List of Laplace transforms

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 (frequency).
Properties.
The Laplace transform of a function formula_1 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 formula_2 and formula_3 and for scalar formula_4, the Laplace transform satisfies
and is, therefore, regarded as a linear operator.
Time shifting.
The Laplace transform of formula_6 is formula_7.
Frequency shifting.
formula_8 is the Laplace transform of formula_9.
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:
Table.
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