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 |
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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 |
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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:
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Read more about this topic: Laplace Transform
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