1.5. Examples

Example 1: Laplace Transform of a Constant
𝑓(𝑡) = 1
ℒ{1} = ∫₀^∞ 𝑒⁻ˣ𝑡 ⋅ 1 𝑑𝑡 = [ ] = , for 𝑠 > 0
−𝑠
𝑠

In this example, we compute the Laplace Transform of a constant function, which is 1. The integral we need to evaluate is from 0 to infinity, where we multiply our constant function by the exponential decay term e^(-st). The resulting integral simplifies to 1/s, as we evaluate it from 0 to infinity under the condition that s is greater than 0. This reflects that the total contribution from the constant value over time results in a finite value in the s-domain.

Think of a steady light bulb that gives off a constant amount of light. If this light bulb is turned on for an indefinite amount of time, the total 'light output' in our analysis can be represented using the Laplace Transform, showing how that output translates into a measurable effect (brightness) in a different scenario (s-domain), where it equals 1/s.

Example 2: Laplace Transform of an Exponential Function
𝑓(𝑡) = 𝑒^{𝑎𝑡}
ℒ{𝑒^{𝑎𝑡}} = ∫₀^∞ 𝑒⁻ˣ𝑡 ⋅ 𝑒^{𝑎𝑡} 𝑑𝑡 = ∫₀^∞ 𝑒^{-(𝑠−𝑎)𝑡} 𝑑𝑡 = , for 𝑠 > 𝑎
𝑠−𝑎

In this example, we consider an exponential function, f(t) = e^(at). To compute its Laplace Transform, we set up the integral similarly, but now we have the exponential terms combined. This allows us to factor out e^(-(s-a)t), leading to a simpler form to integrate. When we evaluate this integral, we find that it converges to 1/(s-a) for s greater than a. This property of exponential functions shows how they decay and are transformed into manageable algebraic fractions.

Imagine a plant that grows exponentially at a certain rate a. If you want to measure its growth over time and summarize how much it contributes at different points, the Laplace Transform helps portray that continuous growth in a simplified manner, converting it into a ratio (1/(s-a)), which is easier to analyze than tracking the actual growth curves in real-time.

Example 3: Laplace Transform of 𝑡^n
𝑓(𝑡) = 𝑡^n, 𝑛 ∈ ℕ
ℒ{𝑡^𝑛}= ∫₀^∞ 𝑒⁻ˣ𝑡 𝑡^n 𝑑𝑡 = , for 𝑠 > 0
𝑛!

In this example, we calculate the Laplace Transform of a polynomial function, t^n, where n is a non-negative integer. The integral we evaluate involves multiplying the polynomial by the exponential decay term. Upon integration, this yields a result of n!/s^(n+1), demonstrating a pattern where higher powers lead to factorial growth in the numerator, which is important for understanding how different types of polynomial systems behave in the s-domain.

Think about measuring the effect of time on a project that involves multiple stages or tasks. If the time required for each stage is represented by different powers of time, the Laplace Transform helps in summarizing and transforming that complex structure into a straightforward equation (n!/s^(n+1)), enabling us to easily compute and predict the overall effect on the project's completion.

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