List of Laplace transforms

Summary

The following is a list of Laplace transforms for many common functions of a single variable.[1] The Laplace transform is an integral transform that takes a function of a positive real variable t (often time) to a function of a complex variable s (frequency).

Properties edit

The Laplace transform of a function   can be obtained using the formal definition of the Laplace transform. However, some properties of the Laplace transform can be used to obtain the Laplace transform of some functions more easily.

Linearity edit

For functions   and   and for scalar  , the Laplace transform satisfies

 

and is, therefore, regarded as a linear operator.

Time shifting edit

The Laplace transform of   is  .

Frequency shifting edit

  is the Laplace transform of  .

Explanatory notes edit

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.

The following functions and variables are used in the table below:

Table edit

Function Time domain
 
Laplace s-domain
 
Region of convergence Reference
unit impulse     all s inspection
delayed impulse     Re(s) > 0 time shift of
unit impulse[2]
unit step     Re(s) > 0 integrate unit impulse
delayed unit step     Re(s) > 0 time shift of
unit step[3]
ramp     Re(s) > 0 integrate unit
impulse twice
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
[4][5]
nth root     Re(s) > 0 Set q = 1/n above.
nth power with frequency shift     Re(s) > −α Integrate unit step,
apply frequency shift
delayed nth power
with frequency shift
    Re(s) > −α Integrate unit step,
apply frequency shift,
apply time shift
exponential decay     Re(s) > −α Frequency shift of
unit step
two-sided exponential decay
(only for bilateral transform)
    α < Re(s) < α Frequency shift of
unit step
exponential approach     Re(s) > 0 Unit step minus
exponential decay
sine     Re(s) > 0 [6]
cosine     Re(s) > 0 [6]
hyperbolic sine     Re(s) > |α| [7]
hyperbolic cosine     Re(s) > |α| [7]
exponentially decaying
sine wave
    Re(s) > −α [6]
exponentially decaying
cosine wave
    Re(s) > −α [6]
natural logarithm     Re(s) > 0 [7]
Bessel function
of the first kind,
of order n
    Re(s) > 0
(n > −1)
[7]
Error function     Re(s) > 0 [7]

See also edit

References edit

  1. ^ Distefano, J. J.; Stubberud, A. R.; Williams, I. J. (1995), Feedback systems and control, Schaum's outlines (2nd ed.), McGraw-Hill, p. 78, ISBN 978-0-07-017052-0
  2. ^ Riley, K. F.; Hobson, M. P.; Bence, S. J. (2010), Mathematical methods for physics and engineering (3rd ed.), Cambridge University Press, p. 455, ISBN 978-0-521-86153-3
  3. ^ Lipschutz, S.; Spiegel, M. R.; Liu, J. (2009), "Chapter 33: Laplace transforms", Mathematical Handbook of Formulas and Tables, Schaum's Outline Series (3rd ed.), McGraw-Hill, p. 192, ISBN 978-0-07-154855-7
  4. ^ Lipschutz, S.; Spiegel, M. R.; Liu, J. (2009), "Chapter 33: Laplace transforms", Mathematical Handbook of Formulas and Tables, Schaum's Outline Series (3rd ed.), McGraw-Hill, p. 183, ISBN 978-0-07-154855-7
  5. ^ "Laplace Transform". Wolfram MathWorld. Retrieved 30 April 2016.
  6. ^ a b c d Bracewell, Ronald N. (1978), The Fourier Transform and its Applications (2nd ed.), McGraw-Hill Kogakusha, p. 227, ISBN 978-0-07-007013-4
  7. ^ a b c d e Williams, J. (1973), Laplace Transforms, Problem Solvers, George Allen & Unwin, p. 88, ISBN 978-0-04-512021-5