Laplace transform of sqrt(t), without using the gamma function  Summary and Q&A
TL;DR
The Laplace transform of the square root of T is equal to the square root of pi divided by 2 multiplied by T to the power of 3/2, with the condition that s must be greater than 0.
Key Insights
 ✊ The Laplace transform of T to the power of 1/2 cannot be directly calculated using existing formulas for nonnegative whole numbers.
 The Laplace transform of 1 over the square root of T serves as a reference for deriving the Laplace transform of the square root of T.
 Multiplying the Laplace transform by T and differentiating with respect to s helps in deriving the Laplace transform of the square root of T.
Transcript
all right we're going to see the Laplace transform function square root of T and yes we can write group TS t to the 1/2 power but we can use the formula for the Laplace transform of T to the nth power because that formula it's only good if N is a nonnegative whole number here this is T to the 1/3 power it's a fraction right so we can now use that ... Read More
Questions & Answers
Q: How can we calculate the Laplace transform of the square root of T?
To calculate the Laplace transform of the square root of T, we can use the Laplace transform of 1 over the square root of T as a reference and apply a theorem that involves multiplying the Laplace transform by T.
Q: What is the result of the Laplace transform of 1 over the square root of T?
The Laplace transform of 1 over the square root of T is equal to the square root of pi divided by the square root of T, with the condition that s must be greater than 0.
Q: What is the theorem that allows us to derive the Laplace transform of the square root of T?
The theorem states that if you have the Laplace transform of a function, multiplying it by P (s) is equivalent to differentiating the function with respect to s once and multiplying by 1. This theorem can be used to derive the Laplace transform of the square root of T.
Q: What is the final expression for the Laplace transform of the square root of T?
The final expression is the square root of pi divided by 2 multiplied by T to the power of 3/2, with the condition that s must be greater than 0.
Summary & Key Takeaways

The Laplace transform of T to the power of 1/2 cannot be directly calculated using existing formulas for nonnegative whole numbers.

However, the Laplace transform of 1 over the square root of T can be used as a reference, as both functions involve the square root term.

By applying a theorem that involves multiplying the Laplace transform by T, the Laplace transform of the square root of T can be derived.

The resulting Laplace transform is equal to the square root of pi divided by 2 multiplied by T to the power of 3/2, with the condition that s must be greater than 0.