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] |
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] |