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Function

Laplace Transform

Remarks

$$f(x)$$

$$\mathcal{L}\{f\}(s) = \int_0^\infty f(x)e^{-sx}dx$$

Definition

$$af(x) +bg(x)$$

$$a\mathcal{L}\{f\}(s) + b\mathcal\{g\}(s)$$

Linearity

$$xf(x)$$

$$-\left(\mathcal{L}\{f\}\right)'(s)$$

$$x^nf(x)$$

$$(-1)^n\left(\mathcal{L}\{f\}\right)^{(n)}(s)$$

$$f'(x)$$

$$s\mathcal{L}\{f\}(s) - f(0)$$

$$f^{(n)}(x)$$

$$s^n\mathcal{L}\{f\}(s) - \sum_{k=1}^ns^{n-k}f^{(k-1)}(0)$$

$$\frac{f(x)}{x}$$

$$\int_s^\infty \mathcal{L}\{f\}(\sigma)d\sigma$$

$$\int_0^xf(t)dt$$

$$\frac{\mathcal{L}\{f\}(s)}{s}$$

$$e^{ax}f(x)$$

$$\mathcal{L}\{f\}(s-a)$$

$$f(x-a)u(x-a)$$

$$e^{-as}\mathcal{L}\{f\}(s)$$

$$f(ax)$$

$$\frac{1}{a}\mathcal{L}\{f\}(s)$$

$$(f*g)(x)$$

$$\mathcal{L}\{f\}(s)\mathcal{L}\{g\}(s)$$

Function

Laplace Transform

Remarks

$$f(x)$$

$$\mathcal{L}\{f\}(s) = \int_0^\infty f(x)e^{-sx}dx$$

Definition

$$\delta(x)$$

$$1$$

$\delta$ : Dirac Delta Function

$$u(x)$$

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

$u(x)$ : Step Function

$$\sqrt[n]{x}u(x)$$

$$\frac{1}{s^{\frac{1}{n}+1}}\Gamma\left(\frac{1}{n}+1\right)$$

$$\sin(a x)u(x)$$

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

$$\cos(ax)u(x)$$

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

$$\sinh(ax)u(x)$$

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

$$\cosh(ax)u(x)$$

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