4, 0 5 (f * 9)(t) =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Convolution of Two Functions
In this exercise, you are tasked with finding the convolution of the functions \( f(t) = e^{-2t} \) and
\[
g(t) =
\begin{cases}
4, & 0 \leq t < 5 \\
0, & t \geq 5
\end{cases}
\]
#### Problem Statement
Compute the convolution \( (f * g)(t) \).
#### Explanation
The function \( f(t) = e^{-2t} \) is an exponentially decaying function. The function \( g(t) \) is a piecewise function that takes the value 4 for \( 0 \leq t < 5 \) and 0 for \( t \geq 5 \).
Convolution involves integrating the product of these functions, considering one of them as a shifted version, usually expressed as:
\[
(f * g)(t) = \int_{0}^{t} f(\tau) \cdot g(t-\tau) \, d\tau
\]
Here, integrate over the appropriate intervals based on the definitions of \( g(t) \).
This process provides insight into how the shape of one function influences another when they are combined over time.
Fill in the result of the convolution in the space provided:
\[
(f * g)(t) = \underline{\hspace{5cm}}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fac965a5e-260b-4b5a-b5dd-dbb137feedb3%2Fdf85a692-e933-43f5-9bb1-c4ff7a59e21d%2Fhv4bpf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Convolution of Two Functions
In this exercise, you are tasked with finding the convolution of the functions \( f(t) = e^{-2t} \) and
\[
g(t) =
\begin{cases}
4, & 0 \leq t < 5 \\
0, & t \geq 5
\end{cases}
\]
#### Problem Statement
Compute the convolution \( (f * g)(t) \).
#### Explanation
The function \( f(t) = e^{-2t} \) is an exponentially decaying function. The function \( g(t) \) is a piecewise function that takes the value 4 for \( 0 \leq t < 5 \) and 0 for \( t \geq 5 \).
Convolution involves integrating the product of these functions, considering one of them as a shifted version, usually expressed as:
\[
(f * g)(t) = \int_{0}^{t} f(\tau) \cdot g(t-\tau) \, d\tau
\]
Here, integrate over the appropriate intervals based on the definitions of \( g(t) \).
This process provides insight into how the shape of one function influences another when they are combined over time.
Fill in the result of the convolution in the space provided:
\[
(f * g)(t) = \underline{\hspace{5cm}}
\]
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