9.1.3 - Proof
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Unit Step Function and Definition
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Today, we’ll start our discussion with the unit step function, also known as the Heaviside function. Can anyone describe what happens to the function at time t = a?
It jumps from 0 to 1!
That's correct! We define it as u(t-a), which equals 0 for t < a and 1 for t ≥ a. It’s often used to represent sudden changes in a system. Now, can someone tell me what happens when a = 0?
It simplifies to just u(t)!
Exactly! This basic step function is foundational for many applications in systems analysis. Remember the acronym U for 'Upswing at a' which helps to recall its behavior. Let's explore the next point!
Laplace Transform of the Unit Step Function
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Now, let’s take a look at how we calculate the Laplace Transform of the unit step function. The integral transform involves integrating u(t-a) * e^{-st}. Why do you think we change the limits of integration?
Because u(t-a) is zero for t less than a?
Exactly! That allows us to focus solely on t >= a. Therefore, we write it as such: we change the lower limit to a and calculate the integral. What’s the result we get?
It's e^{-as}/s!
Well done! So to summarize, we derive that the Laplace Transform of u(t-a) is e^{-as}/s for a ≥ 0. Remember the phrase 'Exponential decay over stability' when thinking of this transformation. Let's move on!
Applications and Graphical Representation
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Let’s talk about the applications of the unit step function, especially in control systems. Can anyone give me an example where we might encounter a step function?
In switching systems, like turning on a light.
Exactly! It models sudden inputs. When we represent this graphically, what do we see?
We see a flat line at 0 before a and then it jumps to 1!
Correct! That jump signifies a critical transition phase in many engineering applications. Remember, graphs help visualize functions effectively. Let's summarize!
Second Shifting Theorem
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Lastly, we discuss the second shifting theorem. When we multiply a function f(t) by u(t-a), how do we express its Laplace Transform?
It becomes e^{-as} times the Laplace Transform of f(t+a)!
Very well said! This is crucial for solving equations with discontinuities. Who can remind me why this property is significant?
It simplifies the solving of differential equations!
Absolutely! This is a key takeaway. The power of the Laplace transform lies in these properties. Always keep in mind 'Shift, then Solve!' Let's summarize everything we've covered.
Introduction & Overview
Read summaries of the section's main ideas at different levels of detail.
Quick Overview
Standard
In this section, we explore the unit step function (Heaviside function), its Laplace Transform, and its significance in modeling discontinuous systems. We provide a detailed proof of the Laplace Transform of the unit step function, discuss its applications in differential equations, and demonstrate how combined functions are transformed.
Detailed
Proof of Laplace Transform of Unit Step Function
The unit step function, also known as the Heaviside function, is crucial in engineering and applied mathematics for modeling discontinuities in systems. We define the unit step function as follows:
- Definition:
- $u(t-a) = 0$, if $t < a$
- $u(t-a) = 1$, if $t \geq a$
Therefore, when $a = 0$, it simplifies to $u(t)$.
2. Laplace Transform of Unit Step Function
The Laplace Transform, denoted by ℒ, is defined through an integral:
$$ℒ\{u(t-a)\} = \int_0^{∞} u(t-a)e^{-st} dt$$
Because $u(t-a) = 0$ for $t < a$, we can adjust the limits giving us:
$$ℒ\{u(t-a) \} = e^{-as} \int_{a}^{∞} e^{-st} dt$$
Thus, we arrive at the standard result:
$$\frac{e^{-as}}{s},\text{ for } a \geq 0$$
3. Applications and Properties
The second shifting theorem states that for a function $f(t)$ multiplied by $u(t-a)$,
$$ℒ\{f(t)u(t-a)\} = e^{-as} \cdot ℒ\{f(t+a)\}$$
This is essential for solving differential equations involving discontinuous functions.
The graphical representation shows a constant value before $t = a$ and a jump at $t = a$, which is pivotal in modeling systems like switching circuits or sudden mechanical forces.
Summary
- The unit step function models discontinuities in systems.
- The derived Laplace Transform is $\frac{e^{-as}}{s}$.
- This function simplifies the solving of piecewise differential equations and aids in control theory applications.
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Laplace Transform Definition
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Chapter Content
Using the Laplace Transform definition:
ℒ{𝑢(𝑡−𝑎)} = ∫ 𝑢(𝑡−𝑎)𝑒^{−𝑠𝑡} 𝑑𝑡 from 0 to ∞
Detailed Explanation
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.
Examples & Analogies
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.
Changing the Limit of Integration
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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
Detailed Explanation
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.
Examples & Analogies
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.
Calculating the Integral
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Therefore:
ℒ{𝑢(𝑡−𝑎)} = ∫{𝑎}^{∞} 𝑒^{−𝑠𝑡} 𝑑𝑡 = [ − rac{1}{s} e^{−𝑠𝑡}]{𝑎}^{∞} = 0 + rac{1}{s} e^{−𝑎𝑠}
Detailed Explanation
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.
Examples & Analogies
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.
Key Concepts
-
Unit Step Function: A function that represents a sudden change from 0 to 1.
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Laplace Transform: A technique for converting time-domain functions into frequency-domain representations.
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Heaviside Function: A name for the unit step function that is used in engineering fields.
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Graphical Representation: Visualizing the unit step function as a flat line that jumps at t = a.
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Second Shifting Theorem: A property allowing transformation of delayed inputs in Laplace Transform.
Examples & Applications
Laplace Transform of u(t-3) is e^{-3s}/s.
Using the second shifting theorem, the Laplace of (t-2)u(t-2) becomes e^{-2s} * (1/s^2).
Memory Aids
Interactive tools to help you remember key concepts
Rhymes
When time meets a, the step shoots high, from zero to one without a goodbye.
Stories
Imagine a light switch, it’s off until someone flips it. That flipping moment represents the transition of the unit step function at time a.
Memory Tools
Use 'USS' for 'Unit Step Shifts' to remember the stepping behavior of the function.
Acronyms
UFTS - Unit Function Transform Shift
recalling the transform property of unit step functions.
Flash Cards
Glossary
- Unit Step Function
A function that is 0 for time less than a specified a and 1 for time greater than or equal to a.
- Laplace Transform
An integral transform that converts a function from the time domain to the complex frequency domain.
- Heaviside Function
Another term for the unit step function, commonly used in control theory.
- Second Shifting Theorem
A theorem used to compute the Laplace Transform of functions multiplied by unit step functions.
- Discontinuity
A point at which a mathematical function is not continuous, such as at a transition from 0 to 1.
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