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About 5 years ago, I did some research into factorials. I came across a problem online that asked me to solve x!=6. The answer was 3, of course, but when I tried to solve x!=3, I found no value of x that would work. It is a decimal, because I set the gamma function equal to three. I need assistance in solving the gamma part.

elipson_-1
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  • I was under the impression that Sterling's approximation to N-Factorial would do the trick, but maybe you mean something else. https://en.wikipedia.org/wiki/Stirling%27s_approximation – NoChance May 19 '19 at 04:26
  • Have a look at https://math.stackexchange.com/questions/3224592/inverse-of-the-gamma-function#comment6636192_3224592 . This is a very good approximation. – Claude Leibovici May 19 '19 at 05:13

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Use a language or system that has both gamma function for all real values and a root solver. Then find a root of $\Gamma(x) - n=0 $.

The Lanczos approximation (see https://en.wikipedia.org/wiki/Lanczos_approximation) can, combined with the reflection formula, allow the computation of the Gamma function at all points in the complex plane where it is defined.

I have done this in the past using HP calculators. By using an Aitken iteration, I was able to get results for complex $n$ as well as real.

It was fun finding $x$ for complex $n$.

Looking up "inverse factorial" gives interesting links like this:

https://arxiv.org/abs/1707.01734

Have fun.

marty cohen
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