# Laplace Transform **A transform that converts a function in the time domain into a complex function in the complex frequency domain.** The *Laplace transform* outputs a *complex-valued* function of the *complex variable* $s$. $s=\sigma+i\omega$ The relationship between the Laplace transform and the [[Fourier transform]] is detailed in the [[comparison of the Laplace and Fourier Transform]]. The *unilateral* or *one-sided* Laplace transform is $F(s)=\mathcal{L}[f(t)]=\int_{0}^{\infty}f(t)e^{-st}\;dt$ The unilateral Laplace transform is defined only over $0<t<\infty$. ## Inverse Laplace transform A primitive method of finding the inverse Laplace transform is to decompose $F(s)$ through [[Partial Fraction Decomposition|partial fractions]] and taking the inverse of each term. ## Properties of the Laplace transform | $f(t)$ | $F(s)$ | Name | |:-----------------------------:|:---------------------------------------------:|:-------------------------:| | $a_{1}f_{1}(t)+a_{2}f_{2}(t)$ | $a_{1}F_{1}(s)+a_{2}F_{2}(s)$ | Linearity | | $f(at)$ | $\frac{1}{a}F(\frac{s}{a})$ | Scaling | | $f(t-a)u(t-a)$ | $e^{-as}F(s)$ | Time shifting | | $e^{-at}f(t)$ | $F(s+a)$ | Frequency shifting | | $f'(t)$ | $sF(s)-f(0^+)$ | First time derivative | | $f''(t)$ | $s^{2}F(s)-sf(0^+)-f'(0^+)$ | Second time derivative | | $f'''(t)$ | $s^{3}F(s)-s^{2}f(0^+)-sf'(0^+)-f''(0^+)$ | Third time derivative | | $f^{(n)}(t)$ | $s^{n}F(s)-\sum_{k=1}^{n}s^{n-k}f^{(k-1)}(0^+)$ | Time [[Derivative\|differentiation]] | | $\int_{0}^{t}f(\tau)\;d\tau$ | $\frac{1}{s}F(s)$ | Time [[Integral\|integration]] | | $tf(t)$ | $-\frac{d}{ds}F(s)$ | Frequency differentiation | | $\frac{f(t)}{t}$ | $\int_{s}^{\infty}F(s)\;ds$ | Frequency integration | | $f(0)$ | $\lim_{s\rightarrow\infty}sF(s)$ | Initial value | | $f(\infty)$ | $\lim_{s\rightarrow 0}sF(s)$ | Final value | | $f_{1}(t)*f_{2}(t)$ | $F_{1}(s)F_{2}(s)$ | [[Convolution]] | ## Laplace transform pairs As with Fourier transform pairs, some Laplace transform pairs involve the singularity functions $u(t)$, the [[unit step function]], and $\delta(t)$, the [[unit impulse function]]. These pairs are defined with $f(t)$ multiplied by a $u(t)$ term, which is omitted below. Thus, they are only defined for $t\ge 0$ and $f(t)=0$ for $t<0$. | $f(t)$ | $F(s)$ | |:----------------------:|:-------------------------------------:| | $\delta(t)$ | $1$ | | $u(t)$ | $\frac{1}{s}$ | | $e^{-at}$ | $\frac{1}{s+a}$ | | $t$ | $\frac{1}{s^{2}}$ | | $t^{n}$ | $\frac{n!}{s^{n+1}}$ | | $te^{-at}$ | $\frac{1}{(s+a)^{2}}$ | | $t^{n}e^{-at}$ | $\frac{n!}{(s+a)^{n+1}}$ | | $\sin \omega t$ | $\frac{\omega}{s^{2}+\omega^{2}}$ | | $\cos \omega t$ | $\frac{s}{s^{2}+\omega^{2}}$ | | $e^{-at}\sin \omega t$ | $\frac{\omega}{(s+a)^{2}+\omega^{2}}$ | | $e^{-at}\cos \omega t$ | $\frac{s+a}{(s+a)^{2}+\omega^{2}}$ |