You can use the \(\Gamma_\text{U}\) function to calculate the upper incomplete gamma function.

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

The following variants of this function are available:

  • \(\text{real } \Gamma_\text{U} \left ( \text{<s>}, \text{<z>} \right )\)

Where \(\text{<s>}\) and \(\text{<z>}\) can be any basic type. Note that the incomplete gamma function is defined over the range \(\Re \left (s \right ) \geq 0\) and \(\Re \left ( z \right ) \geq 0\) and will return NaN or generate an error for which the values are not defined.

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 \(\Gamma_\text{U}\) function is defined by the integral:

\[\Gamma _ \text{U} \left ( s, z \right ) = \int_{z}^{\infty} t ^ { s - 1 } e ^ {-t} dt\]

The upper incomplete gamma function is calculated using continued fractions using the algorithm described in [7].

The examples below show how you can use the \(\Gamma_\text{U}\) function:

\[e = \frac{1}{\Gamma_\text{U} \left ( 1, 1 \right )}\]
\[e = 2.71828\]