Laplace transform of a piecewise function, sect7.2#11
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
📚 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.
🔍 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
💡Piecewise Function
💡Integration by Parts
💡Improper Integral
💡Sine Function
💡Exponential Function
💡Continuous Function
💡Trigonometric Identity
💡Differential Equation
💡Convergent Integral
💡Complex Frequency-Domain
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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