Question

Laplace transforms. Let $f$ be continuous on $[0, \infty) .$ The Laplace transform of $f$ is the function $F$ defined by setting $$F(s)=\int_{0}^{\infty} e^{-s x} f(x) d x$$ The domain of $F$ is the set of numbers $s$ for which the improper integral converges. Find the Laplace transform $F$ of each of the following functions specifying the domain of $F$. $$f(x)=1$$

Name: Problem 61, Chapter 11: Calculus: One Variable
Uploaded: 2021-08-03T02:22:01-08:00
Duration: 1 min 26 s
Channel: Nick Johnson
Description: Laplace transforms. Let f be continuous on [0, ∞) . The Laplace transform of f is the function F defined by setting F(s)=∫0^∞ e^-s x f(x) d x The domain of F is the set of numbers s for which the improper integral converges. Find the Laplace transform F of each of the following functions specifying the domain of F. f(x)=1

   Laplace transforms. Let $f$ be continuous on $[0, \infty) .$ The Laplace transform of $f$ is the function $F$ defined by setting $$F(s)=\int_{0}^{\infty} e^{-s x} f(x) d x$$ The domain of $F$ is the set of numbers $s$ for which the improper integral converges. Find the Laplace transform $F$ of each of the following functions specifying the domain of $F$.
$$f(x)=1$$

Calculus: One Variable

Garret J. Etgen,… 10th Edition

Chapter 11, Problem 61

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Solved by Expert Nick Johnson

08/03/2021

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The Laplace transform is defined as: $$F(s)=\int_{0}^{\infty} e^{-s x} f(x) d x$$ Show more…

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Laplace transforms. Let $f$ be continuous on $[0, \infty) .$ The Laplace transform of $f$ is the function $F$ defined by setting $$F(s)=\int_{0}^{\infty} e^{-s x} f(x) d x$$ The domain of $F$ is the set of numbers $s$ for which the improper integral converges. Find the Laplace transform $F$ of each of the following functions specifying the domain of $F$. $$f(x)=1$$

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Key Concepts

Laplace Transform

The Laplace transform is an integral transform that converts a function defined on the nonnegative real numbers into a function of a complex variable s. It is defined by the integral ??? e^(–sx) f(x) dx, provided the integral converges. This transformation is widely used to simplify the analysis of linear time-invariant systems, solve differential equations, and study systems in the s-domain.

Improper Integral Convergence and Region of Convergence

The convergence of the Laplace integral depends on the value of s. For functions like constant functions, the integral ??? e^(–sx) dx converges only if the exponential term decays sufficiently fast as x tends to infinity, which occurs when s > 0. This set of s values for which the integral converges defines the region of convergence of the Laplace transform. In the case of the constant function f(x)=1, the Laplace transform can be computed as 1/s for s>0.

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