Factorial2

n!!

gives the double factorial of n.

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
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n!! is a product of even numbers for n even, and odd numbers for n odd.
Factorial2 can be evaluated to arbitrary numerical precision.
Factorial2 automatically threads over lists.
Factorial2 can be used with Interval and CenteredInterval objects. »

Examples

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Basic Examples (7)

Evaluate at integer values:

Evaluate at real values:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Series expansion at Infinity:

Series expansion at a singular point:

Scope (29)

Numerical Evaluation (5)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Complex number inputs:

Evaluate efficiently at high precision:

Factorial2 can be used with Interval and CenteredInterval objects:

Specific Values (3)

Values of Factorial2 at fixed points:

Values at zero:

Find the first positive maximum of Factorial2[x]:

Visualization (2)

Plot the Factorial2 function:

Plot the real part of :

Plot the imaginary part of :

Function Properties (10)

Real domain of the double factorial:

Complex domain:

Double factorial has the mirror property :

Factorial2 threads elementwise over lists:

Factorial2 is not an analytic function:

However, it is meromorphic:

Factorial2 is neither non-decreasing nor non-increasing:

Factorial2 is not injective:

Factorial2 is not surjective:

Factorial2 is neither non-negative nor non-positive:

Factorial2 has both singularity and discontinuity for z≤-2:

Factorial2 is neither convex nor concave:

Differentiation (2)

First derivative with respect to z:

Higher derivatives with respect to z:

Plot the higher derivatives with respect to z:

Series Expansions (4)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

Find the series expansion at Infinity:

Find series expansion for an arbitrary symbolic direction :

Taylor expansion at a generic point:

Function Identities and Simplifications (3)

Functional identities:

Recurrence relation:

Relationship between Factorial and Factorial2 on the integers:

Generalizations & Extensions (3)

Infinite arguments give symbolic results:

Series expansion at poles:

Series expansion at infinity (generalized Stirling approximation):

Applications (2)

Plot of the absolute value of Factorial2 in the complex plane:

Pi representation via infinite series in Factorial and Factorial2:

Calculate the first 30 digits of Pi using this series:

Compare with the Pi output:

Properties & Relations (8)

Use FunctionExpand to expand double factorial into Gamma function:

Use FullSimplify to simplify expressions involving double factorials:

Sums involving Factorial2:

Generating function:

Recover the original power series:

Products involving double factorial:

Factorial2 can be represented as a DifferenceRoot:

FindSequenceFunction can recognize the Factorial2 sequence:

The exponential generating function for Factorial2:

Possible Issues (3)

Large arguments can give results too large to be computed explicitly, even approximately:

Smaller values work:

Machine-number inputs can give high‐precision results:

To compute repeated factorial, use instead of :

Neat Examples (3)

Plot Factorial2 at infinity:

Find the numbers of digits 0 through 9 in 10000!!:

Plot the ratio of doubled factorials over double factorial:

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Factorial2

Details

Examples

Basic Examples (7)