I recently discovered that factorials of positive non-integers are definable (according to http://web2.0calc.com ) I see that the pattern for non-integer x! when 1<x works like this: 2.5! = 2.5 • 1.5 • (0.5!) 4.75! = 4.75 • 3.75 • 2.75 • 1.75 • (0.75!) You get the idea. But how are factorials of numbers between zero and one defined? ( 0.5! and 0.75! in the examples) Is the calculator giving me values for an approximate curve fit, or is there a logical mathematical definition? Here are some rules I discovered that may help give clues (or might just be numerical conventions used by the calculator programers). x! is undefined if x is negative x! is undefined if x is complex (real numbers + imaginary numbers) (x•i)! = (x)! if x is real and non-negative Now, as we know, 0!=1 and 1!=1, and according to the calculator, x!<1 if 0<x<1 So I tried to find the value for x that would give me the lowest possible value for x! I got 0.4365094! = 0.87617767556448 The resolution on my calculator ran out, so I couldn't look for any more decimal places (0.4365095! gives me the same thing), but is the actual lowest value rational or irrational? and does it hold any special significance?
You're getting confused - factorials are only defined for the natural numbers (positive integers and zero). What you are thinking of is the gamma function. For the natural numbers the conventional relationship between the gamma function and the factorial is \( \Gamma(n) = (n - 1)!\). The gamma function is defined in the whole complex plane via the integral \( \Gamma(z) = \int_0^\infty dt t^{z-1} e^{-t}\). You can also show that \( \Gamma(z + 1) = z \Gamma(z)\), as you would have expected from the factorials.
Your web calculator is assuming \(x! \; = \; \Gamma(1 + x ) \; = \; x \Gamma( x ) \; = \; x \int_0^{\infty} e^{-t} t^{x-1} dt \; = \; \int_0^{\infty} e^{-t} t^{x} dt\) Indeed this Gamma function is the most frequently used smooth function that has the property \(\Gamma(x+1) = x \Gamma(x)\). http://en.wikipedia.org/wiki/Gamma_function The minimum value of x! (actually \(\Gamma(1+x)\) as prometheus says above) is closer to 0.461632144968362341262659542325721328468196204006446351295988409 and has value closer to 0.8856031944108887002788159005825887332079515336699034488712001659. So your web calculator is not very accurate or precise. A better web calculator: http://www.wolframalpha.com/input/?i=minimum Gamma(1+x) near x = 0.46
I have confirmed that not-only is the WebCalc 2.0 factorial function NOT a very good approximation of \(\Gamma(1+x)\), it is not self-consistent in that special code is used when x is an integer or half-integer. As a consequence, the curve is discontinuous near integers and half-integers \( \begin{array}{r|lll} x & \textrm{WebCalc factorial} \quad & \quad 1 + \frac{ x \ln x}{12} + \left( \frac{11}{12} - \gamma \right) (x^2 - x) & \quad \quad \Gamma(1+x) \equiv x! \\ \hline \\ \hline \\ 0 & 1 & 1 & 1 \\ 0.0001 & 1.0225890285074 & 0.999889305458- & 0.999942288323+ \\ \hline \\ 0.4999 & 0.87776770606149 & 0.886253564156+ & 0.886223695761+ \\ 0.5 & 0.88622692545276 & 0.886256117035386+ & \sqrt{\frac{\pi}{4}} \approx 0.886226925452758+ \\ 0.5001 & 0.87777768683183 & 0.886258678370+ & 0.866230163441- \\ \hline \\ 0.9999 & 0.99597912459293 & 0.999957725378- & 0.999957725685- \\ 1 & 1 & 1 & 1 \end{array}\)
Thank you both for your replies. (I mistakenly thought my notifications were turned on so I didn't notice them until now.) Here's a followup question: I saw from the Wikipedia article on Gamma that the function has a localized minimum at Can we find instances where this value for X min recurs in nature (similar to how other irrational numbers like the golden ratio, tao (2Pi), e, the square root of two, etc. do)? I can't think of patterns in nature where the Gamma function is present, so I wouldn't know where to start looking.
