9.1.3. Proof

Using the Laplace Transform definition:

ℒ{𝑢(𝑡−𝑎)} = ∫ 𝑢(𝑡−𝑎)𝑒^{−𝑠𝑡} 𝑑𝑡 from 0 to ∞

To find the Laplace Transform of the unit step function, we start with its definition as an integral. The Laplace Transform, denoted as ℒ, transforms a function from the time domain into the frequency domain. For the unit step function u(t-a), we use the integral from 0 to infinity of the product of the step function and e raised to the power of -st, which is a common form in the Laplace Transform.

Think of the Laplace Transform like a translator turning a spoken language (time domain) into text (frequency domain). Just like how a translator needs to understand the context of the words, the Laplace Transform needs to process functions in a specific way to capture their essence in a new format.

Since 𝑢(𝑡−𝑎) = 0 for 𝑡 < 𝑎, the lower limit of the integral can be changed to 𝑎:

ℒ{𝑢(𝑡−𝑎)} = ∫_{𝑎}^{∞} 𝑒^{−𝑠𝑡} 𝑑𝑡

When we evaluate the integral for the Laplace Transform, we notice that the unit step function u(t-a) is zero for times before t = a. This means that any contributions to the integral from those times are negated, allowing us to limit our integration from a to infinity instead of from 0. This simplification is vital as it directly affects the form of our final result.

Imagine you are evaluating a business only after a significant event, like a company launch (which is time a). Before that launch, the company’s activities are minimal or non-existent (similar to the step function being zero). This focus on the relevant timeframe makes analyzing the company’s performance much easier.

Therefore:

ℒ{𝑢(𝑡−𝑎)} = ∫{𝑎}^{∞} 𝑒^{−𝑠𝑡} 𝑑𝑡 = [ −rac{1}{s} e^{−𝑠𝑡}]{𝑎}^{∞} = 0 + rac{1}{s} e^{−𝑎𝑠}

Now we calculate the integral. The integral of e^{-st} gives us -1/s e^{-st}. Evaluating this from a to infinity yields two parts: at infinity, this term approaches zero (since e^{-∞} equals 0). Hence we are left with the term evaluated at t = a, which gives us (1/s)e^{-as}. This is how we derive the standard result for the Laplace Transform of the unit step function.

Think of this process like filtering out unnecessary data. By evaluating our function over the specified interval (after time a), we extract only what matters, akin to how businesses analyze performances after campaigns or significant changes to see their real impact.

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