Laplace Transform - Table of Selected Laplace Transforms

Table of Selected Laplace Transforms

The following table provides Laplace transforms for many common functions of a single variable. For definitions and explanations, see the Explanatory Notes at the end of the table.

Because the Laplace transform is a linear operator:

  • The Laplace transform of a sum is the sum of Laplace transforms of each term.
  • The Laplace transform of a multiple of a function is that multiple times the Laplace transformation of that function.

Using this linearity, and various trigonometric, hyperbolic, and Complex number (etc.) properties and/or identities, some Laplace transforms can be obtained from others quicker than by using the definition directly.

The unilateral Laplace transform takes as input a function whose time domain is the non-negative reals, which is why all of the time domain functions in the table below are multiples of the Heaviside step function, u(t). The entries of the table that involve a time delay τ are required to be causal (meaning that τ > 0). 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 region of convergence for causal systems is not the same as that of anticausal systems.

Function Time domain
 Laplace s-domain
 Region of convergence Reference
unit impulse inspection
delayed impulse time shift of
unit impulse
unit step Re(s) > 0 integrate unit impulse
delayed unit step Re(s) > 0 time shift of
unit step
ramp Re(s) > 0 integrate unit
impulse twice
delayed nth power
with frequency shift		 Re(s) > −α Integrate unit step,
apply frequency shift,
apply time shift
nth power
( for integer n )		 Re(s) > 0
(n > −1)		 Integrate unit
step n times
qth power
(for complex q)		 Re(s) > 0
Re(q) > −1
nth root Re(s) > 0 Set q = 1/n above.
nth power with frequency shift Re(s) > −α Integrate unit step,
apply frequency shift
exponential decay Re(s) > −α Frequency shift of
unit step
two-sided exponential decay −α < Re(s) < α Frequency shift of
unit step
exponential approach Re(s) > 0 Unit step minus
exponential decay
sine Re(s) > 0 Bracewell 1978, p. 227
cosine Re(s) > 0 Bracewell 1978, p. 227
hyperbolic sine Re(s) > |α| Williams 1973, p. 88
hyperbolic cosine Re(s) > |α| Williams 1973, p. 88
Exponentially decaying
sine wave		 Re(s) > −α Bracewell 1978, p. 227
Exponentially decaying
cosine wave		 Re(s) > −α Bracewell 1978, p. 227
natural logarithm Re(s) > 0 Williams 1973, p. 88
Bessel function
of the first kind,
of order n		 Re(s) > 0
(n > −1)		 Williams 1973, p. 89
Error function Re(s) > 0 Williams 1973, p. 89
Explanatory notes:
  • u(t) represents the Heaviside step function.
  • represents the Dirac delta function.
  • Γ(z) represents the Gamma function.
  • γ is the Euler–Mascheroni constant.
  • t, a real number, typically represents time,
    although it can represent any independent dimension.
  • s is the complex angular frequency, and Re(s) is its real part.
  • α, β, τ, and ω are real numbers.
  • n is an integer.

Read more about this topic:  Laplace Transform

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