Pi factorial

[Reaper]

Anybody here know how to compute pi factorial? Pi isn't an integer, or a rational number, so I'm at a loss at how to do it....

[timo]

Why would you want to calculate something for something when the something isn't defined for the other something? You could just define it as some thing but something tells me you are probably looking for something like the Gamma function.

[ajb]

You need to use the Gamma function as Atheist says. However, the expression is of course irrational.

 

[math]\Gamma[\pi] = \int_{0}^{\infty}dt \:t^{\pi -1 } e^{-t}[/math]

 

is one definition .

 

The problem now becomes evaluating the above integral.

 

As a very rough guess, you can set [math]\pi[/math] to 3, and then you can easily evaluate the integral and you see that [math]\Gamma[\pi] \simeq 2[/math]. Which gives you the order of magnitude. I can give a more accurate answer later. But you have a go first.

[the tree]

As a very rough guess, you can set [math]\pi[/math] to 3, and then you can easily evaluate the integral and you see that [math]\Gamma[\pi] \simeq 2[/math]. Which gives you the order of magnitude. I can give a more accurate answer later. But you have a go first.

Well it's the case that [math]\Gamma(n+1)=n![/math] so he'd perhaps be looking for something like [math]\Gamma(\pi + 1)\approx 7.188[/math] which at the very least falls between (3!)=6 and (4!)=24 although I couldn't say what use that is. It's really a pretty odd question.

[ajb]

Well it's the case that [math]\Gamma(n+1)=n![/math] so he'd perhaps be looking for something like [math]\Gamma(\pi + 1)\approx 7.188[/math] which at the very least falls between (3!)=6 and (4!)=24 although I couldn't say what use that is. It's really a pretty odd question.

 

Yes, of course.

 

I also don't really understand why you would want to define [math]\pi ![/math] . I mean, it isn't going to crop up as a combinatorial factor.

 

What is true, as you have said is that the gamma function allows you to define factorials for complex numbers in general.

 

Below I have posted a plot of [math]\Gamma[x+1][/math]