Laplace transform of zero
Understand the Problem
The question is asking for the Laplace transform of the constant function zero. The Laplace transform is an integral transform used to convert a function of time into a function of a complex variable, and this question seeks to understand the result of applying that transform to the zero function.
Answer
The Laplace transform of the constant function zero is $0$.
Answer for screen readers
The Laplace transform of the constant function zero is $0$.
Steps to Solve
- Define the Laplace Transform
The Laplace transform of a function $f(t)$ is defined as:
$$ \mathcal{L}{f(t)} = \int_0^\infty e^{-st} f(t) , dt $$
where $s$ is a complex number.
- Set Up the Integral for the Zero Function
For the zero function, we substitute $f(t) = 0$ into the Laplace transform formula:
$$ \mathcal{L}{0} = \int_0^\infty e^{-st} \cdot 0 , dt $$
- Evaluate the Integral
Since the integrand is zero, the integral simplifies to:
$$ \mathcal{L}{0} = \int_0^\infty 0 , dt $$
- Conclude the Calculation
The integral of zero over any interval is zero, so:
$$ \mathcal{L}{0} = 0 $$
The Laplace transform of the constant function zero is $0$.
More Information
The Laplace transform is often used in engineering and physics for solving linear ordinary differential equations. The result that the Laplace transform of zero is zero helps in understanding the behavior of differential equations where zero-input leads to zero-output systems.
Tips
One common mistake is to forget that the integrand is zero; sometimes students may mistakenly try to simplify the integral without recognizing that the function itself is zero.
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