Elementary Laplace Transforms

f(t)    

   F(s)
1       
= $\displaystyle \frac{1}{s} \ \ (s>0)$ (1)
eat       = $\displaystyle \frac{1}{s-a}\ \ (s>a)$ (2)
tn       
= $\displaystyle \frac{n!}{s^{n+1}}\ \ (s>0,n \mbox{ a positive integer})$ (3)
tp      
= $\displaystyle \frac{\Gamma(p+1)}{s^{p+1}}\ \ (s>0,p>-1)$ (4)
sin(at)    
= $\displaystyle \frac{a}{s^2+a^2}\ \ (s>0)$ (5)
cos(at)    
= $\displaystyle \frac{s}{s^2+a^2} \ \ (s>0)$ (6)
eatsin(at)     = $\displaystyle \frac{b}{(s-a)^2+b^2}\ \ (s>a)$ (7)
eatcos(at)   
= $\displaystyle \frac{s-a}{(s-a)^2+b^2}\ \ (s>a)$ (8)
tneat       
= $\displaystyle \frac{n!}{(s-a)^{n+1}}\ \ (s>a)$ (9)
L(f '(t))     
= $\displaystyle s\cdot {\cal L}(f(t))(s) - f(0)$ (11)
u(c - t)       
= $\displaystyle \frac{e^{-cs}}{s}\ \ (s>0)$ (12)
L(ectf(t))      
= $\displaystyle {\cal L}(f(t))(s-c)$ (16)