- Mathematical function, suitable for both symbolic and numeric manipulation.
- Hyperfactorial is defined as for positive integers .
- Hyperfactorial is defined as for positive integers and is otherwise defined as .
- The hyperfactorial function satisfies .
- Hyperfactorial can be evaluated to arbitrary numerical precision.
- Hyperfactorial automatically threads over lists.
Examplesopen allclose all
Basic Examples (5)
Plot over a subset of the reals:
Plot over a subset of the complexes:
Series expansion at the origin:
Series expansion at Infinity:
Numerical Evaluation (4)
Evaluate to high precision:
The precision of the output tracks the precision of the input:
Complex number inputs:
Evaluate efficiently at high precision:
Specific Values (4)
Values at fixed points:
Value at zero:
For a simple parameter, Hyperfactorial gives exact values:
Find the positive minimum:
Plot the Hyperfactorial function:
Plot the real part of :
Plot the imaginary part of :
First derivative with respect to n:
Higher derivatives with respect to n:
Plot the higher derivatives with respect to n:
Series Expansions (3)
Find the Taylor expansion using Series:
Plots of the first three approximations around :
Find the series expansion at Infinity:
Taylor expansion at a generic point:
The discriminant of the Hermite polynomial is related to the hyperfactorial:
Properties & Relations (2)