Laplace transform of a piecewise function, sect7.2#11

blackpenredpen

13 Apr 201709:20

EducationalLearning

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TLDRThe video script details a mathematical process to find the Laplace transform of a piecewise function defined by sin(T) when T is between 0 and π, and 0 when T is greater than π. The method involves setting up integrals from 0 to π and from π to infinity, integrating by parts using the u-substitution method, and solving the resulting equations. The final result is presented as a function of 's', with conditions for convergence and simplification steps explained, showcasing the application of calculus in solving Laplace transforms.

Takeaways

📚 The video is about finding the Laplace transform of a piecewise function defined by two conditions: sin(T) when T is between 0 and π, and 0 when T is greater than π.
🔍 The Laplace transform is applied using the integral definition, which involves breaking the integral into two parts based on the piecewise nature of the function.
📐 The first integral is from 0 to π and involves the function e^(-T) * sin(T), while the second integral, from π to infinity, is straightforwardly zero because the function is zero in this range.
🧩 The process of integration by parts is used to evaluate the integral of e^(-T) * sin(T), which requires differentiation of the exponential function and integration of the sine function.
🔄 Integration by parts is applied iteratively, resulting in a series of terms that involve exponential and trigonometric functions, as well as their derivatives.
⚖️ The final result of the integration is expressed in terms of s, the parameter of the Laplace transform, and involves terms like e^(-sT) and sin(T)/s.
🔢 The limits are then substituted into the result to evaluate the integrals, with special attention to the behavior of the function at the limits of integration.
📉 The second integral from π to infinity contributes nothing to the result, as the function is zero in this range, simplifying the evaluation.
📌 The video emphasizes the importance of correctly applying the limits of integration and the conditions of the piecewise function to the Laplace transform.
📝 The final expression for the Laplace transform is simplified by combining terms with a common denominator, resulting in a clear and concise formula.
🔑 The video concludes by highlighting that the improper integral part does not require a condition on s, making the result valid for all s values.

Q & A

What is the piecewise function given in the script?
-The piecewise function is defined as e^(-T) * sin(T) when T is between 0 and π, and 0 when T is greater than π.
What is the method used to find the Laplace transform of the given piecewise function?
-The Laplace transform is found by using the definition of the Laplace transform and breaking it into two integrals based on the piecewise nature of the function.
What are the limits of integration for the two parts of the Laplace transform?
-The first integral has limits from 0 to π, and the second integral has limits from π to infinity.
Why is the second integral considered to be 'super easy'?
-The second integral is easy because the function being integrated (0) times e^(-T) results in an integral of zero, which is straightforward to evaluate.
What technique is used to integrate e^(-T) * sin(T)?
-Integration by parts is used, with the choice of u as e^(-T) and dv as sin(T), and then applying the method repeatedly until the integral can be evaluated.
What is the role of the function's continuity in the Laplace transform calculation?
-The function's piecewise continuity allows for the application of the Laplace transform definition without issues, ensuring that the integral converges properly.
What is the role of the function's continuity in the Laplace transform calculation?
-The function's piecewise continuity allows for the application of the Laplace transform definition without issues, ensuring that the integral converges properly.

Outlines

00:00

📚 Calculating the Laplace Transform of a Piecewise Function

This paragraph discusses the process of finding the Laplace transform of a piecewise function defined by two different expressions depending on the value of the variable T. The first integral is calculated from 0 to π, with the function being -e^(-T)sin(T), while the second integral, from π to infinity, involves the function being 0, simplifying the process significantly. The speaker emphasizes the importance of recognizing the piecewise continuity of the function and the ease of solving the integrals due to the function's properties. The summary also touches on the application of integration by parts to find the integral of e^(-T)sin(T), detailing the steps involved in the integration process, including differentiation and integration of the function's components.

05:18

🔍 Evaluating the Integral for the Laplace Transform

In this paragraph, the focus is on evaluating the integral obtained from the Laplace transform calculation. The speaker simplifies the integral by recognizing that the second part, from π to infinity, contributes nothing due to the function being zero in that range. The evaluation of the first integral involves plugging in the limits and simplifying the expression, resulting in an expression involving e^(-πs) and cosine terms over a denominator of 1 plus s squared. The speaker also discusses the need to set a condition on the variable s to ensure the integral converges, but notes that for the given improper integral part, no such condition is necessary, making the result valid for all s values.

