Engineering Tables/Laplace Transform Table 2

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ID Function Time domain
Laplace domain
Region of convergence
for causal systems
1 Ideal delay
1a Unit impulse
2 Delayed nth power with frequency shift
2a nth Power
2a.1 qth Power
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 Exponentially-decaying sine
9 Exponentially-decaying cosine
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
Explanatory notes:
  • represents the Heaviside step function.
  • represents the Dirac delta function.
  • represents the Gamma function.
  • is the Euler-Mascheroni 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.