IA (HL) Need assistance Math IA

[Kairking]

I´m currently doing my IA about the growth of Factorials. I´m was very passionate about my investigation until I handed a preview to my teacher. I have just shown the development of the factorials and that there is not a term that defines the growth (such as geometric or arithmetic) but I don´t know the application, and meaning, of what I´ve found nor a good conclusion. I started to question the mathematical complexity in it, as it´s quite simple and I believe that it needs to be more complex.

I´m aware factorials are used in combinations and permutations, but I don´t have an "aim" for the factorials in those terms.

Does anyone have suggestions on what I can focus my findings on? Or you guys think it´s to simple and it´s better to change topic?

[emmapatregnani]

You could use permutation and combination and apply it to game theory like poker or billiard.

[maroctam]

I personally don't understand it but I know that factorials are only "meant" to be possible for discrete values of n, but they have been generalised to all n within real numbers (try typing 1.5! on wolframalpha for example). Maybe you can explore how this was done?

[Guest Skyrior]

^Good idea.

We are treading in the field of complex analysis here. The gamma function defines (n-1)! You can prove the functional equation of the gamma function to obtain around a 5/6 for Criterion E. Proving the functional equation of the gamma function requires at most up to integrating by part.

That will give you answers such as 7.3! etc.

To further extend the possibilities, consider (2+3i)!

To do so, we can use the Riemann zeta function and Euler constant. Take note though that proving the functional equation of Riemann zeta function is not only insanely hard, but probably even beyond undergraduate level. I would suggest you not to tackle it. That said, playing around with the idea of complex plane factorial yields many interesting possibilites. Give it a shot.