17Calculus - Laplace Transforms Table

17Calculus

Laplace Transforms

$$f(t)$$

$$\displaystyle{ F(s) }$$

Basic Functions

$$t^n, ~ n = 1, 2, 3, \ldots$$

$$\displaystyle{ \frac{n!}{s^{n+1}} }$$

$$e^{at}$$

$$\displaystyle{ \frac{1}{s-a} }$$

$$\sin(\alpha t)$$

$$\displaystyle{ \frac{\alpha}{s^2 + \alpha^2} }$$

$$\cos(at)$$

$$\displaystyle{ \frac{s}{s^2 + a^2} }$$

$$\sinh(at)$$

$$\displaystyle{ \frac{a}{s^2 - a^2} }$$

$$\cosh(at)$$

$$\displaystyle{ \frac{s}{s^2 - a^2} }$$

Special Functions

$$\delta(t)$$ unit impulse

$$1$$

$$\delta(t-\tau)$$ shifted unit impulse

$$e^{-\tau s}$$

$$u(t)$$ unit step

$$\displaystyle{ \frac{1}{s} }$$

$$u(t-\tau)$$ shifted unit step

$$\displaystyle{ \frac{1}{s} e^{-\tau s} }$$

Combined Functions

$$e^{at}\sin(\alpha t)$$

$$\displaystyle{ \frac{\alpha}{(s-a)^2 + \alpha^2} }$$

$$f(t)u(t-a)$$

$$\displaystyle{ e^{-sa} \mathcal{L}\{ f(t+a) \} }$$

$$t^n e^{at}, ~ n = 1, 2, 3, \ldots$$

$$\displaystyle{ \frac{n!}{(s-a)^{n+1}} \} }$$

Derivatives and Integrals

$$f'(t)$$

$$sF(s) - f(0)$$

$$f''(t)$$

$$s^2F(s) - sf(0) - f'(0)$$

$$\displaystyle{ f^{(n)}(t) }$$

$$\displaystyle{ s^nF(s) - s^{n-1}f(0) - }$$ $$\displaystyle{ s^{n-2}f'(0) - . . . - f^{(n-1)}(0) }$$

$$\int_0^t{ f(v)~dv }$$

$$\displaystyle{ \frac{F(s)}{s} }$$

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