LGAMMA Function

Description

The LGAMMA function computes the natural logarithm of the absolute value of the Gamma function (Log-Gamma function). Its mathematical definition is:

lgamma(x) = ln(|Γ(x)|)

where Γ(x) is the Gamma function. The Log-Gamma function is very useful for computations involving large-number factorials, because directly computing the Gamma function value may cause numeric overflow, whereas taking the logarithm allows safe computation over a much larger numeric range.

Syntax

lgamma(expr)

Parameters

• expr: A numeric expression for which to compute the Log-Gamma function value. Supports numeric types (implicitly castable to DOUBLE).

Returns

The return type is DOUBLE, representing the ln(|Γ(x)|) value corresponding to the input.

Examples

1. Compute lgamma(1) (ln(|Γ(1)|) = ln(1) = 0):

   SELECT lgamma(1) AS res;
   +-----+
   | res |
   +-----+
   | 0   |
   +-----+
   

2. Compute lgamma(5) (ln(|Γ(5)|) = ln(4!) = ln(24) ≈ 3.178):

   SELECT CAST(lgamma(5) AS decimal(18, 10)) AS res;
   +--------------+
   |     res      |
   +--------------+
   | 3.1780538303 |
   +--------------+
   

3. Compute the Log-Gamma value for a non-integer argument (lgamma(0.5) = ln(√π)):

   SELECT CAST(lgamma(0.5) AS decimal(18, 10)) AS res;
   +--------------+
   |     res      |
   +--------------+
   | 0.5723649429 |
   +--------------+
   

4. When the input is NULL:

   SELECT lgamma(NULL) AS res;
   +------+
   | res  |
   +------+
   | NULL |
   +------+
   

Notes

• When the input parameter is NULL, the result is NULL.
• For zero and negative integers (0, -1, -2, ...), the Gamma function is undefined (tends to infinity), and lgamma returns Infinity (CASTing such results to decimal yields NULL).
• For positive integers n, lgamma(n) = ln((n-1)!), e.g., lgamma(5) = ln(24) ≈ 3.178, lgamma(1) = 0, lgamma(2) = 0.
• Relationship between lgamma and tgamma: lgamma(x) = ln(|tgamma(x)|). When the result of tgamma(x) is very large (e.g., for large arguments), using tgamma directly may overflow to Infinity, whereas lgamma can still return a valid result, making it suitable for large-number factorial computations.
• lgamma(0.5) = ln(√π) ≈ 0.5724, a classic special value of the Log-Gamma function.