Inverse factorials. What are they? How to calculate?

[ktoz]

Hi

I was trying to solve the equation (m + n)!/m! = v for "n" and found an online calculator here (http://www.hostsrv.com/webmab/app1/MSP/quickmath/02/pageGenerate?site=quickmath&s1=equations&s2=solve&s3=advanced#reply) which spit out the following as a solution:

n = factorial(-1)[vm!] - m

I Googled "inverse factorial" and searched on wolfram.com and wikipedia.com but didn't find a clear answer as to what inverse factorials are, much less how to calculate them.

Factorials are so simple to calculate (1 x 2 x 3 ... x n) do inverse factorials have a similarly simple method?

Thanks for any help

[HallsofIvy]

"Inverse factorial" is, of course, the inverse of the factorial functions: Since 1!= 1, factorial-1(1)= 1, 2!= 2 so factorial-1(2)= 2. 3!= 6 so factorial-1(6)= 3, etc. You will notice that there is no number n such that n!= 3, say, so factorial-1(3) is undefined. No, there is no simple formula- that's why the online calculator gave you the function in that form.

(0!= 1 also so the factorial has to be restricted to positive integers in order to have an inverse.)

[mathwonk]

presumably in general it would be the inverse of the gamma function.

[CompuChip]

Mathematica gives a warning
During evaluation of InverseFunction::ifun: Inverse function
are being used. Values may be lost for multivalued inverses. >>
and then proceeds to print Factorial^(-1)(n)

As it gives the same output when replacing Factorial by any (undefined) function f, I don't think there it knows a numerical inverse (let alone a closed form exists) and f^(-1) is just what it (or the online calculator) give in general when they don't know the inverse of a function.

Neither does it know an inverse of the \Gamma function, which doesn't really surprise me.

[J.D.B.]

The gamma inverse function is multivalued for a given number. So there are many different branches .
If you have mathematica you might plot the following code that i made to visualize this function:

G[u_] := 1/Gamma[u];

Manipulate[Quiet[ParametricPlot3D[{
Re[f[x + \[ImaginaryI] y]], Im[f[x + \[ImaginaryI] y]],
variabletype}, {x, xo, xf}, {y, yo, yf},
PlotStyle -> Opacity[.85], ColorFunction -> (Hue[#5/(2 Pi)] &),
ColorFunctionScaling -> False,
RegionFunction ->
Function[{x, y, z},
If[condition, 0 < Im[f[x + I*y]], 0 > Im[f[x + I*y]],
Im[f[x + I*y]] > -10^6]], BoxRatios -> {1, 1, 3}]]
, {{variabletype, x, ""}, {x -> "Real Part", y -> "Imaginary Part"}}
, {{f, Gamma, "Riemann surface of the inverse"}, { Gamma -> "gamma",
G -> "1/Gamma[z]"}}, {{xo, -6, "initial value of x"}, -6,
0}, {{xf, 4, "final value of x"}, 0,
6}, {{yo, -Pi, "initial value of y"}, -2 Pi,
0}, {{yf, Pi, "final value of y"}, 0,
2 Pi}, {{condition, "Show All",
"Complex Plane View"}, {True -> "Upper Part",
False -> "Lower Part", "Show all"}}]
It will give you a the continuation of gamma-1 function over the complex z plane.
It is possible to find the many different values of this function graphicaly.
On the attachement, you will see the 2 first plots are the real part , and the lasts are the imaginary part.

[J.D.B.]

In this link you also have invormation on this function that they call the Arcfactorial

http://en.citizendium.org/wiki/Factorial