Laplace transform of 1/sqrt(t), *SPEED RUN*

TL;DR

The Laas transform of the function 1/t is equal to π√(s)/s, where s is a positive constant.

Transcript

hi I would like to go over the laass transform of the function one/ of t as you can see when T is equal to Z we have one / Z Nam we have a vertical asset when tal zero however this function still have a laas transform and to do this we are going to use the definition of laas transform namely we will have the integral from t isal 0 to T Infinity e t... Read More

Key Insights

• ❓ The Laas transform is a mathematical technique used to analyze functions and systems.
• 🎭 The Laas transform of the function 1/t involves making substitutions and performing integration.
• 🤩 The substitution u = st simplifies the integral and allows for the evaluation of the Laas transform.
• 😌 The value of the integral ∫ e^(w^2) dw is √π/2, which is crucial in finding the Laas transform of 1/t.
• 😃 The final result of the Laas transform of 1/t is expressed as π√(s)/s, where s is a positive constant.
• 👻 The Laas transform allows for the analysis of the frequency characteristics of a function.
• ❓ The analysis of the Laas transform involves understanding the properties of integrals and substitutions.

Questions & Answers

Q: What is the Laas transform of the function 1/t?

The Laas transform of 1/t is equal to π√(s)/s, where s is a positive constant. It can be obtained by performing substitutions and integration.

Q: How is the substitution u = st used in finding the Laas transform?

The substitution u = st is used to simplify the integration step in finding the Laas transform. It allows for the transformation of the integral from t to u, making it easier to evaluate.

Q: What happens when t is equal to zero or infinity in the Laas transform?

When t is equal to zero, the substitution results in u also being zero. Similarly, when t is infinity, u becomes infinity as well. These values need to be taken into account when evaluating the Laas transform integral.

Q: How is the value of the integral ∫ e^(w^2) dw calculated?

The integral ∫ e^(w^2) dw is a famous integral known as the Gaussian integral. Its value is equal to √π/2. This result is used in evaluating the Laas transform of 1/t.

Summary & Key Takeaways

• The Laas transform of the function 1/t can be found by using the definition of the Laas transform and making a substitution.

• The substitution involves letting u = st and using differentiation to solve for dt.

• After performing the necessary substitutions and transformations, the Laas transform of 1/t is expressed as π√(s)/s.