Laplace Transform Find the Laplace transform for the decaying exponential function defined by: [latex]x(t)=\left\{\begin{array}{cc}e^{-a t} & t \geq 0 \\0 & \text { else }\end{array}\right.[/latex]

[latex]\begin{aligned}X(s) &=\int_0^{\infty} x(t) e^{-s t} dt \\&=\int_0^{\infty} e^{-a t} e^{-s t} dt \\&=\int_0^{\infty} e^{-(s+a) t} dt \\&=\left.\frac{-1}{s+a} e^{-(s+a) t}\right|_0 ^{\infty} \\&=\frac{1}{s+a}\end{aligned}[/latex] Note: The last equality is true when R(s) > -a so that [latex]e^{-(s+a) t} \rightarrow 0[/latex] as t → ∞.

Question B.2: Laplace Transform Find the Laplace transform for the decayin...

Electrical Engineering: Concepts and Applications

Laplace Transform

Find the Laplace transform for the decaying exponential function defined by:

x(t)=\left\{\begin{array}{cc}e^{-a t} & t \geq 0 \\0 & \text { else }\end{array}\right.

The "Step-by-Step Explanation" refers to a detailed and sequential breakdown of the solution or reasoning behind the answer. This comprehensive explanation walks through each step of the answer, offering you clarity and understanding.
Our explanations are based on the best information we have, but they may not always be right or fit every situation.

The blue check mark means that this solution has been answered and checked by an expert. This guarantees that the final answer is accurate.
Learn more on how we answer questions.

This Question has been Answered.