1.3. First Shifting Theorem (Laplace Domain Shift)

If ℒ{𝑓(𝑡)}= 𝐹(𝑠), then ℒ{𝑒𝑎𝑡𝑓(𝑡)}= 𝐹(𝑠−𝑎), where 𝑎 ∈ ℝ, 𝑠 > 𝑎, and ℒ denotes the Laplace Transform.

This statement introduces the First Shifting Theorem and establishes the relationship between the time-domain function and its Laplace transform when multiplied by an exponential term. It says that if we have the Laplace transform of 𝑓(𝑡), denoted as 𝐹(𝑠), then multiplying this function by an exponential term 𝑒^{𝑎𝑡} leads to the Laplace transform being shifted from 𝑠 to 𝑠−𝑎. The conditions state that 𝑎 can be any real number and that the value of 𝑠 must be greater than 𝑎 for the transformation to be valid.

Think of this theorem as adjusting the starting point on a timeline. If you're tracking the growth of a population where growth is multiplied by an exponential factor, moving the timeline to the right (shifting) makes it easier to analyze the population growth over time accurately.

Multiplying a time-domain function 𝑓(𝑡) by an exponential term 𝑒𝑎𝑡 results in a horizontal shift in the Laplace domain—shifting 𝑠 to 𝑠−𝑎.

This chunk elaborates on the meaning of the theorem. When we multiply a function in the time domain by an exponential term, it modifies the function's behavior, and in the Laplace domain, this change is represented as a shift. Instead of working with the original 's' variable, we use a new variable 's−𝑎', making it easier to compute the transform for functions that include exponential growth or decay.

Imagine a car moving on a straight road—if you want to consider how far it is from a different starting point (due to an obstacle), you shift your measure to account for that. This adjustment in the timeline reflects similar reasoning: we're moving our analysis point to get better insights into the system we're studying.

Let’s begin with the definition of the Laplace Transform: ℒ{𝑒^{𝑎𝑡}𝑓(𝑡)} = ∫ 𝑒^{−𝑠𝑡} 𝑒^{𝑎𝑡}𝑓(𝑡) 𝑑𝑡 Simplify the exponent: = ∫ 𝑒^{−(𝑠−𝑎)𝑡}𝑓(𝑡) 𝑑𝑡 = ℒ{𝑓(𝑡)} ext{ evaluated at } (𝑠−𝑎) = 𝐹(𝑠−𝑎) Hence, ℒ{𝑒^{𝑎𝑡}𝑓(𝑡)}= 𝐹(𝑠−𝑎} This completes the proof.

In this chunk, we provide the mathematical proof of the First Shifting Theorem. Starting with the definition of the Laplace transform, we replace the original function in the integral with the product of the time-domain function and an exponential. By simplifying the exponent, we manage to show how the transform can be related back to the original function evaluated at the shifted variable 's−𝑎'. This rigorous step demonstrates why the shift occurs.

Think of cooking a recipe where you double the ingredients to make a larger batch. Just as you adjust the measurements while keeping the recipe the same, when we apply the theorem, we're adjusting our 'ingredients' (s variable) to account for the modified function in the way we'd like to analyze it.

• Solving ODEs with exponential forcing functions • Modeling damping or growth in control systems • Electrical engineering problems with exponentials in circuit input signals • Mechanical systems involving exponentially decaying or growing forces

This section discusses the various practical applications of the First Shifting Theorem. It is particularly useful in solving ordinary differential equations (ODEs) that include exponential terms, which frequently arise in modeling physical systems. For instance, in control systems, we often encounter scenarios where the damping or growth of a response can be analyzed effectively using this theorem. Its application spans electrical and mechanical engineering, where systems with exponential signals or forces are common.

Imagine trying to predict the future behavior of a swinging pendulum or a spinning top—their motions often involve exponential changes. Applying this theorem helps engineers or scientists predict when and how these motions will change, akin to having a weather forecast that helps prepare for different conditions.

🟩 Example 1: Let 𝑓(𝑡) = sin(𝑏𝑡), ℒ{sin(𝑏𝑡)} =

s² + b² Then, ℒ{𝑒^{𝑎𝑡}sin(𝑏𝑡)} = (𝑠−𝑎)² + b² 🟩 Example 2: Let 𝑓(𝑡) = 𝑡, ℒ{𝑡} =

s² Then, ℒ{𝑒^{2𝑡} ⋅ 𝑡} = (𝑠−2)² 🟩 Example 3: Find the Laplace Transform of 𝑒^{−3𝑡}cos(4𝑡) We know: ℒ{cos(4𝑡)} =

s² + 16 Using the First Shifting Theorem: ℒ{𝑒^{−3𝑡}cos(4𝑡)} = (𝑠 + 3)² + 16

In this section, we look at specific instances where we apply the First Shifting Theorem to different functions. The examples illustrate how the theorem simplifies the computation of Laplace transforms for functions multiplied by an exponential. Each example showcases a different function, helping to solidify understanding through practical application.

These examples are like different puzzles where we adjust how we look at the pieces. Each time, we shift our perspective (our s variable) based on the type of puzzle (function) we are trying to solve, revealing a clearer solution that fits into the larger picture of understanding systems behavior.

• Confusing signs: If you have 𝑒^{−𝑎𝑡}, remember to shift right, not left. So 𝑠 → 𝑠 + 𝑎, not 𝑠 − 𝑎. • Not checking conditions: Make sure 𝑠 > Re(𝑎) for convergence. • Forgetting the base transform: Always start with a known ℒ{𝑓(𝑡)}= 𝐹(𝑠) before applying the theorem.

In this section, we highlight common pitfalls students may encounter while applying the theorem. Misunderstanding the sign changes can lead to incorrect transformations. It's also crucial to ensure that the conditions for convergence are met and to remember to start with the known Laplace transform of the function before applying the theorem.

Think of these mistakes as navigating a road trip. If you take a wrong turn (confusing signs), it can lead you off course. Similarly, if you forget to check your fuel (conditions for convergence), you might be stranded. Always begin your journey (with the base transform) to ensure a successful trip across the landscape of Laplace transforms.

Aspect Description Theorem ℒ{𝑒^{𝑎𝑡}𝑓(𝑡)}= 𝐹(𝑠− 𝑎) Interpretation Multiplying by 𝑒^{𝑎𝑡} causes a shift in 𝑠 Application Useful in ODEs, systems with exponential inputs Formula Shortcut Replace 𝑠 with 𝑠−𝑎 in the Laplace domain Conditions 𝑓(𝑡) should have Laplace Transform and 𝑠 > 𝑎.

This final chunk provides a concise summary of the key aspects of the First Shifting Theorem. It reiterates the theorem's statement, its interpretation concerning Laplace transforms, its application in solving equations involving exponential inputs, a formula shortcut to simplify calculations, and the conditions that need to be satisfied for its application.

Consider this summary as a cheat sheet for exam prep. Just like you would create a quick reference guide to remember essential formulas and concepts, this summarizes everything you need to know about the First Shifting Theorem, making it easier to recall during problem-solving sessions.

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