Question

laplace transform of f(t) is defined for positive and negative values of t
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Answer 1

Answer:

the Laplace transform, named after its inventor Pierre-Simon Laplace (/ləˈplɑːs/), is an integral transform that converts a function of a real variable {\displaystyle t}t (often time) to a function of a complex variable {\displaystyle s}s (complex frequency). The transform has many applications in science and engineering because it is a tool for solving differential equations. In particular, it transforms differential equations into algebraic equations and convolution into multiplication.

Step-by-step explanation:

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Answer 2

Laplace transform of f(t) is defined for positive and negative values of t.

• The Laplace transform ℒ, of a function f(t) for t > 0 is defined by the following integral over 0 to ∞: ℒ } {f(t)}  = ∫0∞ f(t) e−st dt.
• The resulting expression is a function of s, which we write as F(s).

• The function f(t), which is a function of time, is transformed to a function F(s).
•  The function F(s) is a function of the Laplace variable, "s."  We call this a Laplace domain function.
• So the Laplace Transform takes a time domain function, f(t), and converts it into a Laplace domain function, F(s).
• We use a lowercase letter for the function in the time domain, and un uppercase letter in the Laplace domain.
• For our purposes the time variable, t, and time domain functions will always be real-valued.  The Laplace variable, s, and Laplace domain functions are complex.
• Since the integral goes from 0 to ∞, the time variable, t, must not occur in the Laplace domain result .
• Note that none of the Laplace Transforms in the table have the time variable, t, in them.

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