The following is a list of Laplace transforms for many common functions of a single variable. The Laplace transform is an integral transform that takes a function of a positive real variable (often time) to a function of a complex variable (complex angular frequency).
Properties
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
For functions and and for scalar , the Laplace transform satisfies
and is, therefore, regarded as a linear operator.
Time shifting
The Laplace transform of , where is the Heaviside step function, is .
Frequency shifting
The Laplace transform of is .
Explanatory notes
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, .
The entries of the table that involve a time delay are required to be causal (meaning that ). A causal system is a system where the impulse response is zero for all time prior to . 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:
represents the Dirac delta function. represents the Heaviside step function. Literature may refer to this by other notation, including or . represents the Gamma function. is the Euler–Mascheroni constant. is a real number. It typically represents time, although it can represent any independent dimension. is the complex frequency domain parameter, and is its real part. is an integer. are real numbers. is a complex number.
Table Function Time domain Laplace -domain Region of convergence Reference unit impulse all inspection delayed impulse time shift ofunit impulse unit step integrate unit impulse delayed unit step time shift ofunit step ramp integrate unitimpulse twice th power (for integer ) () Integrate unitstep times th power (for complex ) th root Set above. th power with frequency shift Integrate unit step,apply frequency shift delayed th power with frequency shift Integrate unit step,apply frequency shift,apply time shift exponential decay Frequency shift ofunit step two-sided exponential decay (only for bilateral transform) Frequency shift ofunit step exponential approach Unit step minusexponential decay sine cosine hyperbolic sine hyperbolic cosine exponentially decaying sine wave exponentially decaying cosine wave natural logarithm Bessel function of the first kind, of order n () Error function
See also List of Fourier transforms
References