91Ó°ÊÓ

The incomplete gamma function is$\underset{0}{\overset{∞}{∫}}{t}^{p-1}{e}^{-t}dt=\mathrm{Γ}\text{'}(p,x)$.

Gamma function behaves similarly to a factorial for natural numbers (a discrete set), its extension to positive real numbers (a continuous set) allows it to be used to model situations involving continuous change.

Gamma function$\mathit{Γ}\mathbf{\text{'}}\mathbf{\left(}\mathit{p}\mathbf{\right)}\mathbf{=}\underset{\mathbf{0}}{\overset{\mathbf{∞}}{\mathbf{∫}}}{\mathit{x}}^{\mathbf{p}\mathbf{-}\mathbf{1}}{\mathit{e}}^{\mathbf{-}\mathbf{x}}\mathit{d}\mathit{x}$.

The incomplete gamma function is $\underset{0}{\overset{∞}{∫}}{t}^{p-1}{e}^{-t}dt=\mathrm{Γ}\text{'}(p,x)$.

Partially integrate the function mentioned above.

The function becomes as follows.

$$\begin{gathered} \mathrm{Γ}\text{'}(p.x)={\left(-t^{p-1}{e}^{-t}\right)}_x^{∞}+\left(p-1\right)\underset{x}{\overset{∞}{∫}}{t}^{p-2}{e}^{-t}dt \\ \mathrm{Γ}\text{'}(p.x)=x^{p-1}{e}^{-x}+\left(p-1\right)\underset{x}{\overset{∞}{∫}}{t}^{p-2}{e}^{-t}dt \end{gathered}$$

Integrate partially $\underset{x}{\overset{∞}{∫}}{t}^{p-2}{e}^{-t}dt$.

The integration becomes as follows.

$\underset{x}{\overset{∞}{∫}}{t}^{p-2}{e}^{-t}dt=t^{p-2}{e}^{-t}+\left(p-2\right)\underset{x}{\overset{∞}{∫}}{t}^{p-3}{e}^{-t}dt$

Substitute the value of integration mentioned above in the equation given below.

$$\begin{gathered} \mathrm{Γ}\text{'}(p,x)=x^{p-1}{e}^{-x}+\left(p-1\right)\underset{x}{\overset{∞}{∫}}{t}^{p-2}{e}^{-t}dt \\ \mathrm{Γ}\text{'}(p,x)=x^{p-1}{e}^{-x}+\left(p-1\right)\left({\left(-t^{p-2}{e}^{-t}\right)}_x^{∞}+\left(p-2\right)\underset{x}{\overset{∞}{∫}}{t}^{p-3}{e}^{-t}dt\right) \\ \mathrm{Γ}\text{'}(p,x)=x^{p-1}{e}^{-x}+\left(p-1\right)x^{p-2}{e}^{-x}+\left(p-1\right)\left(p-2\right)\underset{x}{\overset{∞}{∫}}{t}^{p-3}{e}^{-t}dt \end{gathered}$$

Integrate partially.

The integration becomes as follows.

$\underset{x}{\overset{∞}{∫}}{t}^{p-2}{e}^{-t}dt=x^{p-3}{e}^{-x}+\left(p-3\right)\underset{x}{\overset{∞}{∫}}{t}^{p-4}{e}^{-t}dt$

Substitute the value of integration mentioned above in the equation given below.

$$\begin{gathered} \mathrm{Γ}\text{'}(p,x)=x^{p-1}{e}^{-x}+\left(p-1\right)x^{\left(p-2\right)}{e}^{-x}+\left(p-1\right)\left(p-2\right)\underset{x}{\overset{∞}{∫}}{t}^{p-3}{e}^{-t}dt \\ \mathrm{Γ}\text{'}(p,x)=x^{p-1}{e}^{-x}+\left(p-1\right)x^{p-2}{e}^{-x}+\left(p-1\right)\left(p-2\right)\left(x^{p-3}{e}^{-x}+\left(p-3\right)\underset{x}{\overset{∞}{∫}}{t}^{p-4}{e}^{-t}dt\right) \\ \mathrm{Γ}\text{'}(p,x)=x^{p-1}{e}^{-x}+\left(p-1\right)x^{p-2}{e}^{-x}+\left(p-1\right)\left(p-2\right)x^{p-3}{e}^{-x}+\left(p-1\right)\left(p-2\right)\left(p-3\right)\underset{x}{\overset{∞}{∫}}{t}^{p-4}{e}^{-t}dt \end{gathered}$$

Hence, the several terms of asymptotic series is mentioned below:

$\mathrm{Γ}\text{'}(p,x)=x^{p-1}{e}^{-x}\left(1+\frac{\left(p-1\right)}{x}+\frac{\left(p-1\right)\left(p-2\right)}{x^2}+\frac{\left(p-1\right)\left(p-2\right)\left(p-3\right)}{x^3}+.....\right)$