dVanDaHorns wrote:Taalit wrote:Basically what I was wondering is if there is a simple (that is, in terms of 'well-known' constants like e, pi, the euler-mascheroni constant, etc.) representation of values like Γ(1/5). The gamma function is easily solved for integers as it corresponds to a shifted factorial, but no closed forms are known for any fractional arguments of Γ(z) besides half-integers. What I am wondering is if expressions representing these values that way even exist.
Oh. Yeah, there haven't been any breakthroughs in that domain, just quite yet. They are strictly undefined irrational numbers...(although you might be able to approximate them with various defined irrational numbers, if you played around with them enough, but it's not really worth it, unfortunately.)
Gammas of fractions we would usually leave in their simplest form; that is, Γ(x), where x is the fraction you are trying to express.
Well, if I remember well, the formula for half integers, uses the "double factorial" divided by a power of two, with a factor of sqrt(pi) (a very logical way of extending factorials, if you ask me). So I guess you could define multiple factorials analog to the double factorial, and divide them by the power of the corresponding number. And as for the factor, well, probably it would be sqrt(2*pi/n) or something similar.
EDIT: actually, check this:
http://en.wikipedia.org/wiki/Particular ... _arguments
I only was wrong in the factor, it says that at least gamma(1/3) and gamma(1/4) are irrational trascendental constants on their own. It also suggests a formula for gamma(1/4). Now with millions of digits of either constant you can always try to guess some formula at home in your computer, inputting several combinations of constants