You can use the \(\gamma\) function to calculate the lower incomplete gamma function a value.

You can use the \lowergamma backslash command to insert this function.

The following variants of this function are available:

  • \(\text{complex } \gamma \left ( \text{<s>}, \text{<z>} \right )\)

The values \(\text{<s>}\) and \(\text{<z>}\) can be any basic type. The \(\gamma\) function is defined over the range \(\Re \left ( z \right ) \geq 0\) and \(\Re \left ( s \right ) > 0\). The \(\gamma\) function will generate a run-time error or report NaN for values outside of this range.

Note that run-time type conversion allows the result of this function to be assigned to any basic type provided the returned value is compatible with that type.

The lower incomplete gamma function is defined by the integral:

\[\gamma \left ( s, z \right ) = \int_{0}^{z} t ^ { s - 1 } e ^ {-t} dt\]

The \(\gamma\) function calculates the lower incomplete gamma function from the regularlized lower gamma function \(P\) or by the upper gamma function \(\Gamma \left ( s, z \right )\). The approach used depends on the value of \(s\) and \(z\).

The examples below show how you can use the \(\gamma\) function:

\[e = \frac{1}{1 - \gamma \left ( 1, 1 \right ) }\]
\[\text{ e = 2.7182818284590450908 }\]