I need to evaluate the integral in the attached file can anyone help with this or more generally can someone help with solving (/obtaining an indefinate integral) the integral x^0.5Exp[-x/ab]. Thanks
Based on the integral in the document you are looking for an incomplete gamma function. I've only used one of these once so I can be of no further help. Best of luck!
-Dan
This is an excellent example of something you might find in a text book in a section entitled "Functions Defined by Integrals". Don't be discouraged. There are ways to "evaluate" such things. It depends on what it is you are doing. So, why do you need to "evaluate" it and what do you mean by "evaluate"?
Hi thanks for your helpful replies sorry for the late response I am trying to work on other things as well. I need to calculate k for different ion - molecule systems (therefore polarizabilities) for a given E and T, this integral is given in a paper (S. C. Smith et al, International Journal of Mass Spectrometry and Ion Processes, 96 1990 77-96) with the specification that the integral can be easily numerically evaluated and reduces to Langevin k if the hard sphere diameter is small and/or energy (and effective temperature) low or conversly to the hard sphere (energy high and/or hard sphere diameter large). I have used Langevin and hard sphere derived k in the past so am familiar with these and no what order of magnitude I expect k to be. I have some contact with the corresponding author about other work so I may e-mail him after i've had a bit more of a go at it myself. Thanks
Assuming that $\displaystyle ab<0$ then using $\displaystyle t=-x/ab$ converts the integral into,
$\displaystyle -\sqrt{-ab}\int t^{1/2} e^{t} dt$
But,
$\displaystyle \int t^{1/2} e^t dt = \int 2t^{1/2} e^t \left( \frac{1}{2} t^{-1/2} \right) dt $
Let $\displaystyle \mu = t^{1/2}$ to get,
$\displaystyle \int 2\mu ^2 e^{\mu ^2} d \mu = \int \mu \cdot (2\mu e^{\mu ^2} ) d\mu$
Use integration by parts,
$\displaystyle \mu e^{\mu ^2 } - \int e^{\mu ^2 }du$
This is the Error Function,
$\displaystyle \mu e^{\mu ^2 } - \mbox{erf}(\mu) + C$