Ken Ward's Mathematics Pages

Series: How to plot the Gamma Function

Page Contents

1. The Basic Formulae
2. A Closed Formula For Γ(x)
3. Calculating Higher Values of gamma from Γ(x)
4. Calculating Lower Values of Gamma from Γ(x)

We seek to compute the values of gamma and produce a graph:

Note: The tables on this page are to illustrate calculating gamma values, not for reference. Although the gamma values are believed accurate to 5 decimals, they are often shown to 6 because they are meant to illustrate the use of the Stirling Approximation.

The Basic Formulae

We are seeking a formula for gamma in terms of factorials. We wish to use Stirling's approximation to calculated the factorials, so we want to be able to use large factorials.

We will start with the following formulae, which have been derived elsewhere.

[1.01] (from this page: BinomialCoefficientsGamma.htm#[2.07] )

  [1.01b] (from this page: BinomialCoefficientsGamma.htm#2.08)

[1.02] (derived from: BinomialCoefficientsGamma.htm#[4.01] )

A Closed Formula For Γ(x)

When we have no values for gamma, we need to calculate them using a closed formula.

By writing x=x+n in [1.01], we obtain:

[1.03]

By recursively applying [1.01], we obtain:

[1.04]

Because Γ(n+x+1)=(x+n)!, we can write [1.04] as:

[1.05]

We therefore have a closed formula for gamma. This is the same formula that was derived here. In the formula, we also have an arbitrary n which we can make sufficiently large to give us the accuracy we require when applying Stirling's Approximation

[2.01 repeated here]

 (n+x) must be at least one and the factors must not be zero. However, if we know the value of (n+x)!, it can be negative

For instance, when x=4 and n=2, we have Γ(4)=6!/(4·5·6)=3·2·1, as expected.

Making a Basic Table of Gamma Values

If we have a basic table of gamma values, we can use this to calculate other gamma values with the same increment or step value. I chose 0.05, but using a spreadsheet, it is easy to use a much smaller step than this, if we had need of it.

We can calculate gamma to any accuracy we choose (arbitrary precision) by choosing a sufficiently large n. I chose 20. The table is accurate to approximately 5 decimal places. The table contains the raw data for comparison purposes.

They are called 'basic' values because they form the basis of calculating other values.

Calculating Higher Values of gamma from Γ(x)

After calculating a number of gamma values in a given range, we can use these values to find the values of higher gamma. By higher, I mean greater in magnitude. So as we have a basic table for 0-1, the higher values are 2, 3, 4, etc.

By using [1.01] when x=x+n−1, we note:

[2.01]

By applying [1.01] recursively, we find:

[2.02]

For instance, a trivial example, n=3 and x=1, Γ(4)=3·Γ(3)=3·2·1, as before.

However, for creating a table after we have found an initial set of values, all we need is:

[1.01 repeated]

We can use this formula when x≠0.

The values in the following table are calculated using [1.01]. The first values on the left come from table 1.

Calculating Lower Values of Gamma from Γ(x)

With our list of gamma values in a given range, we seek a formula to compute those in lower ranges. This is just our previous formula, [2.02], with Γ(x) made the subject:

[3.01]

For instance, x=4, n=3, Γ(4)=6!/(4·5·6)=3·2·1, as before.

We can use the formula [3.01] to compute Γ(−1.5), by setting n=2 and x=−1.5:

Γ(−1.5)=Γ(2−1.5)/((−1.5)·(−0.5))

=Γ(1/2)/(0.75)

=√(π)/(0.75), because Γ(1/2)=√(π) (See this page)

=2.363272 (six decimals)

However, we can use [1.01] with negative gamma (except negative integers and zero). So we can use the easy formula, when we know Γ(x+1):

[1.01b, repeated]

The first two columns are from the basic values, written in descending order.  The reason becomes apparent from the formula:

Γ(-0.05)=Γ(-0.05+1)/(-0.05). So we compute Γ(-0.05) from Γ(0.95)