Question

I'm trying to understand how to reduce Gamma functions for Laplace transforms, starting from Gamma(n/2)=sqrt(pi/2). More detail with formatting below:

Answer 1

\[\Gamma (\frac{5}{2})=(\frac{3}{2}) \Gamma (\frac{3}{2}) =(\frac{1}{2}) ( \frac{3}{2})\Gamma (\frac{1}{2}) =\frac{3 }{4} \sqrt{\pi }\]

Answer 2

It's the reduction of the formula above that's I'm not understanding.

Answer 3

Ah, I think I have it. Is it because of this? (See image).

Answer 4

\[\Gamma(z+1) = z\cdot \Gamma (z)\]

Answer 5

Which gives me this, right?

Answer 6

that looks right, [ i haven't practiced these in a while ]

Answer 7

for reducing the Gamma function, this form is used: \[\Gamma(z)=(z-1)\Gamma(z-1)\]

Answer 8

Ah, that'll be handy. Thank you, I'll try it on this next problem.

Answer 9

\[\sqrt\pi=\Gamma(1/2)=2\cdot \Gamma(3/2)=2\times\frac23\cdot \Gamma(5/2)=2\times \frac23\times\frac25\cdot\Gamma(7/2)=\cdots
\]

Answer 10

Excellent. We didn't do a lot of this in class, but I didn't want to be blindsided on the final tomorrow, either.

Answer 11

Thanks again.

Answer 12

ps wolfram agrees with your Laplace transform

https://www.wolframalpha.com/input/?i=Laplace+transform+%5Bt%5E%281%2F2%29%2B3t%5D

Answer 13

I love it when Wolfram agrees. :)

Answer 14

if you can remember only one of the forms of the gamma function,

make sure to manipulate it to be in a useful form before you try to use it,
because if you're trying to use is in backwards it gets confusing

Answer 15

Cool. Conveniently, my calculator can do them pretty easily. It should be pretty easy to check my work.