# $$\gamma\text{Inv}$$¶

You can use the $$\gamma\text{Inv}$$ function to calculate the inverse lower gamma function.

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

The following variants of this function are available:

• $$\text{real } \gamma\text{Inv} \left ( \text{<s>}, \text{<y>} \right )$$

The values $$\text{<s>}$$ and $$\text{<y>}$$ can be any scalar value.

The function returns $$z$$ where:

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

The $$\gamma\text{Inv}$$ function calculates the value of $$z$$ using Halley’s method with an initial guess value calculated by

$\begin{split}z _ { initial } = \begin{cases} - \ln \left ( 1 - y \right ) & \text{if } y < 1 \\ 1 & \text{otherwise} \end{cases}\end{split}$

Halley’s method was selected due to the speed of convergence. Once the first derivative of the lower gamma function with respect to $$z$$ is calculated, the additional work to calculate the second derivative is minimal. For small values of $$s$$, Halley’s method may not converge. The Aion implementation of $$\gamma\text{Inv}$$ includes code to handle this scenario.

The examples below show how you can use the $$\gamma\text{Inv}$$ function:

$z = \gamma\text{Inv} \left ( 1, 1 - \frac{1}{e} \right )$
$\text{ z = 1 }$