resultsfor trig functions actually follow from those for exponential functions.

The gamma function above is Γ(x) =∫∞

0xn−1e−x dx. It satisfies Γ(x + 1) = xΓ(x)

and Γ(n + 1) = n! for n a positive integer. Also Γ(1/2) =√

π.