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> single mostly-analytic function I think by "mostly-analytic" here you mean meromorphic. One really cool thing about the gamma function is it is not the only
by tanderson92 11y ago
> single mostly-analytic function
I think by "mostly-analytic" here you mean meromorphic.
One really cool thing about the gamma function is it is not the only meromorphic function which satisfies that recurrence relation on the integers! E.g.: gamma(x) + sin(x*pi) is meromorphic and agrees with the factorial.
- Chinjut 11y agogamma(x) + sin(x * pi) agrees with the factorial [shifted by one, because the gamma function has that stupid shift in its definition for no good reason] on integer arguments, but does not satisfy the recurrence relation on non-integer arguments. That having been said, one can devise infinitely many analytic functions which do satisfy the recurrence relation in general. See my other comment.
- tanderson92 11y agoYes indeed, that's why I only said they agreed "on the integers". Your other example is deviously clever, nicely done.