i need help to understand how this function has been laplace transformed, though it doesn't seem very complicated

[tommycashmoney]

F(T) is the function i need to transform and i have already the solution. i understand the translation rule, the reason why the number e/2 is outside the brackets and that rad(pi) is the laplace-transformed function of t^-1/2. but i don't quite understand how the rules L(H(t)t^a, s) = gamma(a+1)/s^(a+1) has been applied. is anybody able to expand that passage and calculate it to show me how it works? or at least give me some detailed hint on how to solve it.

 

[Deleted member 4993]

View attachment 24082

F(T) is the function i need to transform and i have already the solution. i understand the translation rule, the reason why the number e/2 is outside the brackets and that rad(pi) is the laplace-transformed function of t^-1/2. but i don't quite understand how the rules L(H(t)t^a, s) = gamma(a+1)/s^(a+1) has been applied. is anybody able to expand that passage and calculate it to show me how it works? or at least give me some detailed hint on how to solve it.

Please define H(t)

 

[Romsek]

I'm going to guess that H(t) is the Heaviside function, sometimes known as the unit step.

[MATH]H(t) \begin{cases} 0 &t<0\\1 &0 \leq t \end{cases}[/MATH]

 

[Romsek]

See if this vid helps.

https://www.khanacademy.org/math/di...ce-transform/v/laplace-transform-of-t-n-l-t-n