Laplace Transform Properties

Laplace Transform Properties#

No.	Name	Time Domain \(f(t)\)	Complex Frequency Domain \(F(s)\)
1.	Linearity	\(a_1f_1(t)+a_2f_2(t)+\cdots+a_nf_n(t)\)	\(a_1F_1(s)+a_2F_2(s)+\cdots+a_nF_n(s)\)
2.	Time shifting	\(\displaystyle{f(t-a)}u_0(t-a)\)	\(\displaystyle{e^{-a s}F(s)}\)
3.	Frequency shifting	\(\displaystyle{e^{-as}f(t)}\)	\(\displaystyle{F(s+a)}\)
4.	Time scaling	\(f(a t)\)	\(\displaystyle{\frac{1}{a}F\left(\frac{s}{a}\right)}\)
5.	Time differentiation	\(\displaystyle{\frac{d}{dt}\,f(t)}\)	\(\displaystyle{sF(s)-f(0^-)}\)
6.	Frequency differentiation	\(\displaystyle{tf(t)}\)	\(\displaystyle{-\frac{d}{ds}F(s)}\)
7.	Time integration	\(\displaystyle{\int_{-\infty}^{t}f(\tau)d\tau}\)	\(\displaystyle{\frac{F(s)}{s}+ \frac{f(0^-)}{s}}\)
8.	Frequency integration	\(\displaystyle{\frac{f(t)}{t}}\)	\(\displaystyle{\int_s^\infty F(\sigma)\,d\sigma}\)
9.	Time Periodicity	\(\displaystyle{f(t + nT)}\)	\(\displaystyle{\frac{\int_0^T f(t)e^{-st}\,dt}{1 - e^{-sT}}}\)
10.	Initial value theorem	\(\displaystyle{\lim_{t\rightarrow 0} f(t)}\)	\(\displaystyle{\lim_{s\rightarrow \infty}sF(s) = f(0^-)}\)
11.	Final value theorem	\(\displaystyle{\lim_{t\rightarrow \infty} f(t)}\)	\(\displaystyle{\lim_{s\rightarrow 0}sF(s) = f(\infty)}\)
12.	Time convolution	\(\displaystyle{f_1(t)*f_2(t)}\)	\(\displaystyle{F_1(js) F_2(s)}\)
13.	Frequency convolution	\(\displaystyle{f_1(t)f_2(t)}\)	\(\displaystyle{\frac{1}{j2\pi}F_1(s)*F_2(s)}\)