Mindmap

Keywords

💡Laplace Transform

The Laplace Transform is a mathematical technique used to convert a time-domain function into a complex frequency-domain function. It is a powerful tool in solving differential equations and analyzing linear time-invariant systems. In the video, the Laplace Transform is used to find the transform of a piecewise function, which is a key step in the analysis process.

💡Piecewise Function

A piecewise function is a mathematical function that is defined by different expressions for different intervals of its domain. In the video, the function in question changes its behavior at a specific point (T=π), which is why it is referred to as a piecewise function, and this characteristic is essential for the application of the Laplace Transform.

💡Integration by Parts

Integration by parts is a method in calculus for integrating a product of two functions, typically denoted as ∫udv, where u and dv are parts of the integrand. The video script describes using this method to integrate the product of e^(-T) and sine(T), which is a crucial step in finding the Laplace Transform of the given function.

💡Improper Integral

An improper integral is an integral that has an infinite limit of integration or an integrand that has vertical asymptotes within the interval of integration. In the script, the improper integral is mentioned in the context of integrating from π to infinity, where the function being integrated is zero for T > π, simplifying the calculation.

💡Sine Function

The sine function is a trigonometric function that describes a smooth, periodic oscillation. It is used in the integral within the Laplace Transform calculation, where sine(T) is the function being transformed along with e^(-T).

💡Exponential Function

The exponential function, often represented as e^x, is a mathematical function that grows (or decays) at a rate proportional to its current value. In the video, e^(-T) is the decaying exponential part of the integrand in the Laplace Transform of the sine function.

💡Continuous Function

A continuous function is one where the limit of the function as the input approaches a certain value is the same as the function's value at that point. The script mentions that the function is piecewise continuous, which means it is continuous except possibly at a finite number of points, which is important for the validity of the Laplace Transform.

💡Trigonometric Identity

Trigonometric identities are equations that involve trigonometric functions and are true for all permissible values of the variables. In the script, the identity for the derivative of sine is used (sine' = cosine), which is a fundamental identity in the process of integration by parts.

💡Differential Equation

A differential equation is an equation that relates a function with its derivatives. Although not explicitly mentioned in the script, the process of finding the Laplace Transform of a function is often used to solve differential equations, which is a broader context for the video's content.

💡Convergent Integral

A convergent integral is an integral that has a finite value. The script discusses setting a condition on 's' to ensure the integral converges, which is important for the existence of the Laplace Transform and the validity of the mathematical process.

💡Complex Frequency-Domain

The complex frequency-domain is a representation of signals or systems using complex exponentials, as opposed to the time-domain. The Laplace Transform maps time-domain functions into this domain, which is useful for analyzing the behavior of systems over frequency, as illustrated in the video.

Highlights

Introduction to finding the Laplace transform of a piecewise function.

Function definition: e^(-T) * sin(T) for T between 0 and π, and 0 for T > π.

Use of the Laplace transform definition to set up the integral.

Division of the integral into two parts: from 0 to π and from π to infinity.

Explanation of the integral's first part involving e^(-T) * sin(T).

Second part of the integral is straightforward due to the function being zero.

Mention of the piecewise continuous nature of the function.

Application of integration by parts to solve the integral.

Differentiation of e^(-T) and integration of sin(T) using the u-substitution method.

Iterative differentiation and integration steps leading to the integral solution.

Final expression of the integral involving e^(-T) * (cos(T) - sin(T)/T).

Plugging in the limits of integration for the first part from 0 to π.

Simplification of the second integral from π to infinity, which evaluates to zero.

Calculation of the Laplace transform with the given limits and function values.

Result of the Laplace transform involving e^(-πs)/(s^2 + 1).

Condition on s for the integral to converge, applicable to the improper integral part.

Conclusion that the result holds for all s values, indicating the integral's convergence.

Transcripts

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