Engineering Tables/Laplace Transform Table 2

ID Function Time domain Laplace domain Region of convergence for causal systems 1Ideal delay 1aUnit impulse 2Delayed nth power with frequency shift 2a nth Power 2a.1 qth Power 2a.2Unit step 2bDelayed unit step 2cRamp 2d nth Power with frequency shift 2d.1 Exponential decay 3Exponential approach 4 Sine 5Cosine 6Hyperbolic sine 7Hyperbolic cosine 8 Exponentially-decaying sine 9 Exponentially-decaying cosine 10 nth Root 11 Natural logarithm 12Bessel function of the first kind, of order n 13 Modified Bessel function of the first kind, of order n 14Bessel function of the second kind, of order 0    15Modified Bessel function of the second kind, of order 0     16Error function 17 Constant 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.