Laplace Transform Properties

$L [a f_{1} (t) + b f_{2} (t)] = a F_{1} (s) + b F_{2} (s)$

Linearity

$L [f (a t)] = \frac{1}{a} F (\frac{s}{a})$

Time scaling

$L [f (t - T)] = e^{- s T} F (s)$

Time shift

$L [f (t) e^{s_{0} t}] = F (s - s_{0})$

Frequency shift

$L [f_{1} (t)] * f_{2} (t)] = F_{1} (s) F_{2} (s)$

Convolution

$L [\int_{0}^{t} f (τ) d τ] = \frac{1}{s} F (s)$

Integration

$L [f^{'} (t)] = s F (s) - f (0)$

Derivative

$L [\frac{d ^{n} f ( t )}{d t ^{n}}] = s^{n} F (s) - k = 0 \sum n - 1 s^{n - 1 - k} f^{(k)} (0)$

$n$ th derivative
$n = 1$ : $s F (s) - f (0)$
$n = 2$ : $s^{2} F (s) - s f (0) - f^{'} (0)$
$n = 3$ : $s^{3} F (s) - s^{2} f (0) - s f^{'} (0) - f^{''} (0)$

$L [t f (t)] = - F^{'} (s)$

Multiplication by $t$

$L [f_{1} (t) f_{2} (t)] = \frac{1}{2 πj} \int_{γ - j \infty}^{γ + j \infty} F_{1} (p) F_{2} (s - p) d p$

Multiplication

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Laplace Transform Properties

$L [a f_{1} (t) + b f_{2} (t)] = a F_{1} (s) + b F_{2} (s)$

$L [f (a t)] = \frac{1}{a} F (\frac{s}{a})$

$L [f (t - T)] = e^{- s T} F (s)$

$L [f (t) e^{s_{0} t}] = F (s - s_{0})$

$L [f_{1} (t)] * f_{2} (t)] = F_{1} (s) F_{2} (s)$

$L [\int_{0}^{t} f (τ) d τ] = \frac{1}{s} F (s)$

$L [f^{'} (t)] = s F (s) - f (0)$

$L [\frac{d ^{n} f ( t )}{d t ^{n}}] = s^{n} F (s) - k = 0 \sum n - 1 s^{n - 1 - k} f^{(k)} (0)$

$L [t f (t)] = - F^{'} (s)$

$L [f_{1} (t) f_{2} (t)] = \frac{1}{2 πj} \int_{γ - j \infty}^{γ + j \infty} F_{1} (p) F_{2} (s - p) d p$

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Laplace Transform Properties

L[af1​(t)+bf2​(t)]=aF1​(s)+bF2​(s)

L[f(at)]=a1​F(as​)

L[f(t−T)]=e−sTF(s)

L[f(t)es0​t]=F(s−s0​)

L[f1​(t)]∗f2​(t)]=F1​(s)F2​(s)

L[∫0t​f(τ)dτ]=s1​F(s)

L[f′(t)]=sF(s)−f(0)

L[dtndnf(t)​]=snF(s)−k=0∑n−1​sn−1−kf(k)(0)

L[tf(t)]=−F′(s)

L[f1​(t)f2​(t)]=2πj1​∫γ−j∞γ+j∞​F1​(p)F2​(s−p)dp

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$L [a f_{1} (t) + b f_{2} (t)] = a F_{1} (s) + b F_{2} (s)$

$L [f (a t)] = \frac{1}{a} F (\frac{s}{a})$

$L [f (t - T)] = e^{- s T} F (s)$

$L [f (t) e^{s_{0} t}] = F (s - s_{0})$

$L [f_{1} (t)] * f_{2} (t)] = F_{1} (s) F_{2} (s)$

$L [\int_{0}^{t} f (τ) d τ] = \frac{1}{s} F (s)$

$L [f^{'} (t)] = s F (s) - f (0)$

$L [\frac{d ^{n} f ( t )}{d t ^{n}}] = s^{n} F (s) - k = 0 \sum n - 1 s^{n - 1 - k} f^{(k)} (0)$

$L [t f (t)] = - F^{'} (s)$

$L [f_{1} (t) f_{2} (t)] = \frac{1}{2 πj} \int_{γ - j \infty}^{γ + j \infty} F_{1} (p) F_{2} (s - p) d p$