Table of selected Laplace transforms
From ControlTheoryPro.com

1 Table of selected Laplace transforms
ID  Function  Time domain 
Laplace sdomain 
Region of convergence for causal systems 

1  ideal delay  
1a  unit impulse  
2  delayed nth power with frequency shift 

2a  nth power ( for integer n ) 

2a.1  qth power ( for real q ) 

2a.2  unit step  
2b  delayed unit step  
2c  ramp  
2d  nth power with frequency shift  
2d.1  exponential decay  
3  exponential approach  
4  sine  
5  cosine  
6  hyperbolic sine  
7  hyperbolic cosine  
8  Exponentiallydecaying sine wave 

9  Exponentiallydecaying cosine wave 

10  nth root  
11  natural logarithm  
12  Bessel function of the first kind, of order n 
 
13  Modified Bessel function of the first kind, of order n 

14  Bessel function of the second kind, of order 0 

15  Modified Bessel function of the second kind, of order 0 

16  Error function 
1.1 Explanatory notes
 represents the Heaviside step function.
 represents the Dirac delta function.
 represents the Gamma function.
 is the EulerMascheroni constant.
 , a real number, typically represents time,
although it can represent any independent dimension.  is the complex angular frequency.
 , , , and are real numbers.
 , is an integer.
 A causal system is a system where the impulse response h(t) is zero for all time t prior to t = 0. In general, the ROC for causal systems is not the same as the ROC for anticausal systems. See also causality.