Laplace Transform

What are we talking about?

The Laplace transform is an integral transform that helps to solve differential equations, where it reduces the differential equation into an algebraic problem.

Form

Unilateral Laplace Transform

Bilateral Laplace Transform

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Almost always what is meant by "the" Laplace transform

Properties

If f1 (t) <-> F1 (s) and f2 (t) <-> F2 (s) (<-> implies Laplace transform)

Multiplication by Time

Complex Shift

Integration

Time Reversal

Frequency Shifting

Time Scaling

Linearity

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A table of several important one-sided Laplace transforms

1

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t

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t^n

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t^a

e^(a*t)

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cos (wt)

Applications

It is used to convert derivatives into multiple domain variables and then convert the polynomials back to the differential equation using Inverse Laplace transform.

It is used in the telecommunication field to send signals to both the sides of the medium. For example, when the signals are sent through the phone then they are first converted into a time-varying wave and then superimposed on the medium.

The Laplace transform can be applied to solve the switching transient phenomenon in the series or parallel RL,RC or RLC circuits.

It can be used to solve boundary value problems in heat conduction problems.

It can be used to analyze how the concentration of an ingested drug evolves in the bloodstream