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Inverse and derivative for Factorial and Gamma #171
The polygamma functions, and the derivative of the factorial #2068: the polygamma functions polygamma(n, z) and digamma(z), evaluated to the working precision anywhere in the complex plane. The derivative of x! is x! * digamma(x + 1), so gamma(x) differentiates to (x - 1)! * digamma(x).
The inverse. There is no inverse of x! or Gamma among the functions a CAS writes in closed form. So x! = y is left as the set of x for which it holds, which is what the solver already does for an equation it cannot invert. I'd close this issue on that.
Closing as proposed above. Measured on master at 647aa81a: x! differentiates to x! * digamma(1 + x) and gamma(x) to (x - 1)! * digamma(x), and x! = y solves to { x : x! = y }.
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Decline a factorial's derivative instead of claiming it does not exist (
Decline a factorial's derivative instead of claiming it does not exist (
Name the polygamma functions in the MCP server's syntax resource
The polygamma functions, and the derivative of the factorial (#2068)
Integrate the polygamma functions, and give an odd order its exact va…
Rafael-SOWNet commentedon Oct 22, 2026
The derivative is done, in two PRs.
polygamma(n, z)anddigamma(z), evaluated to the working precision anywhere in the complex plane. The derivative ofx!isx! * digamma(x + 1), sogamma(x)differentiates to(x - 1)! * digamma(x).digamma(x)toln((x - 1)!). An odd order at a whole or half-whole argument is exact:polygamma(1, 1)ispi^2/6.The inverse. There is no inverse of
x!orGammaamong the functions a CAS writes in closed form. Sox! = yis left as the set ofxfor which it holds, which is what the solver already does for an equation it cannot invert. I'd close this issue on that.🤖 Generated with Claude Code
https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura
Rafael-SOWNet commentedon Oct 23, 2026
Closing as proposed above. Measured on master at
647aa81a:x!differentiates tox! * digamma(1 + x)andgamma(x)to(x - 1)! * digamma(x), andx! = ysolves to{ x : x! = y }.🤖 Generated with Claude Code
https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura