DLMF:8.4.E15

MaRDI QIDQ3207

• Label: DLMF:8.4.E15

Digital Library of Mathematical Functions ID 8.4.E15

${\displaystyle{\displaystyle\Gamma\left(-n,z\right)=\frac{(-1)^{n}}{n!}\left(E_{1}\left(z\right)-e^{-z}\sum_{k=0}^{n-1}\frac{(-1)^{k}k!}{z^{k+1}}\right)=\frac{(-1)^{n}}{n!}\left(\psi\left(n+1\right)-\ln z\right)-z^{-n}\sum_{\begin{subarray}{c}k=0\\ k\neq n\end{subarray}}^{\infty}\frac{(-z)^{k}}{k!(k-n)}.}}$

Symbols List

• ${\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}$: psi (or digamma) function
• ${\displaystyle{\displaystyle\mathrm{e}}}$: base of natural logarithm
• ${\displaystyle{\displaystyle E_{1}\left(\NVar{z}\right)}}$: exponential integral
• ${\displaystyle{\displaystyle!}}$:
• ${\displaystyle{\displaystyle\Gamma\left(\NVar{a},\NVar{z}\right)}}$: incomplete gamma function
• ${\displaystyle{\displaystyle\ln\NVar{z}}}$: principal branch of logarithm function
• ${\displaystyle{\displaystyle z}}$: complex variable
• ${\displaystyle{\displaystyle k}}$: nonnegative integer
• ${\displaystyle{\displaystyle n}}$: nonnegative integer
 

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