What is the Laplace transform of a periodic function with period T?

What is the Laplace transform of a periodic function with period T?

The Laplace Transform of the periodic function f(t) with period p, equals the Laplace Transform of one cycle of the function, divided by ( 1 − e − s p ) \displaystyle{\left({1}-{e}^{{-{s}{p}}}\right)} (1−e−sp).

Is f/t periodic?

A function f(t) is periodic if the function values repeat at regular intervals of the independent variable t. The regular interval is referred to as the period. See Figure 1.

What is periodic and nonperiodic?

2 Periodic and aperiodic signals. A periodic signal is one that repeats the sequence of values exactly after a fixed length of time, known as the period. A non-periodic or aperiodic signal is one for which no value of T satisfies Equation 10.11.

What is a period in a periodic function?

The time interval between two waves is known as a Period whereas a function that repeats its values at regular intervals or periods is known as a Periodic Function. In other words, a periodic function is a function that repeats its values after every particular interval.

What is called Laplace transform?

In mathematics, the Laplace transform, named after its inventor Pierre-Simon Laplace (/ləˈplɑːs/), is an integral transform that converts a function of a real variable (often time) to a function of a complex variable. (complex frequency).

Where do we use Laplace transform?

The Laplace transform can also be used to solve differential equations and is used extensively in mechanical engineering and electrical engineering. The Laplace transform reduces a linear differential equation to an algebraic equation, which can then be solved by the formal rules of algebra.

What is the significance of the Laplace transform?

The Laplace transform is an integral transform perhaps second only to the Fourier transform in its utility in solving physical problems. The Laplace transform is particularly useful in solving linear ordinary differential equations such as those arising in the analysis of electronic circuits.

Does this Laplace transform exist?

Existence of the Laplace Transform. A function has a Laplace transform whenever it is ofexponentialorder . That is, there must be a real numbersuch that. As an example, every exponential functionhas aLaplace transform for all finite values of and . Let’slook at this case more closely.

What is the Laplace transform of a constant?

The Laplace transform of a constant is a delta function. Note that this assumes the constant is the function f(t)=c for all t positive and negative. Sometimes people loosely refer to a step function which is zero for negative time and equals a constant c for positive time as a “constant function”.

What is the Laplace transform of f(t)?

The Laplace Transform F (s) of f (t) is defined as In this definition f (t) is assumed to be zero for t < 0. The Laplace variable s (p also used) is a complex variable which can take on all possible vluues. The Laplace Transform is well suited for describing systems with initial values and transients.