Home page for accesible maths 6 Chapter 6 contents

Style control - access keys in brackets

Font (2 3) - + Letter spacing (4 5) - + Word spacing (6 7) - + Line spacing (8 9) - +

6.44 Table of Laplace transforms

We give here the Laplace transforms of some standard functions:

f⁢(x)ℒ⁢(f)⁢(s)ea⁢x1s-a(s>as>a)xn(n=0,1,…n=0,1,\ldots)n!sn+1(s>0s>0)xn⁢ea⁢x(n=0,1,…n=0,1,\ldots)n!(s-a)n+1(s>as>a)sin⁡b⁢xbs2+b2(s>0s>0)cos⁡b⁢xss2+b2(s>0s>0)ea⁢x⁢sin⁡b⁢xb(s-a)2+b2(s>as>a)ea⁢x⁢cos⁡b⁢xs-a(s-a)2+b2(s>as>a)\begin{array}[]{llll}f(x)&&{\mathcal{L}}(f)(s)&\\ \hline e^{ax}&&\frac{1}{s-a}&\hbox{($s>a$)}\\ x^{n}&\hbox{($n=0,1,\ldots$)}&\frac{n!}{s^{n+1}}&\hbox{($s>0$)}\\ x^{n}e^{ax}&\hbox{($n=0,1,\ldots$)}&\frac{n!}{(s-a)^{n+1}}&\hbox{($s>a$)}\\ \sin bx&&\frac{b}{s^{2}+b^{2}}&\hbox{($s>0$)}\\ \cos bx&&\frac{s}{s^{2}+b^{2}}&\hbox{($s>0$)}\\ e^{ax}\sin bx&&\frac{b}{(s-a)^{2}+b^{2}}&\hbox{($s>a$)}\\ e^{ax}\cos bx&&\frac{s-a}{(s-a)^{2}+b^{2}}&\hbox{($s>a$)}\end{array}

The functions on the left are called elementary functions; note that the Laplace transform of any elementary function is a rational function of ss.