Gamma function for mathphys course

[relatively_me]

The given problem is this:

Solve using the gamma function

$\int_0^{\infty}\sqrt{x}\exp{^{-x}}{ dx}$My problem is that I don't know how to use the gamma function. It doesn't make sense to me...any insight would be helpful.

Thanks in advance

 

[relatively_me]

I don't know what's wrong with my code...the upper limit should be

$\infty$

 

[Physics Monkey]

The gamma function can be defined as
$$\Gamma(z) = \int^\infty_0 x^{z-1} e^{-x} \, dx,$$
so find the value z that makes this integral look like yours. I assume this is what is meant by "solve using the gamma function". You can then look up the result in a table.

 

[shmoe]

You might want to review what you do know about the gamma function.

You should be able to write that integral as Gamma(s) for some value of s. What properties of Gamma do you know? Have you evaluated Gamma at any non-integral points before?

 

[benorin]

Hint: $$\int_0^{\infty}\sqrt{x}\exp{^{-x}}{ dx}=\int_0^{\infty}x^{\frac{3}{2}-1}\exp{^{-x}}{ dx}$$

 

[relatively_me]

Well, if z=3/2, then, according to the table I found,

$$\Gamma(\frac{3}{2}) = \int_0^{\infty}x^{\frac{3}{2}-1}\exp{^{-x}}{ dx} = 8.386226 \times 10^{-1}$$

 

[relatively_me]

Thanks for your help...much appreciated...

 

[shmoe]

No need for tables, you can find $$\Gamma(3/2)$$ exactly with the relation:

$$\Gamma(s)\Gamma(1-s)=\frac{\pi}{\sin \pi s}$$

and the functional equation of gamma