laplace

Laplace Transform of Linearity: \mathcal{L}{\beta f(t) + \gamma g(t)}

\beta F(s) + \gamma G(s)

Laplace Transform of a constant: \mathcal{L}{1}

\frac{1}{s}

Laplace Transform of an exponential: \mathcal{L}{e^{bt}}

\frac{1}{s-b}

Laplace Transform of sine: \mathcal{L}{\sin(bt)}

\frac{b}{s^2 + b^2}

Laplace Transform of cosine: \mathcal{L}{\cos(bt)}

\frac{s}{s^2 + b^2}

Time Scaling Property: \mathcal{L}{f(bt)}

\frac{1}{b} F\left(\frac{s}{b}\right)

Frequency Shifting Property: \mathcal{L}{e^{bt} f(t)}

F(s-b)

Laplace Transform of a first derivative: \mathcal{L}{f'(t)}

sF(s) - f(0)

Laplace Transform of an n-th derivative: \mathcal{L}{f^{(n)}(t)}

s^n F(s) - \sum_{k=1}^{n} s^{n-k} f^{(k-1)}(0)

Laplace Transform of an integral: \mathcal{L}{\int_{0}^{t} f(\tau) d\tau}

\frac{1}{s} F(s)

Multiplication by t (Frequency Derivative): \mathcal{L}{tf(t)}

-F'(s)

Laplace Transform of t^k e^{-t}

\frac{k!}{(s+1)^{k+1}}

Laplace Transform of convolution: \mathcal{L}{(f \ast g)(t)}

F(s)G(s)

Laplace Transform of Heaviside step function: \mathcal{L}{\theta(t)}

\frac{1}{s}

Time Shifting Property: \mathcal{L}{f(t-t0)\theta(t-t0)}

e^{-st_0} F(s)

Laplace Transform of Dirac delta function: \mathcal{L}{\delta(t)}

1

Laplace Transform of a power function: \mathcal{L}{t^k}

\frac{k!}{s^{k+1}}