The following is a list of Laplace transforms for many common functions of a single variable. The

Laplace transform In mathematics, the Laplace transform, named after Pierre-Simon Laplace (), is an integral transform that converts a Function (mathematics), function of a Real number, real Variable (mathematics), variable (usually t, in the ''time domain'') to a f ...

is an

integral transform In mathematics, an integral transform is a type of transform that maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily charac ...

that takes a function of a positive real variable (often time) to a function of a complex variable (complex

angular frequency In physics, angular frequency (symbol ''ω''), also called angular speed and angular rate, is a scalar measure of the angle rate (the angle per unit time) or the temporal rate of change of the phase argument of a sinusoidal waveform or sine ...

Properties

The Laplace transform of a function

f(t)

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

f

and

g

and for scalar

a

, the Laplace transform satisfies :

\mathcal\  = a \mathcal\ + b \mathcal\

and is, therefore, regarded as a linear operator.

Time shifting

The Laplace transform of

f(t - a) u(t - a)

e^ F(s)

Frequency shifting

The Laplace transform of

e^ f(t)

F(s - a)

Explanatory notes

The unilateral Laplace transform takes as input a function whose time domain is the

non-negative In mathematics, the sign of a real number is its property of being either positive, negative, or 0. Depending on local conventions, zero may be considered as having its own unique sign, having no sign, or having both positive and negative sign. ...

reals, which is why all of the time domain functions in the table below are multiples of the

Heaviside step function The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function named after Oliver Heaviside, the value of which is zero for negative arguments and one for positive arguments. Differen ...

, . The entries of the table that involve a time delay are required to be

causal Causality is an influence by which one Event (philosophy), event, process, state, or Object (philosophy), object (''a'' ''cause'') contributes to the production of another event, process, state, or object (an ''effect'') where the cause is at l ...

(meaning that ). A causal system is a system where the

impulse response In signal processing and control theory, the impulse response, or impulse response function (IRF), of a dynamic system is its output when presented with a brief input signal, called an impulse (). More generally, an impulse response is the reac ...

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 In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...

. * represents the

. Literature may refer to this by other notation, including

1(t)

H(t)

. * represents the

Gamma function In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined ...

. * is the Euler–Mascheroni constant. * is a

real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...

. It typically represents ''time'', although it can represent ''any'' independent dimension. * is the

complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...

frequency domain parameter, and is its

real part In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...

. * is an

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

. * are real numbers. * is a complex number.

Table

References

{{DEFAULTSORT:Laplace transforms Mathematics-related lists