Integration of exponential function times polynomial of fractional degree

andyyee

1. The problem statement, all variables and given/known data

I'm working out a differential equation problem that I am supposed to solve with the formula [itex]\mathcal{L}\{t^\alpha\} = \frac{\Gamma{(\alpha + 1)}}{s^{\alpha+1}}[/itex]. The problem is [itex]\mathcal{L}\{t^{\frac{1}{2}}\}[/itex] (finding the Laplace transform of the given function).

2. Relevant equations

[itex]\mathcal{L}\{t^\alpha\} = \frac{\Gamma(\alpha + 1)}{s^{\alpha+1}}, \alpha > -1[/itex]

[itex]\Gamma(\alpha) = \int^\infty_0{t^{\alpha-1}e^{-t}dt}, \alpha > 0[/itex]

3. The attempt at a solution

I plug it into the equation, and get [itex]\frac{\Gamma(\frac{3}{2})}{s^\frac{3}{2}} = [/itex] [itex]\frac{\int^\infty_0 {t^\frac{1}{2}e^{-t}dt}}{s^{3/2}}[/itex]. That's where I run into a problem, I have no idea how to solve that integral. I can't use integration by parts, because one term will never disappear or the original integral will not appear again as [itex]\int{vdu}[/itex], so that won't work. I looked it up on Wolfram|Alpha, and it gave me [itex]\frac{1}{2}\sqrt{\pi}\text{erf}{(\sqrt{t})} - e^{-t}\sqrt{t}[/itex] for the indefinite form and [itex]\frac{\sqrt{\pi}}{2}[/itex] for the definite form. It also cited some stuff about integrating the normal distribution and error form, but I don't understand what it is talking about. What I am unsure about is the steps involved in solving the integral, and is there a generalized solution (for the definite integral from 0 to [itex]\infty[/itex]) for other coefficient values in the power of the polynomial?

Thanks
Andrew

LCKurtz

But you don't have to do the integral. You have the LaPlace transform from your formula;

[tex]\frac{\Gamma(\frac 3 2)}{s^\frac 3 2}[/tex]

[itex]\Gamma(3/2)[/itex] is just a constant. Look in your text and see if it doesn't give you [itex]\Gamma(1/2)[/itex] and a relationship between the Gamma value at 3/2 and 1/2.

andyyee

Thanks for the help :-). I completely overlooked the part in my book that mentioned an appendix, which happened to show some basic integration (for gamma of 1/2 using double integrals), as well as ways to solve similar gamma values without integration (simple algebra transformations using the already computed, given gamma of 1/2 and some other formulas). These forums are really great, because although Wolfram|Alpha gives me solutions, it does not tell me why (or to look closer at my book :-).

- Andrew