Understanding the Laplace Transform Property: L{K f(t)} = K F(s)

The question asks about a specific property of the Laplace transform demonstrated by the equation \(L\{K f(t)\} = K F(s)\). Here, \(L\) denotes the Laplace transform operator, \(f(t)\) is a function of time \(t\), \(K\) is a constant, and \(F(s)\) is the Laplace transform of \(f(t)\), i.e., \(F(s) = L\{f(t)\}\).

This equation shows that taking the Laplace transform of a constant multiplied by a function is equivalent to multiplying the Laplace transform of the function by that same constant.

Analyzing the Given Laplace Transform Property

Let's look at the standard definitions of the options provided to see which one matches the given property:

• Shifting theorem: There are typically two shifting theorems. The frequency shifting theorem is \(L\{e^{at}f(t)\} = F(s-a)\). The time shifting theorem is \(L\{f(t-a)u(t-a)\} = e^{-as}F(s)\), where \(u(t-a)\) is the unit step function. Neither of these matches the given equation \(L\{K f(t)\} = K F(s)\).
• Linearity: The linearity property of the Laplace transform states that for constants \(a\) and \(b\), and functions \(f(t)\) and \(g(t)\), \(L\{a f(t) + b g(t)\} = a L\{f(t)\} + b L\{g(t)\} = a F(s) + b G(s)\). The given equation \(L\{K f(t)\} = K F(s)\) is a specific case of this linearity property, where we have only one term \(K f(t)\) (or \(a=K, b=0\)). The linearity property has two parts: additivity \(L\{f(t) + g(t)\} = L\{f(t)\} + L\{g(t)\}\) and homogeneity \(L\{K f(t)\} = K L\{f(t)\}\). The given formula is exactly the homogeneity part of the linearity property.
• Scaling theorem: The term "scaling theorem" in Laplace transforms most commonly refers to the time scaling property, which states that \(L\{f(at)\} = \frac{1}{a}F(\frac{s}{a})\) for \(a > 0\). However, in some contexts, the property \(L\{K f(t)\} = K F(s)\) can also be referred to as a form of scaling, specifically scaling the function \(f(t)\) by a constant \(K\) and observing that the resulting Laplace transform \(F(s)\) is also scaled by the same constant \(K\).
• Distribution theorem: This is not a standard name for a Laplace transform property.

Relating the Property to Scaling Theorem

While the property \(L\{K f(t)\} = K F(s)\) is fundamentally part of the linearity property (specifically homogeneity), it involves scaling the function \(f(t)\) by a constant \(K\). The outcome is that the Laplace transform \(F(s)\) is also scaled by the same constant \(K\). In the context of the provided options, and given that "Scaling theorem" is listed, this property is being referred to as the Scaling theorem. This highlights how scaling a function in the time domain by a constant multiplier translates directly to scaling its Laplace transform in the frequency domain by the same constant multiplier.

Therefore, the property \(L\{K f(t)\} = K F(s)\) is identified as the Scaling theorem among the given options.