How to Calculate the Factorial of a Noninteger

The usual factorial is defined for nonnegative integers by

$$n! = 1\cdot 2\cdot 3\cdots n.$$

That product does not directly make sense when $n$ is, for example, $1/2$ or $2.3$. The standard extension is the Gamma function:

$$x! = \Gamma(x+1).$$

This definition agrees with ordinary factorials and supplies a smooth value for most noninteger inputs.

The Gamma function definition

For $z>0$, the Gamma function can be defined by the improper integral

$$\Gamma(z) = \int_0^\infty t^{z-1}e^{-t}\,dt.$$

Therefore, for a positive real number $x$,

$$x! = \Gamma(x+1) = \int_0^\infty t^x e^{-t}\,dt.$$

The Gamma function satisfies the recurrence relation

$$\Gamma(z+1)=z\Gamma(z).$$

Since $\Gamma(1)=1$, this gives

$$\Gamma(n+1)=n(n-1)\cdots 2\cdot 1\,\Gamma(1)=n!$$

for every nonnegative integer $n$. Thus the Gamma function is not a different answer for integer factorials; it is the standard extension of the same operation.

Worked example: the factorial of one-half

Set $x=1/2$. Then

$$\left(\frac12\right)! = \Gamma\left(\frac32\right).$$

Use the recurrence relation:

$$\begin{aligned} \Gamma\left(\frac32\right) &=\frac12\Gamma\left(\frac12\right)\\ &=\frac12\sqrt{\pi}. \end{aligned}$$

The identity $\Gamma(1/2)=\sqrt{\pi}$ follows from the Gaussian integral. Hence

$$\boxed{\left(\frac12\right)! = \frac{\sqrt{\pi}}{2} \approx 0.8862269255.}$$

Notice that this value is not obtained by trying to multiply all numbers from $1$ to $1/2$. The Gamma function supplies the meaningful extension.

Worked example: the factorial of 3.5

For $x=3.5=7/2$,

$$3.5! = \Gamma(4.5)=\Gamma\left(\frac92\right).$$

Apply the recurrence repeatedly:

$$\begin{aligned} \Gamma\left(\frac92\right) &=\frac72\Gamma\left(\frac72\right)\\ &=\frac72\cdot\frac52\Gamma\left(\frac52\right)\\ &=\frac72\cdot\frac52\cdot\frac32\Gamma\left(\frac32\right)\\ &=\frac72\cdot\frac52\cdot\frac32\cdot\frac12\Gamma\left(\frac12\right)\\ &=\frac{105}{16}\sqrt{\pi}. \end{aligned}$$

Therefore,

$$\boxed{3.5! = \frac{105\sqrt{\pi}}{16} \approx 11.6317283966.}$$

More generally, for a nonnegative integer $n$,

$$\left(n+\frac12\right)! = \Gamma\left(n+\frac32\right) =\frac{(2n+1)!!}{2^{n+1}}\sqrt{\pi},$$

where $(2n+1)!!=1\cdot3\cdot5\cdots(2n+1)$ is an odd double factorial.

Calculating an arbitrary decimal factorial

For a value such as $2.3$, there is usually no simple expression involving elementary functions. Use a numerical implementation of the Gamma function:

$$2.3! = \Gamma(3.3) \approx 2.6834373819.$$

Scientific calculators may have a Gamma function button, often written as Γ, gamma, or GAMMA. In a programming language, use its standard special-function library. For example, Python’s math.gamma gives the generalized factorial directly:

import math

x = 2.3
print(math.gamma(x + 1))

This prints approximately 2.6834373819.

For large arguments, computing the factorial itself can overflow ordinary numerical types. In that situation, use the logarithmic Gamma function instead:

$$\log(x!)=\log\Gamma(x+1).$$

In Python, this is available as math.lgamma(x + 1).

What happens for negative nonintegers?

The Gamma function extends the factorial beyond positive numbers as well. For example,

$$\left(-\frac12\right)! = \Gamma\left(\frac12\right)=\sqrt{\pi}.$$

However, $\Gamma(z)$ has poles at $z=0,-1,-2,\ldots$. Since $x!=\Gamma(x+1)$, the generalized factorial is undefined when

$$x+1=0,-1,-2,\ldots,$$

that is, when $x=-1,-2,-3,\ldots$. Thus $(-1)!$, $(-2)!$, and so on are undefined, while negative nonintegers such as $(-1/2)!$ can have finite values.

The value $0!$ is not a problem: it is

$$0!=\Gamma(1)=1.$$

Why use the Gamma function?

Many functions can be made to agree with the integer factorial values, so merely requiring a function to satisfy $f(n)=n!$ at integers does not uniquely determine an extension. The Gamma function is the conventional choice because it is smooth, satisfies the factorial recurrence, has the integral representation above, and is log-convex on the positive real numbers.

This convention is used in probability, statistics, calculus, and combinatorics. For example, continuous extensions of binomial coefficients use

$$\binom{x}{k}=\frac{\Gamma(x+1)}{\Gamma(k+1)\Gamma(x-k+1)},$$

whenever the Gamma terms are defined.

Common mistakes

• Writing $x!=x\cdot(x-1)\cdots1$ for a noninteger. This finite product only naturally terminates when $x$ is a nonnegative integer. Use $\Gamma(x+1)$ instead.
• Using $\Gamma(x)$ instead of $\Gamma(x+1)$. The Gamma function shifts the factorial by one: $n!=\Gamma(n+1)$, not $\Gamma(n)$.
• Assuming all negative inputs are allowed. Negative nonintegers may be valid, but negative integers are poles and have no finite factorial value.
• Expecting an elementary closed form for every decimal. Values such as $2.3!$ generally require numerical evaluation of the Gamma function.