Check on proof for property of the Laplace transform

In summary, the conversation is about a simple theorem and a proof that someone is asking for confirmation on. The proof involves a variable substitution and the result is the same transform with a changed argument. Upon reflection, the individual is questioning whether this step makes sense. The expert replies that it looks fine.
  • #1
greg_rack
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Homework Statement
Suppose ##F(s)=\mathcal{L}\{f(t)\}## exists for ##s> a\geq 0##.
Show that if c is a positive constant:
$$\mathcal{L} \{f(ct)\}=\frac{1}{c}F(\frac{s}{c}), \ s> ca$$
Relevant Equations
Laplace transform
Could someone check whether my proof for this simple theorem is correct? I get to the result, but with the feeling of having done something very wrong :)
$$\mathcal{L} \{f(ct)\}=\int_{0}^{\infty}e^{-st}f(ct)dt \ \rightarrow ct=u, \ dt=\frac{1}{c}du, \
\mathcal{L} \{f(ct)\}=\frac{1}{c}\int_{0}^{\infty}e^{\frac{-s}{c}u}f(u)du=\frac{1}{c}F(\frac{s}{c})$$
felt very straightforward, but looking back at it the very last step seems weird: in spite of a variable substitution, does it make sense to still get the same transform(only with the argument changed) of ##f(t)##?
 
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  • #2
Looks fine.
 
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FAQ: Check on proof for property of the Laplace transform

What is the Laplace transform?

The Laplace transform is a mathematical operation that transforms a function of time into a function of complex frequency. It is commonly used in engineering and physics to solve differential equations and analyze systems.

What is the property of the Laplace transform?

The property of the Laplace transform is that it can convert a function of time, also known as the time domain, into a function of complex frequency, also known as the frequency domain. This allows for easier analysis and solution of differential equations.

How is the Laplace transform used in solving differential equations?

The Laplace transform is used to convert a differential equation from the time domain into the frequency domain. This makes it easier to solve the equation algebraically, rather than through traditional methods of integration.

What is the inverse Laplace transform?

The inverse Laplace transform is the opposite operation of the Laplace transform. It converts a function of complex frequency back into a function of time. This is useful for finding the original function after performing the Laplace transform.

What are some common applications of the Laplace transform?

The Laplace transform is commonly used in engineering and physics to analyze and solve systems described by differential equations. It is also used in signal processing, control systems, and circuit analysis. Additionally, it has applications in probability and statistics, such as in the Laplace distribution.

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